337 Sentences With "quaternions" | Random Sentence Generator

Only in the last few decades have quaternions experienced a revival.

Halfway through the episode, Urmson is describing the mathematical model his company chose: quaternions.

Quaternions would languish in the shadow of vectors until quantum mechanics revealed their true identity in the 1920s.

This initially alarming property, known as non-commutativity, turns out to be a feature the quaternions share with reality.

Everything you could do with the real and complex numbers, you could do with the quaternions, except for one jarring difference.

"As soon as Hamilton invented the quaternions, everyone and his brother decided to make up their own number system," Baez said.

Except in this case, the Hamilton in question is the Irish physicist and mathematician William Rowan Hamilton, who discovered quaternions, among other seminal contributions.

Still, physicists never adopted quaternions in their day-to-day calculations, because an alternative scheme for dealing with spinors was found based on matrices.

John Graves, a lawyer friend of Hamilton's, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided.

Furey began seriously pursuing this possibility in grad school, when she learned that quaternions capture the way particles translate and rotate in 4-D space-time.

In addition to their adoption in computer graphics, where they serve as efficient tools for calculating rotations, quaternions live on in the geometry of higher-dimensional surfaces.

As Dixon knew, the algebra splits cleanly into two parts: ℂ⊗ℍ and ℂ⊗𝕆, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial).

Complex numbers, suitably paired, form 4-D "quaternions," discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin's Broome Bridge.

Regardless of your math savvy, this project-based training will provide an enjoyable and eye-opening intro to coding with C# and the basics of quaternions and math in Unity.

One not-so-useless construction turned out to be the fourth and final number system that permits a multiplication analog and an associated division, discovered shortly after the quaternions by Hamilton's friend, John Graves.

"I think there's still a lot more to discover about geometry based on the quaternions," said Nigel Hitchin, a geometer at the University of Oxford, "but if you want a new frontier, then it's the octonions."

The first three of these "division algebras" would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein's special theory of relativity.

" By nightfall, Hamilton had already sketched out a scheme for rotating 3-D arrows: He showed that these could be thought of as simplified quaternions created by setting a, the real part, equal to zero and keeping just the imaginary components i, j and k — a trio for which Hamilton invented the word "vector.

Deciding the fourth dimension was entirely too much trouble, Gibbs decapitated Hamilton's creation by lopping off the a term altogether: Gibbs' quaternion-spinoff kept the i, j, k notation, but split the unwieldy rule for multiplying quaternions into separate operations for multiplying vectors that every math and physics undergraduate learns today: the dot product and the cross product.

Another way to describe rotations is using rotation quaternions, also called versors. They are equivalent to rotation matrices and rotation vectors. With respect to rotation vectors, they can be more easily converted to and from matrices. When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions.

William Rowan Hamilton invented quaternions, a mathematical entity in 1843. This article describes Hamilton's original treatment of quaternions, using his notation and terms. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition only by the terminology which is used.

Nick Bobick (February 1998) "Rotating Objects Using Quaternions", Game Developer (magazine) Quaternions have received another boost from number theory because of their relation to quadratic forms.

The product of two quaternions can be found in the article on quaternions. Note, in this case, that a \cdot b and b \cdot a are in general different.

A quaternionic matrix is a matrix whose elements are quaternions.

Quaternions are a concise method of representing the automorphisms of three- and four-dimensional spaces. They have the technical advantage that unit quaternions form the simply connected cover of the space of three-dimensional rotations. For this reason, quaternions are used in computer graphics,Ken Shoemake (1985), Animating Rotation with Quaternion Curves, Computer Graphics, 19(3), 245–254. Presented at SIGGRAPH '85.

Quaternions continued to be a well-studied mathematical structure in the twentieth century, as the third term in the Cayley–Dickson construction of hypercomplex number systems over the reals, followed by the octonions and the sedenions; they are also useful tool in number theory, particularly in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group. The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions all four of whose coordinates are half-integers.

One of the best known noncommutative rings is the division ring of quaternions.

5, pages 380-440. which applied Leonhard Euler's four squares formula, a precursor to the quaternions of William Rowan Hamilton, to the problem of representing rotations in space.John H. Conway, Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry.

In pure mathematics, quaternions show up significantly as one of the four finite-dimensional normed division algebras over the real numbers, with applications throughout algebra and geometry. It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland. In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry.

Linear Algebra by Hüseyin Tevfik Pasha In Constantinople in 1882 Tevfik published Linear Algebra with the presses of A. Y. Boyajain. He begins with the concept of equipollence: :By the expression AB = NO, in Linear Algebra and in the science of Quaternions also, it is understood that the length of AB is equal to that of NO, and also that the direction of line AB is the same as that of NO. (page one) The book has five chapters and an appendix "Complex quantities and quaternions" in 68 pages with contents listed on page 69. Tevfik's book refers on page 11 to Introduction to Quaternions by Kelland and Tait which came out with a second edition in 1882.Philip Kelland & P. G. Tait (1882) Introduction to Quaternions with numerous examples But complex numbers and quaternions are missing.

Hankins (1980) pp. 371-376 In 1843 Hamilton's father had discovered the quaternions, a four-dimensional number system that extends the complex numbers, and he had published Lectures on Quaternions in 1853. From 1858 until his death in 1865 he worked on a second book,Graves (1889) p.

The algebra of the quaternions \H is a real Banach algebra, but it is not a complex algebra (and hence not a complex Banach algebra) for the simple reason that the center of the quaternions is the real numbers, which cannot contain a copy of the complex numbers.

Volume four, published in 1874, began with an discussion of properties of algebraic operations (commutativity, associativity, distribution, and inverses) and used the algebra of quaternions and versors to describe spherical trigonometry.J. Hoüel (1874) Éléments de la Théorie des Quaternions, Gauthier-Villars publisher, link from Google Books However, in 1890 P. G. Tait revealed his dissatisfaction with Hoüel's changes in notation with text that Tait had given for Hoüel's use. Tait wrote:P. G. Tait (1890) An Elementary Treatise on Quaternions, 3rd edition, p.

For a textbook on quaternions, lecturers and students relied on Tait and Kelland's Introduction to Quaternions which had editions in 1873 and 1882. It fell to Knott to prepare a third edition in 1904. By then the Universal Algebra of Alfred North Whitehead (1898) presumed some grounding in quaternions as students encountered matrix algebra. In Knott's introduction to his textbook edition he says "Analytically the quaternion is now known to take its place in the general theory of complex numbers and continuous groups,...".

The vector part of a unit quaternion represents the radius of the 2-sphere corresponding to the axis of rotation, and its magnitude is the sine of half the angle of rotation. Each rotation is represented by two unit quaternions of opposite sign, and, as in the space of rotations in three dimensions, the quaternion product of two unit quaternions will yield a unit quaternion. Also, the space of unit quaternions is "flat" in any infinitesimal neighborhood of a given unit quaternion.

The group Spin(3) is isomorphic to the special unitary group SU(2); it is also diffeomorphic to the unit 3-sphere S3 and can be understood as the group of unit quaternions (i.e. those with absolute value 1). The connection between quaternions and rotations, commonly exploited in computer graphics, is explained in quaternions and spatial rotations. The map from S3 onto SO(3) that identifies antipodal points of S3 is a surjective homomorphism of Lie groups, with kernel {±1}.

Quaternions, one of the ways to describe rotations in three dimensions, consist of a four- dimensional space. Rotations between quaternions, for interpolation, for example, take place in four dimensions. Spacetime, which has three space dimensions and one time dimension is also four-dimensional, though with a different structure to Euclidean space.

Conway has written textbooks and done original work in algebra, focusing particularly on quaternions and octonions.Conway and Smith (2003): "Conway and Smith's book is a wonderful introduction to the normed division algebras: the real numbers, the complex numbers, the quaternions, and the octonions." Together with Neil Sloane, he invented the icosians.

It is a theorem of Frobenius that there are only two real quaternion algebras: 2×2 matrices over the reals and Hamilton's real quaternions. In a similar way, over any local field F there are exactly two quaternion algebras: the 2×2 matrices over F and a division algebra. But the quaternion division algebra over a local field is usually not Hamilton's quaternions over the field. For example, over the p-adic numbers Hamilton's quaternions are a division algebra only when p is 2.

Quaternions are also used in one of the proofs of Lagrange's four- square theorem in number theory, which states that every nonnegative integer is the sum of four integer squares. As well as being an elegant theorem in its own right, Lagrange's four square theorem has useful applications in areas of mathematics outside number theory, such as combinatorial design theory. The quaternion-based proof uses Hurwitz quaternions, a subring of the ring of all quaternions for which there is an analog of the Euclidean algorithm.

Unlike the complex numbers, the split-complex numbers are not algebraically closed, and further contain nontrivial zero divisors and non-trivial idempotents. As with the quaternions, split-quaternions are not commutative, but further contain nilpotents; they are isomorphic to the 2 × 2 real matrices. Split-octonions are non-associative and contain nilpotents.

The ring of Hamiltonian quaternions forms a 4-dimensional algebra over its center, which is isomorphic to the real numbers.

The relationship of quaternions to each other within the complex subplanes of can also be identified and expressed in terms of commutative subrings. Specifically, since two quaternions and commute (i.e., ) only if they lie in the same complex subplane of , the profile of as a union of complex planes arises when one seeks to find all commutative subrings of the quaternion ring. This method of commutative subrings is also used to profile the split-quaternions, which as an algebra over the reals are isomorphic to 2 × 2 real matrices.

A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units. A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of commutative algebra: it is possible for an irreducible quaternion to divide a product ab without dividing either a or b. Every Hurwitz quaternion can be factored as a product of irreducible quaternions.

When is used to describe spatial rotations (cf. quaternions and spatial rotations), it describes a rotation about through an angle of .

On page 665 of Elements of Quaternions Hamilton defines a biquaternion to be a quaternion with complex number coefficients. The scalar part of a biquaternion is then a complex number called a biscalar. The vector part of a biquaternion is a bivector consisting of three complex components. The biquaternions are then the complexification of the original (real) quaternions.

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundle equipped with an action by the algebra of quaternions in such a way that the quaternions I, J, K define integrable almost complex structures. If the almost complex structures are instead not assumed to be integrable, the manifold is called quaternionic, or almost hypercomplex.

In the four-dimensional space of quaternions, there is a sphere of imaginary units. For any point r on this sphere, and x a real number, Euler's formula applies: :\exp(xr) = \cos x + r \sin x, and the element is called a versor in quaternions. The set of all versors forms a 3-sphere in the 4-space.

The cross product can also be described in terms of quaternions. In general, if a vector is represented as the quaternion , the cross product of two vectors can be obtained by taking their product as quaternions and deleting the real part of the result. The real part will be the negative of the dot product of the two vectors.

When considered as the set of unit quaternions, inherits an important structure, namely that of quaternionic multiplication. Because the set of unit quaternions is closed under multiplication, takes on the structure of a group. Moreover, since quaternionic multiplication is smooth, can be regarded as a real Lie group. It is a nonabelian, compact Lie group of dimension 3.

Peter Guthrie Tait in his book An Elementary Treatise on Quaternions edition 1867 on page 244 named Didonia in honour of Delaunay.

His original research was mainly in higher algebra (analysis of plane curves) and in quaternions (considered as the best instrument of research).

In 1843 Hamilton discovered the quaternions, and it was to Graves that he made on 17 October his first written communication of the discovery. In his preface to the Lectures on Quaternions and in a prefatory letter to a communication to the Philosophical Magazine for December 1844 are acknowledgments of his indebtedness to Graves for stimulus and suggestion. Immediately after the discovery of quaternions, before the end of 1843, Graves successfully extended to eight squares Euler's four-square identity, and went on to conceive a theory of "octaves" (now called octonions) analogous to Hamilton's theory of quaternions, introducing four imaginaries additional to Hamilton's i, j and k, and conforming to "the law of the modulus". Octonions are a contemporary if abstruse area of contemporary research of the Standard Model of particle physics.

Finally, invoking the reciprocal of a biquaternion, Girard described conformal maps on spacetime. Among the fifty references, Girard included Alexander Macfarlane and his Bulletin of the Quaternion Society. In 1999 he showed how Einstein's equations of general relativity could be formulated within a Clifford algebra that is directly linked to quaternions. The finding of 1924 that in quantum mechanics the spin of an electron and other matter particles (known as spinors) can be described using quaternions furthered their interest; quaternions helped to understand how rotations of electrons by 360° can be discerned from those by 720° (the “Plate trick”).

Such a set of four numbers is called a quaternion. While the quaternion as described above, does not involve complex numbers, if quaternions are used to describe two successive rotations, they must be combined using the non-commutative quaternion algebra derived by William Rowan Hamilton through the use of imaginary numbers. Rotation calculation via quaternions has come to replace the use of direction cosines in aerospace applications through their reduction of the required calculations, and their ability to minimize round-off errors. Also, in computer graphics the ability to perform spherical interpolation between quaternions with relative ease is of value.

Today, the quaternions are used in computer graphics, control theory, signal processing, and orbital mechanics, mainly for representing rotations/orientations. For example, it is common for spacecraft attitude-control systems to be commanded in terms of quaternions, which are also used to telemeter their current attitude. The rationale is that combining quaternion transformations is more numerically stable than combining many matrix transformations. In control and modelling applications, quaternions do not have a computational singularity (undefined division by zero) that can occur for quarter-turn rotations (90 degrees) that are achievable by many Air, Sea and Space vehicles.

This corresponds to the special case of certain real or complex matrices. The theorem holds for general quaternionic matrices.Due to the non-commutative nature of the multiplication operation for quaternions and related constructions, care needs to be taken with definitions, most notably in this context, for the determinant. The theorem holds as well for the slightly less well-behaved split-quaternions, see .

The latter is adjacent to Broombridge railway station and the canal bridge where Sir William Rowan Hamilton first wrote the fundamental formula for quaternions.

Some of the other hypercomplex systems that Study worked with are dual numbers, dual quaternions, and split-biquaternions, all being associative algebras over R.

The rings of quaternions and split-quaternions can both be represented by certain complex matrices. (When restricted to unit norm, these are the groups and respectively.) Therefore it is not surprising that the theorem holds. There is no such matrix representation for the octonions, since the multiplication operation is not associative in this case. However, a modified Cayley–Hamilton theorem still holds for the octonions, see .

The approach to the topological properties of projective planes via ordering relations is not possible, however, for the planes coordinatized by the complex numbers, the quaternions or the octonion algebra. The point spaces as well as the line spaces of these classical planes (over the real numbers, the complex numbers, the quaternions, and the octonions) are compact manifolds of dimension 2^m,\, 1 \le m \le 4.

In mathematics, a left (or right) quaternionic vector space is a left (or right) H-module where H denotes the noncommutative ring of the quaternions. The space Hn of n-tuples of quaternions is both a left and right H-module using the componentwise left and right multiplication: : q (q_1,q_2,\ldots q_n) = (q q_1,q q_2,\ldots q q_n) : (q_1,q_2,\ldots q_n) q = (q_1 q, q_2 q,\ldots q_n q) for quaternions q and q1, q2, ... qn. Since H is a division algebra, every finitely generated (left or right) H-module has a basis, and hence is isomorphic to Hn for some n.

Hamilton coined the words tensor and scalar, and was the first to use the word vector in the modern sense.Earliest Known Uses of Some of the Words of Mathematics (V) Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by introduction of complex number coefficients. When his work was assembled in 1853, the book Lectures on Quaternions had "formed the subject of successive courses of lectures, delivered in 1848 and subsequent years, in the Halls of Trinity College, Dublin". Hamilton confidently declared that quaternions would be found to have a powerful influence as an instrument of research.

The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the reals is Brauer equivalent to either the reals or the quaternions. Explicitly, the Brauer group of the reals consists of two classes, represented by the reals and the quaternions, where the Brauer group is the set of all CSAs, up to equivalence relation of one CSA being a matrix ring over another. By the Artin–Wedderburn theorem (specifically, Wedderburn's part), CSAs are all matrix algebras over a division algebra, and thus the quaternions are the only non-trivial division algebra over the reals. CSAs – rings over a field, which are simple algebras (have no non-trivial 2-sided ideals, just as with fields) whose center is exactly the field – are a noncommutative analog of extension fields, and are more restrictive than general ring extensions.

Anton III Wierix 1606). The ten quaternions are shown underneath the emperor flanked by the prince-electors (Archbishop of Trier, Archbishop of Cologne, Archbishop of Mainz; King of Bohemia, Count Palatine, Duke of Saxony, Margrave of Brandenburg). A " Quaternion Eagle" (each quaternion being represented by four coats of arms on the imperial eagle's remiges) Hans Burgkmair, c. 1510. Twelve quaternions are shown, as follows (eight dukes being divided into two quaternions called "pillars" and "vicars", respectively c.f. Christian Knorr von Rosenroth, Anführung zur Teutschen Staats-Kunst (1672), p. 669.): Seill ("pillars"), Vicari ("vicars"), Marggrauen (margraves), Lantgrauen (landgraves), Burggrauen (burggraves), Grauen (counts), Semper freie (nobles), Ritter (knights), Stett (cities), Dörfer (villages), Bauern (peasants), Birg (castles). The so- called imperial quaternions (German: Quaternionen der Reichsverfassung "quaternions of the imperial constitution"; from Latin quaterniō "group of four soldiers") were a conventional representation of the Imperial States of the Holy Roman Empire which first became current in the 15th century and was extremely popular during the 16th century.Hans Legband, "Zu den Quaternionen der Reichsverfassung", Archiv für Kulturgeschichte 3 (1905), 495–498. Ernst Schubert, "Die Quaternionen", Zeitschrift für historische Forschung 20 (1993), 1–63.

Algo Li, et al. "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product." International Journal of the Physical Sciences Vol. 5(10), pp.

Left- and right-isoclinic rotations are represented respectively by left- and right- multiplication by unit quaternions; see the paragraph "Relation to quaternions" below. The four rotations are pairwise different except if or . The angle corresponds to the identity rotation; corresponds to the central inversion, given by the negative of the identity matrix. These two elements of SO(4) are the only ones that are simultaneously left- and right-isoclinic.

Each of these methods begins with three independent random scalars uniformly distributed on the unit interval. takes advantage of the odd dimension to change a Householder reflection to a rotation by negation, and uses that to aim the axis of a uniform planar rotation. Another method uses unit quaternions. Multiplication of rotation matrices is homomorphic to multiplication of quaternions, and multiplication by a unit quaternion rotates the unit sphere.

As a distant correspondent, he participated in a vigorous debate about the place of quaternions in physics education.M.J. Crowe (1967) A History of Vector Analysis, U. Notre Dame Press.

Classical quaternion notation had only one concept of multiplication. Multiplication of two real numbers, two imaginary numbers or a real number by an imaginary number in the classical notation system was the same operation. Multiplication of a scalar and a vector was accomplished with the same single multiplication operator; multiplication of two vectors of quaternions used this same operation as did multiplication of a quaternion and a vector or of two quaternions.

Bradstreet wrote four quaternions, "Seasons," "Elements," "Humours," and "Ages," which made possible her "development as a poet in terms of technical craftsmanship as she learned to fashion the form artistically." Bradstreet's first two quaternions were her most successful.Eberwein, Jane Donahue 'Early American Literature' vol 9 no 1 University of North Carolina Press Spring 1974 The central tension in her work is that between delight in the world and belief of its vanity.

Gu Deng was born in Wuxi, Jiangsu. In the end of Qing Dynasty He graduated the department of mathematics, the Gezhi Academy (). In 1909 Gu translated the book about quaternions, this is the first introduction about quaternions in Chinese history. Gu Deng became a teacher in Beijing University and Qinghua University, later he became the president of Beiping Women's College of Humanities and Sciences () and the chairperson of the department of mathematics, Northeastern University.

Peter Guthrie Tait In 1860, Tait succeeded his old master, James D. Forbes, as professor of natural philosophy at the University of Edinburgh, and occupied the Chair until shortly before his death. The first scientific paper under Tait's name only was published in 1860. His earliest work dealt mainly with mathematical subjects, and especially with quaternions, of which he was the leading exponent after their originator, William Rowan Hamilton. He was the author of two text-books on them—one an Elementary Treatise on Quaternions (1867), written with the advice of Hamilton, though not published till after his death, and the other an Introduction to Quaternions (1873), in which he was aided by Philip Kelland (1808–1879), one of his teachers at the University of Edinburgh.

A scientific society, the Quaternion Society was an "International Association for Promoting the Study of Quaternions and Allied Systems of Mathematics". At its peak it consisted of about 60 mathematicians spread throughout the academic world that were experimenting with quaternions and other hypercomplex number systems. The guiding light was Alexander Macfarlane who served as its Secretary initially, and became President in 1909. The Association published a Bibliography in 1904 and a Bulletin (annual report) from 1900 to 1913.

Note that this element lies in the center of Dicn. Indeed, the center consists solely of the identity element and x2. If we add the relation x2 = 1 to the presentation of Dicn one obtains a presentation of the dihedral group Dih2n, so the quotient group Dicn/2> is isomorphic to Dihn. There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations.

In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called. As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.

The discussion is retained in subsequent editions in 1873 and 1890, as well as in his further Introduction to Quaternions Macmillan, London, 1873, pp. 42--43 jointly with Philip Kelland in 1873.

The integral curves or fibers respectively are certain pairwise linked great circles, the orbits in the space of unit norm quaternions under left multiplication by a given unit quaternion of unit norm.

The representation of imperial subjects is also far from complete. The "imperial quaternions" are, rather, a more or less random selection intended to represent pars pro toto the structure of the imperial constitution.

On page 173 Macfarlane expands on his greater theory of quaternion variables. By way of contrast he notes that Felix Klein appears not to look beyond the theory of Quaternions and spatial rotation.

A unit vector in ℝ3 was called a right versor by W. R. Hamilton, as he developed his quaternions ℍ ⊂ ℝ4. In fact, he was the originator of the term vector, as every quaternion q = s + v has a scalar part s and a vector part v. If v is a unit vector in ℝ3, then the square of v in quaternions is –1. Thus by Euler's formula, \exp (\theta v) = \cos \theta + v \sin \theta is a versor in the 3-sphere.

Page numbers are carried from previous publications, and the reader is presumed familiar with quaternions. The first paper is "Principles of the Algebra of Physics" where he first proposes the hyperbolic quaternion algebra, since "a student of physics finds a difficulty in principle of quaternions which makes the square of a vector negative." The second paper is "The Imaginary of the Algebra". Similar to Homersham Cox (1882/83), Macfarlane uses the hyperbolic versor as the hyperbolic quaternion corresponding to the versor of Hamilton.

John Horton Conway has a purely combinatorial proof which consequently also holds for points and lines over the complex numbers, quaternions and octonions.Stasys Jukna, Extremal Combinatorics, Second edition, Springer Verlag, 2011, pages 167 - 168.

Interval arithmetic can be extended, in an analogous manner, to other multidimensional number systems such as quaternions and octonions, but with the expense that we have to sacrifice other useful properties of ordinary arithmetic.

This is related to Hurwitz's theorem, which prohibits the existence of algebraic structures like the quaternions and octonions in dimensions other than 2, 4, and 8. The first exotic spheres ever discovered were seven-dimensional.

He published two works in geometric algebra, Introduction à la Méthode des Quaternions (1881)C.-A. Laisant (1881) Introduction a la Méthode des Quaternions, link from Google Books and Théorie et applications des equipollences (1887).C.-A. Laisant (1887) Theorie et Applications des Equipollence, Gauthier-Villars, link from University of Michigan Historical Math Collection He also co-founded a mathematical journal, L'Intermédiaire des Mathématiciens with Émile Lemoine in 1894, and was in 1888 the president of the Société Mathématique de France.Anciens Présidents de la SMF -- 1873-2006 .

All fields are division rings; more interesting examples are the non-commutative division rings. The best known example is the ring of quaternions H. If we allow only rational instead of real coefficients in the constructions of the quaternions, we obtain another division ring. In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring;Lam (2001), . every division ring arises in this fashion from some simple module.

Adolf Hurwitz He was one of the early students of the Riemann surface theory, and used it to prove many of the foundational results on algebraic curves; for instance Hurwitz's automorphisms theorem. This work anticipates a number of later theories, such as the general theory of algebraic correspondences, Hecke operators, and Lefschetz fixed- point theorem. He also had deep interests in number theory. He studied the maximal order theory (as it now would be) for the quaternions, defining the Hurwitz quaternions that are now named for him.

Analogous identities are Euler's four-square related to quaternions, and Degen's eight-square derived from the octonions which has connections to Bott periodicity. There is also Pfister's sixteen-square identity, though it is no longer bilinear.

Sophus Lie was less than a year old when Hamilton first described quaternions, but Lie's name has become associated with all groups generated by exponentiation. The set of versors with their multiplication has been denoted Sl(1,q) by Robert Gilmore in his text on Lie theory.Robert Gilmore (1974) Lie Groups, Lie Algebras and some of their Applications, chapter 5: Some simple examples, pages 120–35, Wiley Gilmore denotes the real, complex, and quaternion division algebras by r, c, and q, rather than the more common R, C, and H. Sl(1,q) is the special linear group of one dimension over quaternions, the "special" indicating that all elements are of norm one. The group is isomorphic to SU(2,c), a special unitary group, a frequently used designation since quaternions and versors are sometimes considered anachronistic for group theory.

Topologically, SO(3) is the real projective space RP3, with fundamental group Z/2, and only (non-trivial) covering space the hypersphere S3, which is the group Spin(3), and represented by the unit quaternions. Thus quaternions are a preferred method for representing spatial rotations – see quaternions and spatial rotation. However, it is often desirable to represent rotations by a set of three numbers, known as Euler angles (in numerous variants), both because this is conceptually simpler for someone familiar with planar rotation, and because one can build a combination of three gimbals to produce rotations in three dimensions. Topologically this corresponds to a map from the 3-torus T3 of three angles to the real projective space RP3 of rotations, and the resulting map has imperfections due to this map being unable to be a covering map.

In mathematics, a biquaternion algebra is a compound of quaternion algebras over a field. The biquaternions of William Rowan Hamilton (1844) and the related split-biquaternions and dual quaternions do not form biquaternion algebras in this sense.

The fact that the quaternions are the only non- trivial CSA over the reals (up to equivalence) may be compared with the fact that the complex numbers are the only non-trivial field extension of the reals.

Macfarlane (1900) "Hyperbolic Quaternions" Proceedings of the Royal Society at Edinburgh, vol. 23, November 1899 to July 1901 sessions, pp. 169-180+figures plate. Online at Cambridge Journals (paid access), Internet Archive (free), or Google Books (free).

For a fixed r, versors of the form exp(ar) where a ∈ , form a subgroup isomorphic to the circle group. Orbits of the left multiplication action of this subgroup are fibers of a fiber bundle over the 2-sphere, known as Hopf fibration in the case r = i; other vectors give isomorphic, but not identical fibrations. In 2003 David W. Lyons wrote "the fibers of the Hopf map are circles in S3" (page 95). Lyons gives an elementary introduction to quaternions to elucidate the Hopf fibration as a mapping on unit quaternions.

Hamilton invented the term associative to distinguish between the imaginary scalar (known by now as a complex number) which is both commutative and associative, and four other possible roots of negative unity which he designated L, M, N and O, mentioning them briefly in appendix B of Lectures on Quaternions and in private letters. However, non-associative roots of minus one do not appear in Elements of Quaternions. Hamilton died before he worked on these strange entities. His son claimed them to be "bows reserved for the hands of another Ulysses".

Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the universal cover of SO(3).

The octonions and the split-octonions can be obtained from the Cayley–Dickson construction by defining a multiplication on pairs of quaternions. We introduce a new imaginary unit ℓ and write a pair of quaternions (a, b) in the form a + ℓb. The product is defined by the rule: Kevin McCrimmon (2004) A Taste of Jordan Algebras, page 158, Universitext, Springer :(a + \ell b)(c + \ell d) = (ac + \lambda \bar db) + \ell(da + b\bar c) where :\lambda = \ell^2. If λ is chosen to be −1, we get the octonions.

When Slerp is applied to unit quaternions, the quaternion path maps to a path through 3D rotations in a standard way. The effect is a rotation with uniform angular velocity around a fixed rotation axis. When the initial end point is the identity quaternion, Slerp gives a segment of a one-parameter subgroup of both the Lie group of 3D rotations, SO(3), and its universal covering group of unit quaternions, S3. Slerp gives a straightest and shortest path between its quaternion end points, and maps to a rotation through an angle of 2Ω.

This approach made vector calculations available to engineers—and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwell's Treatise on Electricity and Magnetism, separated off their vector part for independent treatment. The first half of Gibbs's Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901, Edwin Bidwell Wilson published Vector Analysis, adapted from Gibb's lectures, which banished any mention of quaternions in the development of vector calculus.

If, instead, it is taken to be +1 we get the split-octonions. One can also obtain the split-octonions via a Cayley–Dickson doubling of the split-quaternions. Here either choice of λ (±1) gives the split-octonions.

Viktor Mulciber, arms tycoon, discusses the Q-weapon, "a weapon based on Time," with the Quarterioneers. Piet Wovre buys the Q weapon from Edouard Gavaert in Brussels. Kit and Umeki's relationship intensifies. They discuss Hamilton's discovery of the Quaternions.

The book has eight chapters: the first on the origins of vector analysis including Ancient Greek and 16th and 17th century influences; the second on the 19th century William Rowan Hamilton and quaternions; the third on other 19th and 18th century vectorial systems including equipollence of Giusto Bellavitis and the exterior algebra of Hermann Grassmann. Chapter four is on the general interest in the 19th century on vectorial systems including analysis of journal publications as well as sections on major figures and their views (e.g., Peter Guthrie Tait as an advocate of Quaternions and James Clerk Maxwell as a critic of Quaternions); the fifth chapter describes the development of the modern system of vector analysis by Josiah Willard Gibbs and Oliver Heaviside. In chapter six, "Struggle for existence", Michael J. Crowe delves into the zeitgeist that pruned down quaternion theory into vector analysis on three-dimensional space.

Alexander Macfarlane stylized his work as "Space Analysis". In 1894 he published his five earlier papersA. Macfarlane (1894) Papers on Space Analysis, B. Westerman, New York, weblink from archive.org and a book review of Alexander McAulay's Utility of Quaternions in Physics.

Wedderburn's little theorem: All finite division rings are commutative and therefore finite fields. (Ernst Witt gave a simple proof.) Frobenius theorem: The only finite-dimensional associative division algebras over the reals are the reals themselves, the complex numbers, and the quaternions.

Saint Boisil (d. 664) of Melrose Abbey was Cuthbert's teacher. Bede's prose life of Cuthbert records that during Boisil's last illness, he and Cuthbert read daily one of the seven gatherings or quaternions of Boisil's manuscript of the Gospel of John.

In 1895, Professor P. Molenbroek of The Hague, Holland, and Shinkichi Kimura studying at Yale put out a call for scholars to form the society in widely circulated journals: Nature,S. Kimura & P. Molenbroek (1895) Friends and Fellow Workers in Quaternions Nature 52:545–6 (#1353) Science,S. Kimura & P. Molenbroek (1895) To those Interested in Quaternions and Allied Systems of Mathematics Science 2nd Ser, 2:524-25 and the Bulletin of the American Mathematical Society."Notes" Bulletin of the American Mathematical Society 2:53, 182; 5:317 Giuseppe Peano also announced the society formation in his Rivista di Matematica.

Intuitively recognizing its power, he snatches up the magnificent weapon which Hamilton tenders us all, and at once dashes off to the jungle on the quest of big game.PG Tait (1893) Nature 28 December McAulay took up the position of Professor of Physics in Tasmania from 1896 until 1929, at which time his son Alexander Leicester McAulay took over the position for the next thirty years. Following William Kingdon Clifford who had extended quaternions to dual quaternions, McAulay made a special study of this hypercomplex number system. In 1898 McAulay published, through Cambridge University Press, his Octonions: a Development of Clifford's Biquaternions.

Quaternion Plaque on Broom Bridge The other great contribution Hamilton made to mathematical science was his discovery of quaternions in 1843. However, in 1840, Benjamin Olinde Rodrigues had already reached a result that amounted to their discovery in all but name. Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional plane) to higher spatial dimensions. He failed to find a useful 3-dimensional system (in modern terminology, he failed to find a real, three-dimensional skew-field), but in working with four dimensions he created quaternions.

This work has proven central to the modern study of classical field theories such as electromagnetism, and to the development of quantum mechanics. In pure mathematics, he is best known as the inventor of quaternions. William Rowan Hamilton's scientific career included the study of geometrical optics, classical mechanics, adaptation of dynamic methods in optical systems, applying quaternion and vector methods to problems in mechanics and in geometry, development of theories of conjugate algebraic couple functions (in which complex numbers are constructed as ordered pairs of real numbers), solvability of polynomial equations and general quintic polynomial solvable by radicals, the analysis on Fluctuating Functions (and the ideas from Fourier analysis), linear operators on quaternions and proving a result for linear operators on the space of quaternions (which is a special case of the general theorem which today is known as the Cayley–Hamilton theorem). Hamilton also invented "icosian calculus", which he used to investigate closed edge paths on a dodecahedron that visit each vertex exactly once.

There he made the acquaintance of Thomas Andrews, whom he joined in researches on the density of ozone and the action of the electric discharge on oxygen and other gases, and by whom he was introduced to Sir William Rowan Hamilton and quaternions.

His subsequent work in computer science focused on computer graphics and visualization of exotic mathematical objects, including widely used images of the Calabi-Yau quintic cross-sections used to represent the hidden dimensions of 10-dimensional string theory. He is the author of Visualizing Quaternions.

Almost thirty years later his identity was rediscovered by John T. Graves and Arthur Cayley as obeyed by the norm of octonions. These were an extension of Hamilton's quaternions. In 1898 Adolf Hurwitz proved that such identities involving 2k squares can exist only for .

However, this is a minority view within the string community. Since E7 is in some sense F4 quaternified and E8 is F4 octonified, the 12 and 16 dimensional theories, if they did exist, may involve the noncommutative geometry based on the quaternions and octonions respectively.

It was concluded that there was no need for the greater physical insights provided by the quaternions if the theory was purely local in nature. Local vector analysis has become the dominant way of using Maxwell's equations ever since. However, this strictly vectorial approach has led to a restrictive topological understanding in some areas of electromagnetism, for example, a full understanding of the energy transfer dynamics in Tesla's oscillator-shuttle-circuit can only be achieved in quaternionic algebra or higher SU(2) symmetries. It has often been argued that quaternions are not compatible with special relativity, but multiple papers have shown ways of incorporating relativity.

The homography group of the ring of integers Z is modular group . Ring homographies have been used in quaternion analysis, and with dual quaternions to facilitate screw theory. The conformal group of spacetime can be represented with homographies where A is the composition algebra of biquaternions.

In 1901 Study published Geometrie der DynamenE. Study (1903) Geometrie der Dynamen, from Historical Math Monographs at Cornell University also using dual quaternions. In 1913 he wrote a review article treating both E(3) and elliptic geometry. This article, "Foundations and goals of analytical kinematics"E.

In two dimensions the even subalgebra is isomorphic to the complex numbers, ℂ, while in three it is isomorphic to the quaternions, ℍ. More generally the even subalgebra can be used to generate rotations in any dimension, and can be generated by bivectors in the algebra.

5(10), pp. 1530-1536, 4 September 2010. By formulating the matrices as dual quaternions, it is possible to get a linear equation by which is solvable in a linear format. An alternative way applies the least-squares method to the Kronecker product of the matrices .

3Blue1Brown is a math YouTube channel created by Grant Sanderson. The channel focuses on higher mathematics with a distinct visual perspective. Topics covered include linear algebra, calculus, neural networks, the Riemann hypothesis, Fourier transform, quaternions and topology. As of October 2020, the channel has 3.14 million subscribers.

Diagram showing the magnitude and direction of the cross product of two vectors, in the notation introduced by Gibbs British scientists, including Maxwell, had relied on Hamilton's quaternions in order to express the dynamics of physical quantities, like the electric and magnetic fields, having both a magnitude and a direction in three-dimensional space. Following W. K. Clifford in his Elements of Dynamic (1888), Gibbs noted that the product of quaternions could be separated into two parts: a one- dimensional (scalar) quantity and a three-dimensional vector, so that the use of quaternions involved mathematical complications and redundancies that could be avoided in the interest of simplicity and to facilitate teaching. In his Yale classroom notes he defined distinct dot and cross products for pairs of vectors and introduced the now common notation for them. Through the 1901 textbook Vector Analysis prepared by E. B. Wilson from Gibbs notes, he was largely responsible for the development of the vector calculus techniques still used today in electrodynamics and fluid mechanics.Wheeler 1998, pp.

Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. A closer non-commutative analog are central simple algebras (CSAs) – ring extensions over a field, which are simple algebra (no non-trivial 2-sided ideals, just as for a field) and where the center of the ring is exactly the field. For example, the only finite field extension of the real numbers is the complex numbers, while the quaternions are a central simple algebra over the reals, and all CSAs over the reals are Brauer equivalent to the reals or the quaternions. CSAs can be further generalized to Azumaya algebras, where the base field is replaced by a commutative local ring.

The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals ℝ, the complexes ℂ, the quaternions ℍ, and the octonions 𝕆, and the Frobenius theorem says the only real associative division algebras are ℝ, ℂ, and ℍ. In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8. It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices.

Brougham (Broom) Bridge, Dublin, which says: > Here as he walked by on the 16th of October 1843 Sir William Rowan Hamilton > in a flash of genius discovered the fundamental formula for quaternion > multiplication > > & cut it on a stone of this bridge. In mathematics, quaternions are a non-commutative number system that extends the complex numbers. Quaternions and their applications to rotations were first described in print by Olinde Rodrigues in all but name in 1840, but independently discovered by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. They find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations.

By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge: : Hamilton called a quadruple with these rules of multiplication a quaternion, and he devoted the remainder of his life to studying and teaching them. From 1844 to 1850 Philosophical Magazine communicated Hamilton's exposition of quaternions.W.R. Hamilton(1844 to 1850) On quaternions or a new system of imaginaries in algebra, Philosophical Magazine, link to David R. Wilkins collection at Trinity College, Dublin In 1853 he issued Lectures on Quaternions, a comprehensive treatise that also described biquaternions.

Octonions were developed independently by Arthur Cayley in 1845 Penrose 2004 pg 202 and John T. Graves, a friend of Hamilton's. Graves had interested Hamilton in algebra, and responded to his discovery of quaternions with "If with your alchemy you can make three pounds of gold [the three imaginary units], why should you stop there?" Two months after Hamilton's discovery of quaternions, Graves wrote Hamilton on December 26, 1843 presenting a kind of double quaternionSee Penrose Road to Reality pg. 202 'Graves discovered that there exists a kind of double quaternion...' that is called an octonion, and showed that they were what we now call a normed division algebra; Graves called them octaves.

Michael J. Crowe devotes chapter six of his book A History of Vector Analysis to the various published views, and notes the hyperbolic quaternion: :Macfarlane constructed a new system of vector analysis more in harmony with Gibbs–Heaviside system than with the quaternion system. ...he...defined a full product of two vectors which was comparable to the full quaternion product except that the scalar part was positive, not negative as in the older system. In 1899 Charles Jasper Joly noted the hyperbolic quaternion and the non-associativity property while ascribing its origin to Oliver Heaviside. The hyperbolic quaternions, as the Algebra of Physics, undercut the claim that ordinary quaternions made on physics.

The densest lattice packing of unit spheres in four dimensions (called the D4 lattice) places a sphere at every point whose coordinates are either all integers or all half-integers. This packing is closely related to the Hurwitz integers: quaternions whose real coefficients are either all integers or all half-integers..

A division ring is a ring such that every non-zero element is a unit. A commutative division ring is a field. A prominent example of a division ring that is not a field is the ring of quaternions. Any centralizer in a division ring is also a division ring.

The facility of the algebra in expressing geometric relationships led to broad acceptance of the method, several compositions by other authors, and stimulation of applied algebra generally. As mathematical terminology has grown since that time, and usage of some terms has changed, the traditional expressions are referred to classical Hamiltonian quaternions.

For a generalised idea of quaternions, one should look into Rotors. Euler angles can also be used, though not with each angle uniformly distributed (; ). For the axis–angle form, the axis is uniformly distributed over the unit sphere of directions, , while the angle has the nonuniform distribution over noted previously .

For , Spin(n) is simply connected and thus the universal covering group for . By far the most famous example of a spin group is , which is nothing but , or the group of unit quaternions. The Pin and Spin groups are found within Clifford algebras, which themselves can be built from orthogonal matrices.

Many different methods and solutions developed to solve the problem, broadly defined as either separable, simultaneous solutions. Each type of solution has specific advantages and disadvantages as well as formulations and applications to the problem. A common theme throughout all of the methods is the common use of quaternions to represent rotations.

However, in Section 189(e) of An Elementary Treatise on Quaternions,Clarendon Press, Oxford, 1867, pp. 133--135 also in 1867, Tait treats the problem (in effect, echoing Davies' remarks in the Gentleman's Diary in 1831 with regard to Question 1265, but now in the setting of quaternions): : If perpendiculars be erected outwards at the middle points of the sides of a triangle, each being proportional to the corresponding side, the mean point of their extremities coincides with that of the original triangle. Find the ratio of each perpendicular to half the corresponding side of the old triangle that the new triangle may be equilateral. Tait concludes that the mean points of equilateral triangles erected outwardly on the sides of any triangle form an equilateral triangle.

The initiation ritual and all group meetings take place in the "Old College", the original building in which James C. Furman taught the university's first courses in Greenville in 1851. It is also widely known that Quaternions are given lifetime access to this building upon initiation which also houses the controls for the 59 bell Burnside Carillon inside Furman's iconic bell tower. Famous Quaternions have included U.S. Secretary of Education Richard Riley, South Carolina Governor Mark Sanford, and Clement Haynsworth, a nominee for the U.S. Supreme Court. There are also a number of strongly rumored secret societies with less documentation including The Magnolia Society, which has apparently formed within the past decade and taps men and women from all classes into something like an elitist supper club.

There are two special degenerate versor cases, called the unit-scalars. These two scalars (negative and positive unity) can be thought of as scalar quaternions. These two scalars are special limiting cases, corresponding to versors with angles of either zero or π. Unlike other versors, these two cannot be represented by a unique arc.

Algorithms based on multidimensional rotations and modified quaternions have been developed to identify topological relationships between protein structures without the need for a predetermined alignment. Such algorithms have successfully identified canonical folds such as the four-helix bundle. The SuperPose method is sufficiently extensible to correct for relative domain rotations and other structural pitfalls.

To apply angular changes, the orientation is modified by a delta angle/axis rotation. The resulting orientation must be re-normalized to prevent floating-point error from successive transformations from accumulating. For matrices, re- normalizing the result requires converting the matrix into its nearest orthonormal representation. For quaternions, re-normalization requires performing quaternion normalization.

The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 16 December 1843. He first published his result slightly later than Arthur Cayley's article. The octonions were discovered independently by Cayley.

To learn about when algebras (in the technical sense) have a division operation, refer to the page on division algebras. In particular Bott periodicity can be used to show that any real normed division algebra must be isomorphic to either the real numbers R, the complex numbers C, the quaternions H, or the octonions O.

Rotation in space is achieved by use of quaternions, and Lorentz transformations of spacetime by use of biquaternions. Early in the 20th century, hypercomplex number systems were examined. Later their automorphism groups led to exceptional groups such as G2. In the 1890s logicians were reducing the primitive notions of synthetic geometry to an absolute minimum.

The projective line over a division ring results in a single auxiliary point . Examples include the real projective line, the complex projective line, and the projective line over quaternions. These examples of topological rings have the projective line as their one-point compactifications. The case of the complex number field C has the Möbius group as its homography group.

Wilhelm Blaschke (1960) Kinematics and Quaternions, page 47, translated by D. H. Delphenich Kotelnikov began instructing at the university in 1893. His habilitation thesis was The Projective Theory of Vectors (1899). In Kiev, Kotelnikov was professor and head of the department of pure mathematics until 1904. Returning to Kazan, he headed the mathematics department until 1914.

For , if Q has diagonalization with non-zero a and b (which always exists if Q is non- degenerate), then is isomorphic to a K-algebra generated by elements x and y satisfying , and . Thus is isomorphic to the (generalized) quaternion algebra . We retrieve Hamilton's quaternions when , since . As a special case, if some x in V satisfies , then .

Special relativity is set in Minkowski space. General relativity uses curved spaces, which may be thought of as with a curved metric for most practical purposes. None of these structures provide a (positive- definite) metric on . Euclidean also attracts the attention of mathematicians, for example due to its relation to quaternions, a 4-dimensional real algebra themselves.

This gives one common way of representing the orientation using an axis–angle representation. Other widely used methods include rotation quaternions, Euler angles, or rotation matrices. More specialist uses include Miller indices in crystallography, strike and dip in geology and grade on maps and signs. Unit vector may also be used to represent an object's normal vector orientation.

Similarly, the quaternions and the octonions are respectively four- and eight- dimensional real vector spaces, and Cn is a 2n-dimensional real vector space. The vector space Fn has a standard basis: :e_1 = (1, 0, \ldots, 0) :e_2 = (0, 1, \ldots, 0) :\vdots :e_n = (0, 0, \ldots, 1) where 1 denotes the multiplicative identity in F.

Physicist John C. Baez notes, > Nothing in the assumptions mentions the continuum: the hypotheses are purely > algebraic. It therefore seems quite magical that [the division ring over > which the Hilbert space is defined] is forced to be the real numbers, > complex numbers or quaternions. Writing a decade after Solèr's original publication, Pitowsky calls her theorem "celebrated".

The quaternions used are actually biquaternions. The book is highly readable and well-referenced with contemporary sources in the footnotes. Several reviews were published. Nature expressed some misgivings:Anon. (1914) Review: Theory of Relativity Nature 94:387 (#2354) :A systematic exposition of the principle of relativity necessarily consists very largely in the demonstration of invariant properties of certain mathematical relations.

In 1876 he was elected trustee; in 1882 he became treasurer; and in 1907 he was elected Chancellor. Chace's lifelong passion was mathematics. He wrote many articles on mathematical subjects, including one called "A Certain Class of Cubic Surfaces Treated by Quaternions" in the Journal of Mathematics. He attended the International Mathematical Congress at Cambridge, England in 1912.

But to extend split-complex arithmetic to four dimensions leads to hyperbolic quaternions, and opens the door to abstract algebra's hypercomplex numbers. Reviewing the expressions of Lorentz and Einstein, one observes that the Lorentz factor is an algebraic function of velocity. For readers uncomfortable with transcendental functions cosh and sinh, algebraic functions may be more to their liking.

The Euclidean algorithm may be applied to some noncommutative rings such as the set of Hurwitz quaternions. Let and represent two elements from such a ring. They have a common right divisor if and for some choice of and in the ring. Similarly, they have a common left divisor if and for some choice of and in the ring.

Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dihn. For this reason the dicyclic group is also known as the binary dihedral group. Note that the dicyclic group does not contain any subgroup isomorphic to Dihn.

Simultaneous solutions are based on solving for both and at the same time (rather than basing the solution of one part off of the other as in separable solutions), propagation of error is significantly reduced.Algo Li, et al. "Simultaneous robot-world and hand-eye calibration using dual-quaternions and Kronecker product." International Journal of the Physical Sciences Vol.

Peter Guthrie Tait carried the quaternion standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇. In 1878, Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product.

It has the property that the absolute value of a quaternion is equal to the square root of the determinant of the matrix image of . The set of unit quaternions is then given by matrices of the above form with unit determinant. This matrix subgroup is precisely the special unitary group . Thus, as a Lie group is isomorphic to .

For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2×2 matrices over the p-adics. To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x2 \+ y2 = −1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation :x2 \+ y2 = −1 is solvable in the p-adic numbers. Therefore the quaternion :xi + yj + k has norm 0 and hence doesn't have a multiplicative inverse. One way to classify the F-algebra isomorphism classes of all quaternion algebras for a given field, F is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over F and isomorphism classes of their norm forms.

A Riemannian symmetric space which is additionally equipped with a parallel subbundle of End(TM) isomorphic to the imaginary quaternions at each point, and compatible with the Riemannian metric, is called quaternion-Kähler symmetric space. An irreducible symmetric space G/K is quaternion-Kähler if and only if isotropy representation of K contains an Sp(1) summand acting like the unit quaternions on a quaternionic vector space. Thus the quaternion-Kähler symmetric spaces are easily read off from the classification. In both the compact and the non- compact cases it turns out that there is exactly one for each complex simple Lie group, namely AI with p = 2 or q = 2 (these are isomorphic), BDI with p = 4 or q = 4, CII with p = 1 or q = 1, EII, EVI, EIX, FI and G.

Tait's articles include those he wrote for the ninth edition of the Encyclopædia Britannica on light, mechanics, quaternions, radiation, and thermodynamics, and the biographical notices of Hamilton and James Clerk Maxwell. He died in Edinburgh on 4 July 1901. He is buried in the second terrace down from Princes Street in the burial ground of St John's Episcopal Church, Edinburgh.

She was privately tutored for one year before she entered Vassar College in 1865, where she met the astronomer Maria Mitchell. During her time at Vassar College, her father died and her brother was lost at sea. She obtained her degree in 1868. From 1869 to 1870, she took some courses about quaternions and celestial mechanics by Benjamin Peirce (at Harvard).

A cross product for 7-dimensional vectors can be obtained in the same way by using the octonions instead of the quaternions. The nonexistence of nontrivial vector- valued cross products of two vectors in other dimensions is related to the result from Hurwitz's theorem that the only normed division algebras are the ones with dimension 1, 2, 4, and 8.

Iterative solutions are another method used to solve the problem of error propagation. One example of an iterative solution is a program based on minimizing . As the program iterates, it will converge on a solution to independent to the initial robot orientation of . Solutions can also be two-step iterative processes, and like simultaneous solutions can also decompose the equations into dual quaternions.

Irish commemorative coin celebrating the 200th Anniversary of his birth. Hamilton retained his faculties unimpaired to the very last, and steadily continued the task of finishing the Elements of Quaternions which had occupied the last six years of his life. He died on 2 September 1865, following a severe attack of gout. He is buried in Mount Jerome Cemetery in Dublin.

Hamilton needed a way to distinguish between two different types of double quaternions, the associative biquaternions and the octaves. He spoke about them to the Royal Irish Society and credited his friend Graves for the discovery of the second type of double quaternion. Hamilton 1853 pg 740See a hard copy of Lectures on quaternions, appendix B, half of the hyphenated word double quaternion has been cut off in the online Edition See Hamilton's talk to the Royal Irish Academy on the subject observed in reply that they were not associative, which may have been the invention of the concept. He also promised to get Graves' work published, but did little about it; Cayley, working independently of Graves, but inspired by Hamilton's publication of his own work, published on octonions in March 1845 – as an appendix to a paper on a different subject.

The sign convention used above corresponds to the signs obtained by multiplying two quaternions. Other sign conventions can be obtained by changing any a_k to -a_k, and/or any b_k to -b_k. If the a_k and b_k are real numbers, the identity expresses the fact that the absolute value of the product of two quaternions is equal to the product of their absolute values, in the same way that the Brahmagupta–Fibonacci two-square identity does for complex numbers. This property is the definitive feature of composition algebras. Hurwitz's theorem states that an identity of form, :(a_1^2+a_2^2+a_3^2+...+a_n^2)(b_1^2+b_2^2+b_3^2+...+b_n^2) = c_1^2+c_2^2+c_3^2+...+c_n^2 where the c_i are bilinear functions of the a_i and b_i is possible only for n = 1, 2, 4, or 8.

When R is the field of real numbers, then the diagonal subring of M2(R) is isomorphic to split- complex numbers. When R is the field of complex numbers, then the diagonal subring is isomorphic to bicomplex numbers. When R = ℍ, the division ring of quaternions, then the diagonal subring is isomorphic to the ring of split- biquaternions, presented in 1873 by William K. Clifford.

William Rowan Hamilton (1805–1865), Irish physicist, astronomer, and mathematician, first foreign member of the American National Academy of Sciences. While maintaining opposing position about how geometry should be studied, Hamilton always remained on the best terms with Cayley. Hamilton proved that for a linear function of quaternions there exists a certain equation, depending on the linear function, that is satisfied by the linear function itself.

Alexander McAulay (9 December 1863 – 6 July 1931) was the first professor of mathematics and physics at the University of Tasmania, Hobart, Tasmania. He was also a proponent of dual quaternions, which he termed "octonions" or "Clifford biquaternions". McAulay was born on 9 December 1863 and attended Kingswood School in Bath. He proceeded to Caius College, Cambridge, there taking up a study of the quaternion algebra.

In particular, the only simple rings that are a finite- dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions. Any quotient of a ring by a maximal ideal is a simple ring. In particular, a field is a simple ring. A ring R is simple if and only its opposite ring Ro is simple.

Jakob Carl Spener, teutsches ivs pvblicvm; oder, des Heil. Römisch-Teutschen Reichs vollständige Staats-Rechts-Lehre, George Marcus Knoche (1723), 124f. (note a); the extended list of quaternions is here traced to Onofrio Panvinio, De Comitiis Imperatoriis (Basel 1558). It is likely that this system was first introduced under Emperor Sigismund, who is assumed to have commissioned the frescoes in Frankfurt city hall in 1414.

Next he used complex quaternions (biquaternions) to represent the Lorentz group of special relativity, including the Thomas precession. He cited five authors, beginning with Ludwik Silberstein, who used a potential function of one quaternion variable to express Maxwell's equations in a single differential equation. Concerning general relativity, he expressed the Runge–Lenz vector. He mentioned the Clifford biquaternions (split- biquaternions) as an instance of Clifford algebra.

The Julia sets and Mandelbrot sets can be extended to the Quaternions, but they must use cross sections to be rendered visually in 3 dimensions. This Julia set is cross sectioned at the plane. Like functions of a complex variable, functions of a quaternion variable suggest useful physical models. For example, the original electric and magnetic fields described by Maxwell were functions of a quaternion variable.

The Quaternions can be generalized into further algebras called quaternion algebras. Take to be any field with characteristic different from 2, and and to be elements of ; a four- dimensional unitary associative algebra can be defined over with basis and , where , and (so ). Quaternion algebras are isomorphic to the algebra of 2×2 matrices over or form division algebras over , depending on the choice of and .

In computer science, a 4D vector is a 4-component vector data type. Uses include homogeneous coordinates for 3-dimensional space in computer graphics, and red green blue alpha (RGBA) values for bitmap images with a color and alpha channel (as such they are widely used in computer graphics). They may also represent quaternions (useful for rotations) although the algebra they define is different.

"Space and Time" (Wikisource).Scott Walter (1999) Non- Euclidean Style of Special Relativity Already in the 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions, though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry. The non-Euclidean planar algebras support kinematic geometries in the plane.

There are only three finite- dimensional associative division algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the algebra of octonions. The octonions are connected to a wide variety of exceptional objects. For example, the exceptional formally real Jordan algebra is the Albert algebra of 3 by 3 self-adjoint matrices over the octonions.

In addition to deepening topological and geometrical insight, it has purely algebraic consequences, such as the classification of finite-dimensional real division algebras: R, C, the quaternions H and the octonions O. The cotangent bundle of a differentiable manifold consists, at every point of the manifold, of the dual of the tangent space, the cotangent space. Sections of that bundle are known as differential one-forms.

In kinematics, a turn is a rotation less than a full revolution. A turn may be represented in a mathematical model that uses expressions of complex numbers or quaternions. In the complex plane every non-zero number has a polar coordinate expression where and a is in . A turn of the complex plane arises from multiplying by an element that lies on the unit circle: : z ↦ uz.

In 1853 William Rowan Hamilton published his Lectures on Quaternions which included presentation of biquaternions. The following passage from page 673 shows how Hamilton uses biquaternion algebra and vectors from quaternions to produce hyperboloids from the equation of a sphere: ::... the equation of the unit sphere , and change the vector to a bivector form, such as . The equation of the sphere then breaks up into the system of the two following, :::, ; ::and suggests our considering and as two real and rectangular vectors, such that :::. ::Hence it is easy to infer that if we assume , where is a vector in a given position, the new real vector will terminate on the surface of a double-sheeted and equilateral hyperboloid; and that if, on the other hand, we assume , then the locus of the extremity of the real vector will be an equilateral but single-sheeted hyperboloid.

About 1818 Danish scholar Ferdinand Degen displayed the Degen's eight-square identity, which was later connected with norms of elements of the octonion algebra: :Historically, the first non- associative algebra, the Cayley numbers ... arose in the context of the number-theoretic problem of quadratic forms permitting composition…this number-theoretic question can be transformed into one concerning certain algebraic systems, the composition algebras... In 1919 Leonard Dickson advanced the study of the Hurwitz problem with a survey of efforts to that date, and by exhibiting the method of doubling the quaternions to obtain Cayley numbers. He introduced a new imaginary unit , and for quaternions and writes a Cayley number . Denoting the quaternion conjugate by , the product of two Cayley numbers is :(q + Qe)(r + Re) = (qr - R'Q) + (Rq + Q r')e . The conjugate of a Cayley number is , and the quadratic form is , obtained by multiplying the number by its conjugate.

Rotating a line about another, to which it is Clifford parallel, creates a Clifford surface. The Clifford parallels through points on the surface all lie in the surface. A Clifford surface is thus a ruled surface since every point is on two lines, each contained in the surface. Given two square roots of minus one in the quaternions, written r and s, the Clifford surface through them is given byH.

As for mathematics, the hyperbolic quaternion is another hypercomplex number, as such structures were called at the time. By the 1890s Richard Dedekind had introduced the ring concept into commutative algebra, and the vector space concept was being abstracted by Giuseppe Peano. In 1899 Alfred North Whitehead promoted Universal algebra, advocating for inclusivity. The concepts of quasigroup and algebra over a field are examples of mathematical structures describing hyperbolic quaternions.

When thought of as a Lie group is often denoted or . It turns out that the only spheres that admit a Lie group structure are , thought of as the set of unit complex numbers, and , the set of unit quaternions. One might think that , the set of unit octonions, would form a Lie group, but this fails since octonion multiplication is nonassociative. The octonionic structure does give one important property: parallelizability.

Two commemorative stamps were issued by Ireland in 1943 to mark the centenary of the announcement of quaternions. A 10 Euros commemorative silver Proof coin was issued by the Central Bank of Ireland in 2005 to commemorate 200 years since his birth. The newest maintenance depot for the Dublin LUAS tram system has been named after him. It is located adjacent to the Broombridge stop on the Green Line.

Lowell was born on December 13, 1856, in Boston, Massachusetts, the second son of Augustus Lowell and Katherine Bigelow Lowell. His mother was a cousin of architect Charles H. Bigelow. A member of the Brahmin Lowell family, his siblings included the poet Amy Lowell, the astronomer Percival Lowell, and Elizabeth Lowell Putnam, an early activist for prenatal care. They were the great-grandchildren of John Lowell and, on their mother's side, the grandchildren of Abbott Lawrence.Lowell, Delmar R., The Historic Genealogy of the Lowells of America from 1639 to 1899 (Rutland VT: The Tuttle Company, 1899), 283 Lowell Lecture Hall Lowell graduated from Noble and Greenough School in 1873 and attended Harvard College where he presented a thesis for honors in mathematics that addressed using quaternions to treat quadricsA.L. Lowell (1878) Surfaces of the second order, as treated by quaternions, Proceedings of the American Academy of Arts and Sciences 13:222–50, from Biodiversity Heritage Library and graduated in 1877.

Corrado Segre (1912) continued the development with that ring. Arthur Conway, one of the early adopters of relativity via biquaternion transformations, considered the quaternion-multiplicative-inverse transformation in his 1911 relativity study.Arthur Conway (1911) "On the application of quaternions to some recent developments of electrical theory", Proceedings of the Royal Irish Academy 29:1–9, particularly page 9 In 1947 some elements of inversive quaternion geometry were described by P.G. Gormley in Ireland.

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing.

He was head of the Graduate Department at Harvard from 1872 to 1895 (becoming its dean when it was converted to the Graduate School). He was the Dean of the Faculty of Arts and Sciences from 1895 to 1898. Among his publications are Mathematical Tables Chiefly to Four Figures (1896) and A Text-Book of Analytic Geometry; On the Basis of Professor Peirce’s Treatise (1857). He was considered a world authority on quaternions.

Lambek's PhD thesis investigated vector fields using the biquaternion algebra over Minkowski space, as well as semigroup immersion in a group. The second component was published by the Canadian Journal of Mathematics. He later returned to biquaternions when in 1995 he contributed "If Hamilton had prevailed: Quaternions in Physics", which exhibited the Riemann–Silberstein bivector to express the free-space electromagnetic equations. Lambek supervised 17 doctoral students, and has 75 doctoral descendants as of 2020.

Its most famous director was William Rowan Hamilton, who, amongst other things, discovered quaternions, the first non-commutative algebra, while walking from the observatory to the city with his wife. He is also renowned for his Hamiltonian formulation of dynamics. In the late 20th century, the city encroached ever more on the observatory, which compromised the seeing. The telescope, no longer state of the art, was used mainly for public 'open nights'.

Using the appropriate "angle", and a radial vector, any one of these planes can be given a polar decomposition. Any one of these decompositions, or Lie algebra renderings, may be necessary for rendering the Lie subalgebra of a 2 × 2 real matrix. There is a classical 3-parameter Lie group and algebra pair: the quaternions of unit length which can be identified with the 3-sphere. Its Lie algebra is the subspace of quaternion vectors.

A Moufang 3-gon can be identified with the incidence graph of a Moufang projective plane. In this identification, the points and lines of the plane correspond to the vertices of the building. Real forms of Lie groups give rise to examples which are the three main types of Moufang 3-gons. There are four real division algebras: the real numbers, the complex numbers, the quaternions, and the octonions, of dimensions 1,2,4 and 8, respectively.

Apart from the highest tiers of the emperor, kings, prince-bishops and the prince electors, the estates are represented in groups of four. The number of quaternions was usually ten, in descending order of precedence Dukes (Duces), Margraves (Marchiones), Landgraves (Comites Provinciales), Burggraves (Comites Castrenses), Counts (Comites), Knights (Milites), Noblemen (Liberi), Cities (Metropoles), Villages (Villae) and Peasants (Rustici). The list could be shortened or expanded, by the mid-16th century to as many as 45.

Eugene Salamin is a mathematician who discovered (independently with Richard Brent) the Salamin-Brent algorithm, used in high-precision calculation of pi. Eugene Salamin worked on alternatives to increase accuracy and minimize computational processes through the use of quaternions. Benefits may include: # the design of spatio-temporal databases; # numerical mathematical methods that traditionally prove unsuccessful due to buildup of computational error; # therefore, may be applied to applications involving genetic algorithms and evolutionary computation, in general.

The cause of gimbal lock is representing an orientation as three axial rotations with Euler angles. A potential solution therefore is to represent the orientation in some other way. This could be as a rotation matrix, a quaternion (see quaternions and spatial rotation), or a similar orientation representation that treats the orientation as a value rather than three separate and related values. Given such a representation, the user stores the orientation as a value.

P.R. Girard's 1984 essay The quaternion group and modern physics discusses some roles of quaternions in physics. The essay shows how various physical covariance groups, namely , the Lorentz group, the general theory of relativity group, the Clifford algebra and the conformal group, can easily be related to the quaternion group in modern algebra. Girard began by discussing group representations and by representing some space groups of crystallography. He proceeded to kinematics of rigid body motion.

To satisfy the last three equations, either or , and are all 0. The latter is impossible because a is a real number and the first equation would imply that Therefore, and In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere. Only negative real quaternions have infinitely many square roots.

Applying an operation on a scene graph requires some way of dispatching an operation based on a node's type. For example, in a render operation, a transformation group node would accumulate its transformation by matrix multiplication, vector displacement, quaternions or Euler angles. After which a leaf node sends the object off for rendering to the renderer. Some implementations might render the object directly, which invokes the underlying rendering API, such as DirectX or OpenGL.

In mathematical field of representation theory, a quaternionic representation is a representation on a complex vector space V with an invariant quaternionic structure, i.e., an antilinear equivariant map :j\colon V\to V which satisfies :j^2=-1. Together with the imaginary unit i and the antilinear map k := ij, j equips V with the structure of a quaternionic vector space (i.e., V becomes a module over the division algebra of quaternions).

Another noted aspect was the multi-layered levels, as compared to equivalent 3D action-adventure games of the time which were limited to a flat-floor system. The game pioneered the use of quaternions for smooth 3D rotation. Lara's movements were hand-animated and coordinated rather than created using motion capture. The reason for this was that the team wanted uniformity in her movement, which was not possible with motion capture technology of the time.

Heaviside rewrote them in the form commonly used today, just four expressions. In addition, Heaviside was responsible for considerable progress in electrical telegraphy, telephony, and the study of the propagation of electromagnetic waves. Independent of Gibbs, Heaviside assembled a set of mathematical tools known as vector calculus to replace the quaternions, which were in vogue at the time but which Heaviside dismissed as "antiphysical and unnatural." Faraday also investigated how electrical currents affected chemical solutions.

Multiplication is often defined for natural numbers, then extended to whole numbers, fractions, and irrational numbers. However, abstract algebra has a more general definition of multiplication as a binary operation on some objects that may or may not be numbers. Notably, one can multiply complex numbers, vectors, matrices, and quaternions. Some educators believe that seeing multiplication exclusively as repeated addition during elementary education can interfere with later understanding of these aspects of multiplication.

Using Solèr's theorem, the field K over which the vector space is defined can be proven, with additional hypotheses, to be either the real numbers, complex numbers, or the quaternions, as is needed for Gleason's theorem to hold. By invoking Gleason's theorem, the form of a probability function on lattice elements can be restricted. Assuming that the mapping from lattice elements to probabilities is noncontextual, Gleason's theorem establishes that it must be expressible with the Born rule.

During Dorpat years, Molien had mathematical correspondence with Frobenius and Hurwitz. Since the University of Dorpat had only one professorship in pure mathematics, Molien had to stay for years at docent's position. Being a docent in Dorpat, Molien prepared and gave different lecture courses: Theory of analytic and elliptic functions, modern geometry and algebra, theory of algebraic equations, number theory, projective geometry, theory of quaternions, history of mathematics and others. Some of these courses were new for the university.

The codex contains the entire of the New Testament (Gospels, Acts, Catholic, Pauline epistles, Revelation) on 400 parchment leaves (size ), they are split in two volumes. The text is written in one column per page, 26 lines per page. The leaves are arranged in quaternions, but separately numbered for each volume. The text is divided according to the (chapters), whose numbers are given at the margin, and their (titles of chapters) at the top of the pages.

By the 1830s the canal carried 80,000 tons of freight and 40,000 passengers a year.Quaternion plaque on Brougham (Broom) Bridge, Dublin Ferns' Lock In 1843, while walking with his wife along the Royal Canal, Sir William Rowan Hamilton realised the formula for quaternions and carved his initial thoughts into a stone on the Broom Bridge over the canal. The annual Hamilton Walk commemorates this event. In 1845 the canal was bought by the Midland Great Western Railway Company.

At the International Congress of Mathematicians (ICM) in 1912 at Cambridge, Silberstein spoke on "Some applications of quaternions". Though the text was not published in the proceedings of the Congress, it did appear in the Philosophical Magazine of May, 1912, with the title "Quaternionic form of relativity".Ludwik Silberstein, "Quaternionic form of relativity", Philosophical Magazine 23:790–809. The following year Macmillan published The Theory of Relativity, which is now available on-line in the Internet Archive (see references).

Study (1913), Delphinich translator, "Foundations and goals of analytical kinematics" from Neo-classical physics develops the field of kinematics, in particular exhibiting an element of E(3) as a homography of dual quaternions. Study's use of abstract algebra was noted in A History of Algebra (1985) by B. L. van der Waerden. On the other hand, Joe Rooney recounts these developments in relation to kinematics.Joe Rooney William Kingdon Clifford, Department of Design and Innovation, the Open University, London.

In 1892 Corrado Segre introduced. (see especially pages 455–67) bicomplex numbers in Mathematische Annalen, which form an algebra isomorphic to the tessarines. Corrado Segre read W. R. Hamilton's Lectures on Quaternions (1853) and the works of W. K. Clifford. Segre used some of Hamilton's notation to develop his system of bicomplex numbers: Let h and i be elements that square to −1 and that commute. Then, presuming associativity of multiplication, the product hi must square to +1.

If A is a commutative semigroup, then one has :\forall x, y \isin A \quad (xy)^2 = xy xy = xx yy = x^2 y^2 . In the language of quadratic forms, this equality says that the square function is a "form permitting composition". In fact, the square function is the foundation upon which other quadratic forms are constructed which also permit composition. The procedure was introduced by L. E. Dickson to produce the octonions out of quaternions by doubling.

Dickstein wrote many mathematical books and founded the journal Wiadomości Mathematyczne (Mathematical News), now published by the Polish Mathematical Society. He was a bridge between the times of Cauchy and Poincaré and those of the Lwów School of Mathematics. He was also thanked by Alexander Macfarlane for contributing to the Bibliography of Quaternions (1904) published by the Quaternion Society. He was also one of the personalities, who contributed to the foundation of the Warsaw Public Library in 1907.

The Quaternion Eagle, hand-coloured woodcut (c. 1510) by Hans Burgkmair. One rendition of the coat of the empire was the "Quaternion Eagle" (so named after the imperial quaternions) printed by David de Negker of Augsburg, after a 1510 woodcut by Hans Burgkmair. It showed a selection of 56 shields of various Imperial States in groups of four on the feathers of a double-headed eagle supporting, in place of a shield, Christ on the Cross.

Some number systems that are not included in the complex numbers may be constructed from the real numbers in a way that generalize the construction of the complex numbers. They are sometimes called hypercomplex numbers. They include the quaternions H, introduced by Sir William Rowan Hamilton, in which multiplication is not commutative, the octonions, in which multiplication is not associative in addition to not being commutative, and the sedenions, in which multiplication is not alternative, neither associative nor commutative.

When he died, Hamilton was working on a definitive statement of quaternion science. His son William Edwin Hamilton brought the Elements of Quaternions, a hefty volume of 762 pages, to publication in 1866. As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing 1899 and the second in 1901. The subject index and footnotes in this second edition improved the Elements accessibility.

However, it is often assumed to have a structure of -vector space over the reals. That is, the codomain may be a Euclidean space, a coordinate vector, the set of matrices of real numbers of a given size, or an -algebra, such as the complex numbers or the quaternions. The structure -vector space of the codomain induces a structure of -vector space on the functions. If the codomain has a structure of -algebra, the same is true for the functions.

Historically, division rings were sometimes referred to as fields, while fields were called commutative fields. The only division rings that are finite-dimensional -vector spaces are itself, (which is a field), the quaternions (in which multiplication is non-commutative), and the octonions (in which multiplication is neither commutative nor associative). This fact was proved using methods of algebraic topology in 1958 by Michel Kervaire, Raoul Bott, and John Milnor. The non-existence of an odd-dimensional division algebra is more classical.

The subject of physical mathematics is concerned with physically motivated mathematics and is different from mathematical physics. The details of physical units and their manipulation were addressed by Alexander Macfarlane in Physical Arithmetic in 1885.Alexander Macfarlane (1885) Physical Arithmetic via Internet Archive The science of kinematics created a need for mathematical represention of motion and has found expression with complex numbers, quaternions, and linear algebra. At Cambridge University the Mathematical Tripos tested students on their knowledge of "mixed mathematics".

Hugh Hawkins, Pioneer: A History of the Johns Hopkins University, 1874-1889 (1960), p. 135. Story was instrumental in starting two publication projects: The Johns Hopkins University Circulars was a student paper detailing classes and attendees. American Journal of Mathematics was also started as a joint effort of Sylvester and Story, but soon Story was replaced by Thomas Craig as managing editor. In 1893 Story became an associate professor; he taught courses on quaternions, elliptic functions, invariant theory, mathematical astronomy and mathematical elasticity.

The coquaternions were initially introduced (under that name)James Cockle (1849), On Systems of Algebra involving more than one Imaginary, Philosophical Magazine (series 3) 35: 434,5, link from Biodiversity Heritage Library in 1849 by James Cockle in the London-Edinburgh- Dublin Philosophical Magazine. The introductory papers by Cockle were recalled in the 1904 BibliographyA. Macfarlane (1904) Bibliography of Quaternions and Allied Systems of Mathematics, from Cornell University Historical Math Monographs, entries for James Cockle, pp. 17–18 of the Quaternion Society.

J. Hunt (1991) The Maxwellians, pages 165,6, Cornell University Press the use of scalar and vector potentials is now standard in the solution of Maxwell's equations. As Barrett and Grimes (1995) describe: > Maxwell expressed electromagnetism in the algebra of quaternions and made > the electromagnetic potential the centerpiece of his theory. In 1881 > Heaviside replaced the electromagnetic potential field by force fields as > the centerpiece of electromagnetic theory. According to Heaviside, the > electromagnetic potential field was arbitrary and needed to be > "assassinated".

Although there are no real square roots of -1, the complex number i satisfies i2 = −1, and as such can be considered as a square root of −1. The only other complex number whose square is −1 is −i because by the fundamental theorem of algebra, there are exactly two square roots of any nonzero complex number. In the algebra of quaternions (where the fundamental theorem does not apply), which contain the complex plane, the equation x2 = −1 has infinitely many solutions.

Furthermore, the fraternity's chapter at Furman carries a unique flag which bears a red rose in the upper right-hand corner. On campus today the only known active secret society is The Quaternion Club, although many are rumored to exist. Quaternion, which dates back to 1903, taps four juniors and four seniors each year in the late winter or early spring. The selection process is guarded but is thought to be controlled by current Quaternions currently in residence at the school.

In mathematics, Solèr's theorem is a result concerning certain infinite- dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia Solèr, the result is significant for quantum logic and the foundations of quantum mechanics. In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates.

Hamilton was stung into protesting Graves' priority in discovery, if not publication; nevertheless, octonions are known by the name Cayley gave them – or as Cayley numbers. The major deduction from the existence of octonions was the eight squares theorem, which follows directly from the product rule from octonions, had also been previously discovered as a purely algebraic identity, by Carl Ferdinand Degen in 1818. This sum-of- squares identity is characteristic of composition algebra, a feature of complex numbers, quaternions, and octonions.

The doubling method was formalized by A. A. Albert who started with the real number field ℝ and the square function, doubling it to obtain the complex number field with quadratic form , and then doubling again to obtain quaternions. The doubling procedure is called the Cayley–Dickson process and the structures produced are composition algebras. The square function can be used with ℂ as the start for another use of the Cayley–Dickson process leading to bicomplex, biquaternion, and bioctonion composition algebras.

Real and pseudoreal representations of a group G can be understood by viewing them as representations of the real group algebra R[G]. Such a representation will be a direct sum of central simple R-algebras, which, by the Artin-Wedderburn theorem, must be matrix algebras over the real numbers or the quaternions. Thus a real or pseudoreal representation is a direct sum of irreducible real representations and irreducible quaternionic representations. It is real if no quaternionic representations occur in the decomposition.

Although the Euclidean algorithm is used to find the greatest common divisor of two natural numbers (positive integers), it may be generalized to the real numbers, and to other mathematical objects, such as polynomials, quadratic integers and Hurwitz quaternions. In the latter cases, the Euclidean algorithm is used to demonstrate the crucial property of unique factorization, i.e., that such numbers can be factored uniquely into irreducible elements, the counterparts of prime numbers. Unique factorization is essential to many proofs of number theory.

Frank Morley consistently referred to elements of the unit circle as turns in the book Inversive Geometry, (1933) which he coauthored with his son Frank Vigor Morley. The Latin term for turn is versor, which is a quaternion that can be visualized as an arc of a great circle. The product of two versors can be compared to a spherical triangle where two sides add to the third. For the kinematics of rotation in three dimensions, see quaternions and spatial rotation.

Quaternions was also one of the themes of his address as president of the mathematical section of the British Association for the Advancement of Science in 1871. He also produced original work in mathematical and experimental physics. In 1864, he published a short paper on thermodynamics, and from that time his contributions to that and kindred departments of science became frequent and important. In 1871, he emphasised the significance and future importance of the principle of the dissipation of energy (second law of thermodynamics).

He had learned about quaternions from James Mills Peirce at Harvard, but Dean A. W. Phillips persuaded him to take Gibbs's course on vectors, which treated similar problems from a rather different perspective. After Wilson had completed the course, Morris approached him about the project of producing a textbook. Wilson wrote the book by expanding his own class notes, providing exercises, and consulting with others (including his father).Edwin Bidwell Wilson (1931) "Reminiscences of Gibbs by a student and colleague" Bulletin of the American Mathematical Society.

The best-known examples of associative division algebras are the finite-dimensional real ones (that is, algebras over the field R of real numbers, which are finite-dimensional as a vector space over the reals). The Frobenius theorem states that up to isomorphism there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4). Wedderburn's little theorem states that if D is a finite division algebra, then D is a finite field.Lam (2001), [ p.

In this section, Hamilton's quaternions are constructed as the even sub algebra of the Clifford algebra Cl(R). Let the vector space V be real three-dimensional space R3, and the quadratic form Q be the negative of the usual Euclidean metric. Then, for v, w in R3 we have the bilinear form (or scalar product) : v \cdot w = v_1 w_1 + v_2 w_2 + v_3 w_3. Now introduce the Clifford product of vectors v and w given by : v w + w v = -2 (v \cdot w) .

The dual numbers can also be thought of as the exterior algebra of a one-dimensional vector space; the general case of dimensions leads to the Grassmann numbers. The algebra of dual numbers is a ring that is a local ring since the principal ideal generated by is its only maximal ideal. Dual numbers form the coefficients of dual quaternions. Like the complex numbers and split-complex numbers, the dual numbers form an algebra that is 2-dimensional over the field of real numbers.

In this case, each axis cuts the unit circle in the plane (which is the one-dimensional sphere) at two points that are each other's antipodes. For N = 4, the Bingham distribution is a distribution over the space of unit quaternions. Since a unit quaternion corresponds to a rotation matrix, the Bingham distribution for N = 4 can be used to construct probability distributions over the space of rotations, just like the Matrix-von Mises–Fisher distribution. These distributions are for example used in geology, crystallography and bioinformatics.

200–5 this paper set straight wayward theorists that expected to find associativity in systems like hyperbolic quaternions. Knott wrote: ::[T]he assumption that the square of a unit vector is positive unity leads to an algebra whose characteristic quantities are non- associative.C.G. Knott (1893) Recent Innovations in Vector Theory, Nature 47 #1225 Evidently Knott overlooked the existence of the ring of coquaternions. Nevertheless, Crowe states that Knott "wrote with care and thoroughness" and that "only Knott was well acquainted with his opponents system".

Broombridge is a railway station beside a LUAS Tram stop serving Cabra, Dublin 7, Ireland. It lies on the southern bank of the Royal Canal at the western end of what had been Liffey Junction station on the erstwhile Midland Great Western Railway (MGWR). It takes its name from Broome Bridge, which crosses the canal, where William Rowan Hamilton developed the mathematical notion of quaternions. A plaque on the adjacent canal bridge and the name of the LUAS Maintenance depot on site, Hamilton Depot, commemorates this.

What are called Maxwell's equations today, are in fact a simplified version of the original equations reformulated by Heaviside, FitzGerald, Lodge and Hertz. The original equations used Hamilton's more expressive quaternion notation, a kind of Clifford algebra, which fully subsumes the standard Maxwell vectorial equations largely used today. In the late 1880s there was a debate over the relative merits of vector analysis and quaternions. According to Heaviside the electromagnetic potential field was purely metaphysical, an arbitrary mathematical fiction, that needed to be "murdered".

In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them. The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications.

The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reportedThomas Kirkman (1848) "On Pluquaternions and Homoid Products of n Squares", London and Edinburgh Philosophical Magazine 1848, p 447 Google books link on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.

In Macfarlane's paper there is an effort to produce "trigonometry on the surface of the equilateral hyperboloids" through the algebra of hyperbolic quaternions, now re-identified in an associative ring of eight real dimensions. The effort is reinforced by a plate of nine figures on page 181. They illustrate the descriptive power of his "space analysis" method. For example, figure 7 is the common Minkowski diagram used today in special relativity to discuss change of velocity of a frame of reference and relativity of simultaneity.

P.G. Gormley (1947) "Stereographic projection and the linear fractional group of transformations of quaternions", Proceedings of the Royal Irish Academy, Section A 51:67–85 In 1968 Isaak Yaglom's Complex Numbers in Geometry appeared in English, translated from Russian. There he uses P(D) to describe line geometry in the Euclidean plane and P(M) to describe it for Lobachevski's plane. Yaglom's text A Simple Non-Euclidean Geometry appeared in English in 1979. There in pages 174 to 200 he develops Minkowskian geometry and describes P(M) as the "inversive Minkowski plane".

A quasi-sphere is a submanifold of a pseudo- Euclidean space consisting of the points for which the displacement vector from a reference point satisfies the equation :, where and . Since in permitted, this definition includes hyperplanes; it is thus a generalization of generalized circles and their analogues in any number of dimensions. This inclusion provides a more regular structure under conformal transformations than if they are omitted. This definition has been generalized to affine spaces over complex numbers and quaternions by replacing the quadratic form with a Hermitian form.

This is in part because of the relationship between screws and dual quaternions which have been used to interpolate rigid-body motions.Selig, J. M. (2011) "Rational Interpolation of Rigid Body Motions," Advances in the Theory of Control, Signals and Systems with Physical Modeling, Lecture Notes in Control and Information Sciences, Volume 407/2011 213–224, Springer. Based on screw theory, an efficient approach has also been developed for the type synthesis of parallel mechanisms (parallel manipulators or parallel robots). Fundamental theorems include Poinsot's theorem (Louis Poinsot, 1806) and Chasles' theorem (Michel Chasles, 1832).

A rotation matrix, on the other hand, provides a full description of the attitude at the expense of requiring nine values instead of three. The use of a rotation matrix can lead to increased computational expense and they can be more difficult to work with. Quaternions offer a decent compromise in that they do not suffer from gimbal lock and only require four values to fully describe the attitude. Changing orientation of a rigid body is the same as rotating the axes of a reference frame attached to it.

In 1848 Kirkman wrote "On Pluquaternions and Homoid Products of n Squares".London and Edinburgh Philosophical Magazine 1848, p 447 Google books link Generalizing the quaternions and octonions, Kirkman called a pluquaternion Qa a representative of a system with a imaginary units, a > 3. Kirkman's paper was dedicated to confirming Cayley's assertions concerning two equations among triple-products of units as sufficient to determine the system in case a = 3 but not a = 4.A. J. Crilly (2006) Arthur Cayley: Mathematician Laureate of the Victorian Era, Johns Hopkins University Press, p.

So a = λ · 1\. This gives an isomorphism from A to C. The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals R, the field of complex numbers C, and the division algebra of quaternions H. This result was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his proof. Gelfand (independently) published a proof of the complex case a few years later.

His work for Mathematische Annalen has highlighted the importance of Bessel functions of the third kind, which were later known as Hankel functions. His 1867 exposition on complex numbers and quaternions is particularly memorable. For example, Fischbein notes that he solved the problem of products of negative numbers by proving the following theorem: "The only multiplication in R which may be considered as an extension of the usual multiplication in R+ by respecting the law of distributivity to the left and the right is that which conforms to the rule of signs."See .

In 1995 Ian Porteous published Clifford Algebras and the Classical Groups which was reviewed by Peter R. Law.Peter R. Law, Review of Clifford Algebras and the Classical Groups, In praise, Law says "Porteous' presentation of the subject matter sets a standard by which others may be judged." The book has 24 chapters including 8:quaternions, 13:The classical groups, 15:Clifford algebras, 16:Spin groups, 17:Conjugation, 20:Topological spaces, 21:Manifolds, 22:Lie groups. In the preface Porteous acknowledges the contribution of his master's degree student Tony Hampson and anticipatory work by Terry Wall.

The City Council hall is decorated with a cycle of limewood figures carved by master Jakob Ruß. The figures illustrate the hierarchy of the imperial estates, from princes to peasants, arranged into groups of four - the so- called "Imperial quaternions". The decorative programme offers an impression of the power structure in the time of its installation (1490–1494). The Granary served as the center of Überlingen's once great grain trade and, since its complete renovation in 1998, is one of the most visually appealing cultural monuments of the city.

A few low-dimensional cases are: :Cl(R) is naturally isomorphic to R since there are no nonzero vectors. :Cl(R) is a two- dimensional algebra generated by e1 that squares to −1, and is algebra- isomorphic to C, the field of complex numbers. :Cl(R) is a four-dimensional algebra spanned by The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the quaternions H. :Cl(R) is an 8-dimensional algebra isomorphic to the direct sum , the split-biquaternions.

The model divided the classes of the Empire into fictitious groups of four, the quaternions, whose members shared one common feature: hence, the group of worldly electors, the margraves, et cetera. This, however, often caused misleading and inappropriate constellations due to the endeavour to come up to the quaternion. Nevertheless, the success of the model was not affected. Early modern drinking culture, in which toasting was a very important custom, resulted in the beaker being connected to the "Imperial Eagle" as an expression of solidarity between the owner and the Empire.

Arthur Cayley, F.R.S. (1821–1895) is widely regarded as Britain's leading pure mathematician of the 19th century. Cayley in 1848 went to Dublin to attend lectures on quaternions by Hamilton, their discoverer. Later Cayley impressed him by being the second to publish work on them. Cayley proved the theorem for matrices of dimension 3 and less, publishing proof for the two- dimensional case. As for matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”.

Another way of looking at this group is with quaternion multiplication. Every rotation in four dimensions can be achieved by multiplying by a pair of unit quaternions, one before and one after the vector. These quaternion are unique, up to a change in sign for both of them, and generate all rotations when used this way, so the product of their groups, S3 × S3, is a double cover of SO(4), which must have six dimensions. Although the space we live in is considered three-dimensional, there are practical applications for four-dimensional space.

A rotation matrix in dimension 3 (which has nine elements) has three degrees of freedom, corresponding to each independent rotation, for example by its three Euler angles or a magnitude one (unit) quaternion. In SO(4) the rotation matrix is defined by two quaternions, and is therefore 6-parametric (three degrees of freedom for every quaternion). The rotation matrices have therefore 6 out of 16 independent components. Any set of 6 parameters that define the rotation matrix could be considered an extension of Euler angles to dimension 4.

Quaternion Slerps are commonly used to construct smooth animation curves by mimicking affine constructions like the de Casteljau algorithm for Bézier curves. Since the sphere is not an affine space, familiar properties of affine constructions may fail, though the constructed curves may otherwise be entirely satisfactory. For example, the de Casteljau algorithm may be used to split a curve in affine space; this does not work on a sphere. The two-valued Slerp can be extended to interpolate among many unit quaternions, but the extension loses the fixed execution-time of the Slerp algorithm.

The "latitude" on the hypersphere will be half of the corresponding angle of rotation, and the neighborhood of any point will become "flatter" (i.e. be represented by a 3D Euclidean space of points) as the neighborhood shrinks. This behavior is matched by the set of unit quaternions: A general quaternion represents a point in a four-dimensional space, but constraining it to have unit magnitude yields a three-dimensional space equivalent to the surface of a hypersphere. The magnitude of the unit quaternion will be unity, corresponding to a hypersphere of unit radius.

Rodrigues was born into a well-to-do Sephardi Jewish family in Bordeaux.Simon Altmann, "Rotations, Quaternions and Double Groups"(Clarendon Press, Oxford, 1986, ): "The family is often said to have been of Spanish origin, but the spelling of the family name rather suggests Portuguese descent (as indeed asserted by the 'Enciclopedia Universal Illustrada Espasa-Calpe')". For more information on the Rodrigues as Portuguese Jews in Bordeaux see also the Jewish Encyclopedia He was awarded a doctorate in mathematics on 28 June 1815 by the University of Paris.Altmann and Ortiz(2005), p.

The seven-dimensional cross product has the same relationship to the octonions as the three-dimensional product does to the quaternions. The seven-dimensional cross product is one way of generalising the cross product to other than three dimensions, and it is the only other bilinear product of two vectors that is vector-valued, orthogonal, and has the same magnitude as in the 3D case. In other dimensions there are vector-valued products of three or more vectors that satisfy these conditions, and binary products with bivector results.

Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by matrices, or direct sums of systems of matrices.Emil Artin later generalized Wedderburn's result so it is known as the Artin–Wedderburn theorem From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn's thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.

Study showed an early interest in systems of complex numbers and their application to transformation groups with his article in 1890.E. Study (1890) D.H. Delphenich translator, "On systems of complex numbers and their applications to the theory of transformation groups" He addressed this popular subject again in 1898 in Klein's encyclopedia. The essay explored quaternions and other hypercomplex number systems. This 34 page article was expanded to 138 pages in 1908 by Élie Cartan, who surveyed the hypercomplex systems in Encyclopédie des sciences mathématiques pures et appliqueés.

However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner. The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere.

In mathematics, a stunted projective space is a construction on a projective space of importance in homotopy theory, introduced by . Part of a conventional projective space is collapsed down to a point. More concretely, in a real projective space, complex projective space or quaternionic projective space :KPn, where K stands for the real numbers, complex numbers or quaternions, one can find (in many ways) copies of :KPm, where m < n. The corresponding stunted projective space is then :KPn,m = KPn/KPm, where the notation implies that the KPm has been identified to a point.

However, proved a complex-number analogue of the Sylvester–Gallai theorem: whenever the points of a Sylvester–Gallai configuration are embedded into a complex projective space, the points must all lie in a two-dimensional subspace. Equivalently, a set of points in three- dimensional complex space whose affine hull is the whole space must have an ordinary line, and in fact must have a linear number of ordinary lines. Similarly, showed that whenever a Sylvester–Gallai configuration is embedded into a space defined over the quaternions, its points must lie in a three- dimensional subspace.

For differentiable manifolds, this can happen in dimensions 2, 6, 14, 30, 62, and possibly 126, and in no other dimensions. The final case of dimension 126 remains open. These five or six framed cobordism classes of manifolds having Kervaire invariant 1 are exceptional objects related to exotic spheres. The first three cases are related to the complex numbers, quaternions and octonions respectively: a manifold of Kervaire invariant 1 can be constructed as the product of two spheres, with its exotic framing determined by the normed division algebra.

"Please excuse my not mailing this," the letter concluded, "but I don't know your new address." Unable to focus on research problems, Feynman began tackling physics problems, not for utility, but for self-satisfaction. One of these involved analyzing the physics of a twirling, nutating disk as it is moving through the air, inspired by an incident in the cafeteria at Cornell when someone tossed a dinner plate in the air. He read the work of Sir William Rowan Hamilton on quaternions, and attempted unsuccessfully to use them to formulate a relativistic theory of electrons.

More precisely, William Rowan Hamilton put forth the quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called algebraic motors. These noncommutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types. One sign of re-organization was the use of direct sums to describe algebraic structure. The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928).

Attitude is part of the description of how an object is placed in the space it occupies. Attitude and position fully describe how an object is placed in space. (For some applications such as in robotics and computer vision, it is customary to combine position and attitude together into a single description known as Pose.) Attitude can be described using a variety of methods; however, the most common are Rotation matrices, Quaternions, and Euler angles. While Euler angles are oftentimes the most straightforward representation to visualize, they can cause problems for highly-maneuverable systems because of a phenomenon known as Gimbal lock.

In 1878 he obtained the chair of mathematics at Dartmouth and served until 1893. According to "The Early History of the [Dartmouth] Mathematics Department 1769–1961": :The one example of mathematical competency was furnished by Arthur Sherburne Hardy who wrote a book on quaternions, an adequate, if not inspiring text. It was something for Dartmouth to offer a course in such an abstruse field, and the course was actually given a few times when a student and an instructor could be found simultaneously. In 1893 Professor Hardy failed in his ambition to be elected President of Dartmouth College.

Stephanos (1881) He also represented Laguerre's oriented spheres by quaternions (1883).Stephanos (1883) Lines, circles, planes, or spheres with radii of certain orientation are called by Laguerre half-lines, half-circles (cycles), half-planes, half-spheres, etc. A tangent is a half- line cutting a cycle at a point where both have the same direction. The transformation by reciprocal directions transforms oriented spheres into oriented spheres and oriented planes into oriented planes, leaving invariant the "tangential distance" of two cycles (the distance between the points of each one of their common tangents), and also conserves the lines of curvature.

In this section, dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form. Let the vector space V be real four- dimensional space R4, and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R3. For v, w in R4 introduce the degenerate bilinear form : d(v, w) = v_1 w_1 + v_2 w_2 + v_3 w_3 . This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane. The Clifford product of vectors v and w is given by :v w + w v = -2 \,d(v, w).

Through the Imperial Eagles the ideal of the durable unity of the Holy Roman Empire took decorative shape and demonstrates the emotional relationship of a broad public to the Empire. The imperial eagle was mostly pictured in the form of a quaternion eagle which related the theory of quaternions to one of the most important symbols of the Empire. As the structure of the Empire was in need of explanation, even for contemporaries, the quaternion model was meant to depict the fabric of the Empire. It was developed in the 14th century and remained popular until the end of the Empire.

The powers of , obtained by substitution from powers of , are defined by repeated matrix multiplication; the constant term of gives a multiple of the power 0, which is defined as the identity matrix. The theorem allows to be expressed as a linear combination of the lower matrix powers of . When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton.

Thus he was aware of the diversity to be encountered in modern mathematical structures, and that quaternions stand as a milestone on the way to others. He became more active in the Royal Society of Edinburgh, serving on the Council from 1894 to 1905, moving up to a Secretary to Ordinary Meetings in 1905 and finally becoming its general secretary in 1912 until his death in 1922.St. Andrews University. School of Mathematics and Statistics, Cargill Gilston Knott Knott also took an active social role in his community including Sunday school teaching and church affairs with the United Free Church of Scotland.

Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends. Let p0 and p1 be the first and last points of the arc, and let t be the parameter, 0 ≤ t ≤ 1\. Compute Ω as the angle subtended by the arc, so that , the n-dimensional dot product of the unit vectors from the origin to the ends.

Going from the real numbers to an arbitrary field, Moufang 3-gons can be divided into three cases as above. The split case in the first diagram exists over any field. The second case extends to all associative, non-commutative division algebras; over the reals these are limited to the algebra of quaternions, which has degree 2 (and dimension 4), but some fields admit central division algebras of other degrees. The third case involves ‘alternative’ division algebras (which satisfy a weakened form of the associative law), and a theorem of Richard Bruck and Erwin Kleinfeld shows that these are Cayley-Dickson algebras.

Adams' Hopf invariant one theorem, named after Frank Adams, states that S0, S1, S3, S7 are the only spheres that are H-spaces. Each of these spaces forms an H-space by viewing it as the subset of norm-one elements of the reals, complexes, quaternions, and octonions, respectively, and using the multiplication operations from these algebras. In fact, S0, S1, and S3 are groups (Lie groups) with these multiplications. But S7 is not a group in this way because octonion multiplication is not associative, nor can it be given any other continuous multiplication for which it is a group.

In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, although the "lines" of elliptic geometry are geodesic curves and, unlike the lines of Euclidean geometry, are of finite length. The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.

That is, it must integrate the vehicle's attitude changes in pitch, roll and yaw, as well as gross movements. Gimballed systems could usually do well with update rates of 50–60 Hz. However, strapdown systems normally update about 2000 Hz. The higher rate is needed to let the navigation system integrate the angular rate into an attitude accurately. The data updating algorithms (direction cosines or quaternions) involved are too complex to be accurately performed except by digital electronics. However, digital computers are now so inexpensive and fast that rate gyro systems can now be practically used and mass-produced.

Grassmann had published his results in 1844, but Saint-Venant claimed that he had first developed these ideas in 1832. One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme :... developed some of Hermann Grassmann's algebras and Hamilton's quaternions. Hankel was the first to recognise the significance of Grassmann's long-neglected writings ...Hankel entry in the Dictionary of Scientific Biography. New York: 1970–1990 In 1872 Victor Schlegel published the first part of his System der Raumlehre which used Grassmann's approach to derive ancient and modern results in plane geometry.

Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form , where B is an elementary abelian 2-group, and D is a periodic abelian group with all elements of odd order. Dedekind groups are named after Richard Dedekind, who investigated them in , proving a form of the above structure theorem (for finite groups). He named the non-abelian ones after William Rowan Hamilton, the discoverer of quaternions. In 1898 George Miller delineated the structure of a Hamiltonian group in terms of its order and that of its subgroups.

Broome Bridge Plaque on Broome Bridge Broom Bridge (Irish: Droichead Broome),Logainm.ie also called Broome Bridge, and sometimes Brougham Bridge, is a bridge along Broombridge Road which crosses the Royal Canal in Cabra, Dublin, Ireland. Broome Bridge is named after William Broome, one of the directors of the Royal Canal company who lived nearby. It is famous for being the location where Sir William Rowan Hamilton first wrote down the fundamental formula for quaternions on 16 October 1843, which is to this day commemorated by a stone plaque on the northwest corner of the underside of the bridge.

In mathematics, the classical groups are defined as the special linear groups over the reals , the complex numbers and the quaternions together with specialHere, special means the subgroup of the full automorphism group whose elements have determinant 1. automorphism groups of symmetric or skew- symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. p. 94. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups.

In abstract algebra, in particular in the theory of nondegenerate quadratic forms on vector spaces, the structures of finite-dimensional real and complex Clifford algebras for a nondegenerate quadratic form have been completely classified. In each case, the Clifford algebra is algebra isomorphic to a full matrix ring over R, C, or H (the quaternions), or to a direct sum of two copies of such an algebra, though not in a canonical way. Below it is shown that distinct Clifford algebras may be algebra-isomorphic, as is the case of Cl2,0(R) and Cl1,1(R), which are both isomorphic to the ring of two-by-two matrices over the real numbers.

In 1883 he published an article "Some general theorems in quaternion integration".A. McAulay (1883) Messenger of Mathematics 13:26 to 37 McAulay took his degree in 1886, and began to reflect on the instruction of students in quaternion theory. In an article "Establishment of the fundamental properties of quaternions"McAulay (1888) Messenger of Mathematics 18:131 to 136 he suggested improvements to the texts then in use. He also wrote a technical articleA. McAulay (1888) "The transformation of multiple surface integrals into multiple line integrals", Messenger of Mathematics 18:139 to 45 on integration. Departing for Australia, he lectured at Ormond College, University of Melbourne from 1893 to 1895.

Simplified rigid body physics is relatively cheap and easy to implement, which is why it appeared in interactive games and simulations earlier than most other techniques. Rigid bodies are assumed to undergo no deformation during simulation so that rigid body motion between time steps can be described as a translation and rotation, traditionally using affine transformations stored as 4x4 matrices. Alternatively, quaternions can be used to store rotations and vectors can be used to store the objects offset from the origin. The most computationally expensive aspects of rigid body dynamics are collision detection, correcting interpenetration between bodies and the environment, and handling resting contact.

Around the same time Euler showed that there is also a similar four-square identity. Later it turned out to be related to the norm of quaternions discovered by William Rowan Hamilton. In 1818 Degen presented to the Academy of Sciences in St. Petersburg where Euler had worked, his eight-square identity of exactly the same structure as the two previous identities.A. Rice and E. Brown, Commutativity and collinearity: A historical case study of the interconnection of mathematical ideas. Part I , Journal of the British Society for the History of Mathematics 31 (1), 1–14 (2016). The following year he was elected as a «corresponding member» to the same academic society.

Papers three and four are "Fundamental Theorems of Analysis Generalized for Space" and "On the definition of the Trigonometric Functions", which he had presented the previous year in Chicago at the Congress of Mathematicians held in connection with the World's Columbian Exhibition. He follows George Salmon in exhibiting the hyperbolic angle, argument of hyperbolic functions. The fifth paper is "Elliptic and Hyperbolic Analysis" which considers the spherical law of cosines as the fundamental theorem of the sphere, and proceeds to analogues for the ellipsoid of revolution, general ellipsoid, and equilateral hyperboloids of one and two sheets, where he provides the hyperbolic law of cosines. In 1900 Alexander published "Hyperbolic Quaternions"A.

The bivector was first defined in 1844 by German mathematician Hermann Grassmann in exterior algebra as the result of the exterior product of two vectors. Just the previous year, in Ireland, William Rowan Hamilton had discovered quaternions. It was not until English mathematician William Kingdon Clifford in 1888 added the geometric product to Grassmann's algebra, incorporating the ideas of both Hamilton and Grassmann, and founded Clifford algebra, that the bivector as it is known today was fully understood. Around this time Josiah Willard Gibbs and Oliver Heaviside developed vector calculus, which included separate cross product and dot products that were derived from quaternion multiplication.

The possibility of gravitational waves was also discussed by Heaviside using the analogy between the inverse-square law in gravitation and electricity.A gravitational and electromagnetic analogy,Electromagnetic Theory, 1893, 455-466 Appendix B. This was 25 years before Einstein's paper on this subject With quaternion multiplication, the square of a vector is a negative quantity, much to Heaviside’s displeasure. As he advocated abolishing this negativity, he has been credited by C. J. Joly with developing hyperbolic quaternions, though in fact that mathematical structure was largely the work of Alexander Macfarlane. He invented the Heaviside step function, using it to calculate the current when an electric circuit is switched on.

William Thomson, Lord Kelvin (1904) Molecular dynamics and the wave theory of light, twenty lectures transcribed by A.S. Hathaway, Cambridge University Press, link from Internet Archive In 1987 Hathaway's original transcription from 1884 was published when Johns Hopkins Center for the History and Philosophy of Science decided to commemorate the centennial of Kelvin's lectures.Robert Kargon and Peter Achinstein (1987) Kelvin’s Baltimore Lectures and Modern Theoretical Physics: historical and philosophical perspectives, MIT Press In Terre Haute, Indiana Hathaway taught at Rose Polytechnic Institute until 1920 and published A Primer on Quaternions in 1896. He became the U.S. national secretary for the international Quaternion Society in 1899.

To establish priority of Grassmann's ideas, Gibbs convinced Grassmann's heirs to seek the publication in Germany of the essay "Theorie der Ebbe und Flut" on tides that Grassmann had submitted in 1840 to the faculty at the University of Berlin, in which he had first introduced the notion of what would later be called a vector space (linear space).Wheeler 1998, pp. 113–116 As Gibbs had advocated in the 1880s and 1890s, quaternions were eventually all but abandoned by physicists in favor of the vectorial approach developed by him and, independently, by Oliver Heaviside. Gibbs applied his vector methods to the determination of planetary and comet orbits.

Returning to our picture of the unit two-sphere sitting on the Euclidean plane: Consider a geodesic in the plane, based at the origin, and map this to a geodesic in the two-sphere of the same length, based at the south pole. Under this map all points of the circle of radius are sent to the north pole. Since the open unit disk is homeomorphic to the Euclidean plane, this is again a one-point compactification. The exponential map for 3-sphere is similarly constructed; it may also be discussed using the fact that the 3-sphere is the Lie group of unit quaternions.

One of the features of Hamilton's quaternion system was the differential operator del which could be used to express the gradient of a vector field or to express the curl. These operations were applied by Clerk Maxwell to the electrical and magnetic studies of Michael Faraday in Maxwell's Treatise on Electricity and Magnetism (1873). Though the del operator continues to be used, the real quaternions fall short as a representation of spacetime. On the other hand, the biquaternion algebra, in the hands of Arthur W. Conway and Ludwik Silberstein, provided representational tools for Minkowski space and the Lorentz group early in the twentieth century.

A physicist who took the issues involved seriously was Pierre Duhem, writing at the beginning of the twentieth century. He wrote an extended analysis of the approach he saw as characteristically British, in requiring field theories of theoretical physics to have a mechanical-physical interpretation. That was an accurate characterisation of what Dirac (himself British) would later argue against. The national characteristics specified by Duhem do not need to be taken too seriously, since he also claimed that the use of abstract algebra, namely quaternions, was also characteristically British (as opposed to French or German); as if the use of classical analysis methods alone was important one way or the other.

In fact, most of the important Lie groups (but not all) can be expressed as matrix groups. If this idea is generalised to matrices with complex numbers as entries, then we get further useful Lie groups, such as the unitary group U(n). We can also consider matrices with quaternions as entries; in this case, there is no well-defined notion of a determinant (and thus no good way to define a quaternionic "volume"), but we can still define a group analogous to the orthogonal group, the symplectic group Sp(n). Furthermore, the idea can be treated purely algebraically with matrices over any field, but then the groups are not Lie groups.

Distribution of misorientation angles for a randomly texture polycrystal, from Mackenzie(1958) Discrete misorientations or the misorientation distribution can be fully described as plots in the Euler angle, axis/angle, or Rodrigues vector space. Unit quaternions, while computationally convenient, do not lend themselves to graphical representation because of their four-dimensional nature. For any of the representations, plots are usually constructed as sections through the fundamental zone; along φ2 in Euler angles, at increments of rotation angle for axis/angle, and at constant ρ3 (parallel to <001>) for Rodrigues. Due to the irregular shape of the cubic-cubic FZ, the plots are typically given as sections through the cubic FZ with the more restrictive boundaries overlaid.

It enables all analytical work to be with reals, although the geometry becomes non-Euclidean. The article reviewed was "The space-time manifold of relativity, the non-Euclidean geometry of mechanics, and electromagnetics".E. B. Wilson & G. N. Lewis (1912) Proceedings of the American Academy of Arts and Sciences 48: 389–507 However, when the textbook The Theory of Relativity by Ludwik Silberstein in 1914 was made available as an English understanding of Minkowski space, the algebra of biquaternions was applied, but without references to the British background or Macfarlane or other quaternionists of the Society. The language of quaternions had become international, providing content to set theory and expanded mathematical notation, and expressing mathematical physics.

When interpreted as quaternions, the 120 vertices of the 600-cell form a group under quaternionic multiplication. This group is often called the binary icosahedral group and denoted by 2I as it is the double cover of the ordinary icosahedral group I. It occurs twice in the rotational symmetry group RSG of the 600-cell as an invariant subgroup, namely as the subgroup 2IL of quaternion left-multiplications and as the subgroup 2IR of quaternion right- multiplications. Each rotational symmetry of the 600-cell is generated by specific elements of 2IL and 2IR; the pair of opposite elements generate the same element of RSG. The centre of RSG consists of the non-rotation Id and the central inversion −Id.

The non-triviality of the (additional) conjugacy conclusion can be illustrated with the Klein four-group V as the non-example. Any of the three proper subgroups of V (all of which have order 2) is normal in V; fixing one of these subgroups, any of the other two remaining (proper) subgroups complements it in V, but none of these three subgroups of V is a conjugate of any other one, because V is Abelian. The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as C_4 and the quaternions.

Macfarlane was also the author of a popular 1916 collection of mathematical biographies (Ten British Mathematicians), a similar work on physicists (Lectures on Ten British Physicists of the Nineteenth Century, 1919). Macfarlane was caught up in the revolution in geometry during his lifetime,1830–1930: A Century of Geometry, L Boi, D. Flament, JM Salanskis editors, Lecture Notes in Physics No. 402, Springer-Verlag in particular through the influence of G. B. Halsted who was mathematics professor at the University of Texas. Macfarlane originated an Algebra of Physics, which was his adaptation of quaternions to physical science. His first publication on Space Analysis preceded the presentation of Minkowski Space by seventeen years.

A tesseract is an example of a four-dimensional object. Whereas outside mathematics the use of the term "dimension" is as in: "A tesseract has four dimensions", mathematicians usually express this as: "The tesseract has dimension 4", or: "The dimension of the tesseract is 4" or: 4D. Although the notion of higher dimensions goes back to René Descartes, substantial development of a higher-dimensional geometry only began in the 19th century, via the work of Arthur Cayley, William Rowan Hamilton, Ludwig Schläfli and Bernhard Riemann. Riemann's 1854 Habilitationsschrift, Schläfli's 1852 Theorie der vielfachen Kontinuität, and Hamilton's discovery of the quaternions and John T. Graves' discovery of the octonions in 1843 marked the beginning of higher-dimensional geometry.

It is also closely related (double covered) with the set of quaternions with their internal product, as well as to the set of rotation vectors (though here the relation is harder to describe, see below for details), with a different internal composition operation given by the product of their equivalent matrices. Rotation vectors notation arise from the Euler's rotation theorem which states that any rotation in three dimensions can be described by a rotation by some angle about some axis. Considering this, we can then specify the axis of one of these rotations by two angles, and we can use the radius of the vector to specify the angle of rotation. These vectors represent a ball in 3D with an unusual topology.

A device designed and patented in 1971 by Dale A. Adams and reported in The Amateur Scientist in December 1975, solves this problem with a rotating disk above a base from which a cable extends up, over, and onto the top of the disk. As the disk rotates the plane of this cable is rotated at exactly half the rate of the disk so the cable experiences no net twisting. What makes the device possible is the peculiar connectivity of the space of 3D rotations, as discovered by P. A. M. Dirac and illustrated in his Plate trick (also known as the string trick or belt trick). Its covering Spin(3) group can be represented by unit quaternions, also known as versors.

Numbers from various number systems, like integers, rationals, complex numbers, quaternions, octonions, ... may have multiple attributes, that fix certain properties of a number. If a number system bears the structure of an ordered ring, for example, the integers, it must contain a number that does not change any number when it is added to it (an additive identity element). This number is generally denoted as Because of the total order in this ring, there are numbers greater than zero, called the positive numbers. For other properties required within a ring, for each such positive number there exists a number less than which, when added to the positive number, yields the result These numbers less than are called the negative numbers.

A discussion of quaternions from these years is The success of vector calculus, and of the book Vector Analysis by Gibbs and Wilson, had the effect that the insights of Hamilton and Clifford were overlooked for a long time, since much of 20th century mathematics and physics was formulated in vector terms. Gibbs used vectors to fill the role of bivectors in three dimensions, and used "bivector" to describe an unrelated quantity, a use that has sometimes been copied. Today the bivector is largely studied as a topic in geometric algebra, a Clifford algebra over real or complex vector spaces with a nondegenerate quadratic form. Its resurgence was led by David Hestenes who, along with others, applied geometric algebra to a range of new applications in physics.

Due to Kunihiko Kodaira's classification of complex surfaces, we know that any compact hyperkähler 4-manifold is either a K3 surface or a compact torus T^4. (Every Calabi–Yau manifold in 4 (real) dimensions is a hyperkähler manifold, because SU(2) is isomorphic to Sp(1).) As was discovered by Beauville, the Hilbert scheme of k points on a compact hyperkähler 4-manifold is a hyperkähler manifold of dimension 4k. This gives rise to two series of compact examples: Hilbert schemes of points on a K3 surface and generalized Kummer varieties. Non-compact, complete, hyperkähler 4-manifolds which are asymptotic to H/G, where H denotes the quaternions and G is a finite subgroup of Sp(1), are known as asymptotically locally Euclidean, or ALE, spaces.

In the final chapter concerned with textbooks, they used Klein's and Molk's encyclopedia projects§ 10.10: Complex analysis in the German and French Encyclopädie, pages 691 to 759 in Hidden Harmony – Geometric Fantasies, Springer to contrast the approaches in Germany (Weierstrass and Riemann) and France (Cauchy). In 1900 an element of an algebra over a field (usually ℝ or ℂ) was known as a hypercomplex number, exemplified by quaternions ℍ which contributed the dot product and cross product useful in analytic geometry, and the del operator in analysis. Explorative articles on hypercomplex numbers, mentioned by Bottazzini and Gray, written by Eduard Study (1898) and Elie Cartan (1908), served as advertisements to twentieth century algebrists, and they soon retired the term hypercomplex by displaying the structure of algebras.

Hamilton solved this problem using the icosian calculus, an algebraic structure based on roots of unity with many similarities to the quaternions (also invented by Hamilton). This solution does not generalize to arbitrary graphs. Despite being named after Hamilton, Hamiltonian cycles in polyhedra had also been studied a year earlier by Thomas Kirkman, who, in particular, gave an example of a polyhedron without Hamiltonian cycles.. Even earlier, Hamiltonian cycles and paths in the knight's graph of the chessboard, the knight's tour, had been studied in the 9th century in Indian mathematics by Rudrata, and around the same time in Islamic mathematics by al-Adli ar-Rumi. In 18th century Europe, knight's tours were published by Abraham de Moivre and Leonhard Euler..

A simple ring can always be considered as a simple algebra over its center. Rings which are simple as rings but not as modules do exist: the full matrix ring over a field does not have any nontrivial ideals (since any ideal of Mn(R) is of the form Mn(I) with I an ideal of R), but has nontrivial left ideals (namely, the sets of matrices which have some fixed zero columns). According to the Artin–Wedderburn theorem, every simple ring that is left or right Artinian is a matrix ring over a division ring. In particular, the only simple rings that are a finite-dimensional vector space over the real numbers are rings of matrices over either the real numbers, the complex numbers, or the quaternions.

The projective plane over such a division algebra then gives rise to a Moufang 3-gon. These projective planes correspond to the building attached to SL3(R), SL3(C), a real form of A5 and to a real form of E6, respectively. In the first diagram the circled nodes represent 1-spaces and 2-spaces in a three-dimensional vector space. In the second diagram the circled nodes represent 1-space and 2-spaces in a 3-dimensional vector space over the quaternions, which in turn represent certain 2-spaces and 4-spaces in a 6-dimensional complex vector space, as expressed by the circled nodes in the A5 diagram. The fourth case — a form of E6 — is exceptional, and its analogue for Moufang 4-gons is a major feature of Weiss’s book.

One of the first examples of an exotic sphere found by was the following: Take two copies of B4×S3, each with boundary S3×S3, and glue them together by identifying (a,b) in the boundary with (a, a2ba−1), (where we identify each S3 with the group of unit quaternions). The resulting manifold has a natural smooth structure and is homeomorphic to S7, but is not diffeomorphic to S7. Milnor showed that it is not the boundary of any smooth 8-manifold with vanishing 4th Betti number, and has no orientation-reversing diffeomorphism to itself; either of these properties implies that it is not a standard 7-sphere. Milnor showed that this manifold has a Morse function with just two critical points, both non-degenerate, which implies that it is topologically a sphere.

Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions). ;Integers :N\times M is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by :N\times (-M) = (-N)\times M = - (N\times M) and :(-N)\times (-M) = N\times M :The same sign rules apply to rational and real numbers.

That same year he patented, in England, the coaxial cable. In 1884 he recast Maxwell's mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell's equations. The four re-formulated Maxwell's equations describe the nature of electric charges (both static and moving), magnetic fields, and the relationship between the two, namely electromagnetic fields. Between 1880 and 1887, Heaviside developed the operational calculus using p for the differential operator, (which Boole had previously denoted by D"A Treatise on Differential Equations", 1859), giving a method of solving differential equations by direct solution as algebraic equations.

In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres, planes, circles and other geometrical objects, and equations connecting them, all transform covariantly. A geometric object (a -sphere) can be synthesized as the wedge product of linearly independent vectors representing points on the object; conversely, the object can be decomposed as the repeated wedge product of vectors representing distinct points in its surface. Some intersection operations also acquire a tidy algebraic form: for example, for the Euclidean base space , applying the wedge product to the dual of the tetravectors representing two spheres produces the dual of the trivector representation of their circle of intersection.

Note the negative sign is introduced to simplify the correspondence with quaternions. Denote a set of mutually orthogonal unit vectors of R4 as e1, e2, e3 and e4, then the Clifford product yields the relations :e_m e_n = -e_n e_m, \,\,\, m e n, and :e_1 ^2 = e_2^2 =e_3^2 = -1, \,\, e_4^2 = 0. The general element of the Clifford algebra has 16 components. The linear combination of the even degree elements defines the even subalgebra with the general element : H = h_0 + h_1 e_2 e_3 + h_2 e_3 e_1 + h_3 e_1 e_2 + h_4 e_4 e_1 + h_5 e_4 e_2 + h_6 e_4 e_3 + h_7 e_1 e_2 e_3 e_4. The basis elements can be identified with the quaternion basis elements i, j, k and the dual unit ε as : i = e_2 e_3, j = e_3 e_1, k = e_1 e_2, \,\, \varepsilon = e_1 e_2 e_3 e_4.

The notation was first coined by William Rowan Hamilton in Elements of Quaternions (1866) and subsequently used by Irving Stringham in works such as Uniplanar Algebra (1893), or by James Harkness and Frank Morley in their Introduction to the Theory of Analytic Functions (1898). It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. It is mostly used as a convenient shorthand notation to simplify some expressions, for example in conjunction with Fourier and Hartley transforms, or when exponential functions shouldn't be used for some reason in math education. In information technology, the function sees dedicated support in various high-performance math libraries (such as Intel's Math Kernel Library (MKL)), available for many compilers, programming languages (including C, C++, Common Lisp, D, Fortran, Haskell, Julia), and operating systems (including Windows, Linux, macOS and HP-UX).

The absolute value of a number may be thought of as its distance from zero. In mathematics, the absolute value or modulus of a real number , denoted , is the non-negative value of without regard to its sign. Namely, if is positive, and if is negative (in which case is positive), and . For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. Generalisations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example, an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.

A Julia set Three-dimensional slices through the (four-dimensional) Julia set of a function on the quaternions In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f).

Atiyah and Ward used the Penrose correspondence to reduce the classification of all instantons on the 4-sphere to a problem in algebraic geometry. With Hitchin he used ideas of Horrocks to solve this problem, giving the ADHM construction of all instantons on a sphere; Manin and Drinfeld found the same construction at the same time, leading to a joint paper by all four authors. Atiyah reformulated this construction using quaternions and wrote up a leisurely account of this classification of instantons on Euclidean space as a book. Atiyah's work on instanton moduli spaces was used in Donaldson's work on Donaldson theory. Donaldson showed that the moduli space of (degree 1) instantons over a compact simply connected 4-manifold with positive definite intersection form can be compactified to give a cobordism between the manifold and a sum of copies of complex projective space.

Elements of SO(8) can be described with unit octonions, analogously to how elements of SO(2) can be described with unit complex numbers and elements of SO(4) can be described with unit quaternions. However the relationship is more complicated, partly due to the non-associativity of the octonions. A general element in SO(8) can be described as the product of 7 left-multiplications, 7 right-multiplications and also 7 bimultiplications by unit octonions (a bimultiplication being the composition of a left- multiplication and a right-multiplication by the same octonion and is unambiguously defined due to octonions obeying the Moufang identities). It can be shown that an element of SO(8) can be constructed with bimultiplications, by first showing that pairs of reflections through the origin in 8-dimensional space correspond to pairs of bimultiplications by unit octonions.

The emergence of differential geometry as a distinct discipline is generally credited to Carl Friedrich Gauss and Bernhard Riemann. Riemann first described manifolds in his famous habilitation lecture before the faculty at Göttingen.B. Riemann (1867). He motivated the idea of a manifold by an intuitive process of varying a given object in a new direction, and presciently described the role of coordinate systems and charts in subsequent formal developments: : Having constructed the notion of a manifoldness of n dimensions, and found that its true character consists in the property that the determination of position in it may be reduced to n determinations of magnitude, ... – B. Riemann The works of physicists such as James Clerk Maxwell,Maxwell himself worked with quaternions rather than tensors, but his equations for electromagnetism were used as an early example of the tensor formalism; see .

According to Hamilton, on 16 October he was out walking along the Royal Canal in Dublin with his wife when the solution in the form of the equation : suddenly occurred to him; Hamilton then promptly carved this equation using his penknife into the side of the nearby Broom Bridge (which Hamilton called Brougham Bridge). This event marks the discovery of the quaternion group. A plaque under the bridge was unveiled by the Taoiseach Éamon de Valera, himself a mathematician and student of quaternions,De Valera School of Mathematics and Statistics University of St Andrews, Scotland on 13 November 1958. Since 1989, the National University of Ireland, Maynooth has organised a pilgrimage called the Hamilton Walk, in which mathematicians take a walk from Dunsink Observatory to the bridge, where no trace of the carving remains, though a stone plaque does commemorate the discovery.

By contrast with case in most of the rest of the Old Testament, the Amiatinus psalms text is commonly considered an inferior witness of Jerome's Versio juxta Hebraicum; the presence of the 'Columba' series of psalm headings, also found in the Cathach of St. Columba, demonstrates that an Irish psalter must have been its source; but the text differs in many places from the best Irish manuscripts. The New Testament is preceded by the Epistula Hieronymi ad Damasum, Prolegomena to the four Gospels. The Codex Amiatinus qualifies as an illuminated manuscript as it has some decoration including two full-page miniatures, but these show little sign of the usual insular style of Northumbrian art and are clearly copied from Late Antique originals. It contains 1,040 leaves of strong, smooth vellum, fresh-looking today despite their great antiquity, arranged in quires of four sheets, or quaternions.

Grassmann then showed that once geometry is put into the algebraic form he advocated, the number three has no privileged role as the number of spatial dimensions; the number of possible dimensions is in fact unbounded. Fearnley-Sander (1979) describes Grassmann's foundation of linear algebra as follows:Hermann Grassmann and the Creation of Linear Algebra Following an idea of Grassmann's father, A1 also defined the exterior product, also called "combinatorial product" (in German: äußeres ProduktTr. outer product or kombinatorisches ProduktTr. combinatorial product), the key operation of an algebra now called exterior algebra. (One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.) In 1878, William Kingdon Clifford joined this exterior algebra to William Rowan Hamilton's quaternions by replacing Grassmann's rule epep = 0 by the rule epep = 1.

The Sylvester–Gallai theorem also does not directly apply to geometries in which points have coordinates that are pairs of complex numbers or quaternions, but these geometries have more complicated analogues of the theorem. For instance, in the complex projective plane there exists a configuration of nine points, Hesse's configuration (the inflection points of a cubic curve), in which every line is non-ordinary, violating the Sylvester–Gallai theorem. Such a configuration is known as a Sylvester–Gallai configuration, and it cannot be realized by points and lines of the Euclidean plane. Another way of stating the Sylvester–Gallai theorem is that whenever the points of a Sylvester–Gallai configuration are embedded into a Euclidean space, preserving colinearities, the points must all lie on a single line, and the example of the Hesse configuration shows that this is false for the complex projective plane.

Timely, meticulous, rigorous, and often the final word on a given topic, they have been of immense value to the development and definition of these two projects. In addition to the mathematical principles they frequently include working algorithms (often with source code when relevant). Amongst them are, for Hipparcos, the three-step astrometric reduction, optimization of the scanning law, notes on the imaging properties used for the multiple star analysis, assessment of chromatic effects, attitude developments, and many others. For Gaia, his technical notes cover the mathematical and statistical aspects of the Gaia instrument and processing (including the attitude determination and its mathematical representation with quaternions and splines), the modelling of the point/line spread functions, the CCD geometric calibrations, broad band photometry design, maximum likelihood determination of the CCD image centroiding, differential equations and optimal properties of the scanning law, along with the subtle systematic effects in astrometry caused by instrumental misalignments.

In Alfred North Whitehead's book A Treatise on Universal Algebra, published in 1898, the term universal algebra had essentially the same meaning that it has today. Whitehead credits William Rowan Hamilton and Augustus De Morgan as originators of the subject matter, and James Joseph Sylvester with coining the term itself. At the time structures such as Lie algebras and hyperbolic quaternions drew attention to the need to expand algebraic structures beyond the associatively multiplicative class. In a review Alexander Macfarlane wrote: "The main idea of the work is not unification of the several methods, nor generalization of ordinary algebra so as to include them, but rather the comparative study of their several structures."Alexander Macfarlane (1899) Review:A Treatise on Universal Algebra (pdf), Science 9: 324–8 via Internet Archive At the time George Boole's algebra of logic made a strong counterpoint to ordinary number algebra, so the term "universal" served to calm strained sensibilities.

The walk was launched in 1990 by Prof Tony O'Farrell of the Department of Mathematics at St Patrick's College, Maynooth.Irish mathematicians receive Maths Week Ireland Award The Irish Times, October 18, 2018 It starts at Dunsink observatory, where Hamilton lived and was the Director from 1827 to 1865, and ends at the spot where he recorded his discovery by carving the following equation on Broom Bridge: :i^2=j^2=k^2=ijk=-1\, These are the basic relations which define the quaternions. The original inscription by Hamilton is no longer there, but a plaque unveiled by the Taoiseach Éamon de Valera in 1958 marks the spot where he recorded his discovery. Many prominent mathematicians have attended the event; they include Wolf Prize winner Roger Penrose (2013), Abel Prize and Copley Medal winner Andrew Wiles (2003), Fields Medallists Timothy Gowers (2004) and Efim Zelmanov (2009), and Nobel Prize winners Murray Gell-Mann (2002), Steven Weinberg (2005) and Frank Wilczek (2007).

It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 November 1833, On Conjugate Functions or Algebraic Couples, and subsequently published in the seventeenth volume of the Transactions of the Royal Irish Academy. To this memoir were prefixed A Preliminary and Elementary Essay on Algebra as the Science of Pure Time, and some General Introductory Remarks. In the concluding paragraphs of each of these three papers Hamilton acknowledges that it was "in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms", that he was conducted to "the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers" were foundational for his own work, culminating in the discovery of quaternions. For many years Graves and Hamilton maintained a correspondence on the interpretation of imaginaries.

A finite- dimensional unital associative algebra (over any field) is a division algebra if and only if it has no zero divisors. Whenever A is an associative unital algebra over the field F and S is a simple module over A, then the endomorphism ring of S is a division algebra over F; every associative division algebra over F arises in this fashion. The center of an associative division algebra D over the field K is a field containing K. The dimension of such an algebra over its center, if finite, is a perfect square: it is equal to the square of the dimension of a maximal subfield of D over the center. Given a field F, the Brauer equivalence classes of simple (contains only trivial two-sided ideals) associative division algebras whose center is F and which are finite-dimensional over F can be turned into a group, the Brauer group of the field F. One way to construct finite-dimensional associative division algebras over arbitrary fields is given by the quaternion algebras (see also quaternions).

Vector spaces stem from affine geometry, via the introduction of coordinates in the plane or three- dimensional space. Around 1636, French mathematicians René Descartes and Pierre de Fermat founded analytic geometry by identifying solutions to an equation of two variables with points on a plane curve.. To achieve geometric solutions without using coordinates, Bolzano introduced, in 1804, certain operations on points, lines and planes, which are predecessors of vectors.. This work was made use of in the conception of barycentric coordinates by Möbius in 1827.. The foundation of the definition of vectors was Bellavitis' notion of the bipoint, an oriented segment one of whose ends is the origin and the other one a target. Vectors were reconsidered with the presentation of complex numbers by Argand and Hamilton and the inception of quaternions by the latter.. They are elements in R2 and R4; treating them using linear combinations goes back to Laguerre in 1867, who also defined systems of linear equations. In 1857, Cayley introduced the matrix notation which allows for a harmonization and simplification of linear maps.

In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K. As an example, note that any simple algebra is a central simple algebra over its center. For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below). Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras.

This formulation uses the negative sign so the correspondence with quaternions is easily shown. Denote a set of orthogonal unit vectors of R3 as e1, e2, and e3, then the Clifford product yields the relations : e_2 e_3 = -e_3 e_2, \,\,\, e_3 e_1 = -e_1 e_3,\,\,\, e_1 e_2 = -e_2 e_1, and : e_1 ^2 = e_2^2 = e_3^2 = -1. The general element of the Clifford algebra Cl(R) is given by : A = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_2 e_3 + a_5 e_3 e_1 + a_6 e_1 e_2 + a_7 e_1 e_2 e_3. The linear combination of the even degree elements of Cl(R) defines the even subalgebra Cl(R) with the general element : q = q_0 + q_1 e_2 e_3 + q_2 e_3 e_1 + q_3 e_1 e_2. The basis elements can be identified with the quaternion basis elements i, j, k as : i= e_2 e_3, j = e_3 e_1, k = e_1 e_2, which shows that the even subalgebra Cl(R) is Hamilton's real quaternion algebra. To see this, compute : i^2 = (e_2 e_3)^2 = e_2 e_3 e_2 e_3 = - e_2 e_2 e_3 e_3 = -1, and : ij = e_2 e_3 e_3 e_1 = -e_2 e_1 = e_1 e_2 = k.

In mathematics, the Hurwitz problem, named after Adolf Hurwitz, is the problem of finding multiplicative relations between quadratic forms which generalise those known to exist between sums of squares in certain numbers of variables. There are well-known multiplicative relationships between sums of squares in two variables : (x^2+y^2)(u^2+v^2) = (xu-yv)^2 + (xv+yu)^2 \ , (known as the Brahmagupta–Fibonacci identity), and also Euler's four-square identity and Degen's eight-square identity. These may be interpreted as multiplicativity for the norms on the complex numbers, quaternions and octonions respectively.Charles W. Curtis (1963) "The Four and Eight Square Problem and Division Algebras" in Studies in Modern Algebra edited by A.A. Albert, pages 100–125, Mathematical Association of America, Solution of Hurwitz’s Problem on page 115 The Hurwitz problem for the field K is to find general relations of the form : (x_1^2+\cdots+x_r^2) \cdot (y_1^2+\cdots+y_s^2) = (z_1^2 + \cdots + z_n^2) \ , with the z being bilinear forms in the x and y: that is, each z is a K-linear combination of terms of the form xiyj.

The problem of gimbal lock appears when one uses Euler angles in applied mathematics; developers of 3D computer programs, such as 3D modeling, embedded navigation systems, and video games must take care to avoid it. In formal language, gimbal lock occurs because the map from Euler angles to rotations (topologically, from the 3-torus T3 to the real projective space RP3 which is the same as the space of 3d rotations SO3) is not a local homeomorphism at every point, and thus at some points the rank (degrees of freedom) must drop below 3, at which point gimbal lock occurs. Euler angles provide a means for giving a numerical description of any rotation in three- dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space (rotations) can be realized by a change in the source space (Euler angles). This is a topological constraint – there is no covering map from the 3-torus to the 3-dimensional real projective space; the only (non-trivial) covering map is from the 3-sphere, as in the use of quaternions.