In geometry, the rhombille tiling,. also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.
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One of the rhombic triacontahedron's rhombi All of the faces of the rhombic triacontahedron are golden rhombi A golden rhombus is a rhombus whose diagonals are in the golden ratio. The rhombic triacontahedron is a convex polytope that has a very special property: all of its faces are golden rhombi. In the rhombic triacontahedron the dihedral angle between any two adjacent rhombi is 144°, which is twice the isosceles angle of a golden triangle and four times its most acute angle..
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The advantage of pattern block art is that it can be changed around, added, or turned into something else. All six of the shapes (green triangles, blue (thick) rhombi, red trapezoids, yellow hexagons, orange squares, and tan (thin) rhombi) are applied to make mosaics.
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A well-studied group of tangential polygons are the tangential quadrilaterals, which include the rhombi and kites.
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The Moser spindle embedded as a unit distance graph in the plane, together with a seven-coloring of the plane. As a unit distance graph, the Moser spindle is formed by two rhombi with 60 and 120 degree angles, so that the sides and short diagonals of the rhombi form equilateral triangles. The two rhombi are placed in the plane, sharing one of their acute-angled vertices, in such a way that the remaining two acute-angled vertices are a unit distance apart from each other. The eleven edges of the graph are the eight rhombus sides, the two short diagonals of the rhombi, and the edge between the unit-distance pair of acute-angled vertices.
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An n-gonal form has 3n vertices, 6n edges, and 2+3n faces: 2 regular n-gons, n rhombi, and 2n triangles.
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The rhombille tiling is the dual of the trihexagonal tiling. It is one of many different ways of tiling the plane by congruent rhombi. Others include a diagonally flattened variation of the square tiling (with translational symmetry on all four sides of the rhombi), the tiling used by the Miura-ori folding pattern (alternating between translational and reflectional symmetry), and the Penrose tiling which uses two kinds of rhombi with 36° and 72° acute angles aperiodically. When more than one type of rhombus is allowed, additional tilings are possible, including some that are topologically equivalent to the rhombille tiling but with lower symmetry.
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The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces.
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One of the ways of making a dodecagon with pattern blocks Pattern blocks consist of various wooden shapes (green triangles, red trapezoids, yellow hexagons, orange squares, tan (long) rhombi, and blue (wide) rhombi) that are sized in such a way that students will be able to see relationships among shapes. For example, three green triangles make a red trapezoid; two red trapezoids make up a yellow hexagon; a blue rhombus is made up of two green triangles; three blue rhombi make a yellow hexagon, etc. Playing with the shapes in these ways help children develop a spatial understanding of how shapes are composed and decomposed, an essential understanding in early geometry. Pattern blocks are also used by teachers as a means for students to identify, extend, and create patterns.
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All tetrahedra can be inscribed in a parallelepiped. A tetrahedron is orthocentric if and only if its circumscribed parallelepiped is a rhombohedron. Indeed, in any tetrahedron, a pair of opposite edges is perpendicular if and only if the corresponding faces of the circumscribed parallelepiped are rhombi. If four faces of a parallelepiped are rhombi, then all edges have equal lengths and all six faces are rhombi; it follows that if two pairs of opposite edges in a tetrahedron are perpendicular, then so is the third pair, and the tetrahedron is orthocentric. A tetrahedron ABCD is orthocentric if and only if the sum of the squares of opposite edges is the same for the three pairs of opposite edges:Reiman, István, "International Mathematical Olympiad: 1976-1990", Anthem Press, 2005, pp. 175-176.
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For instance, the regular octagon can be tiled by two squares and four 45° rhombi. In a generalization of Monsky's theorem, proved that no zonogon has an equidissection into an odd number of equal-area triangles.
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The bisected hexagonal tiling. The dual graph of a simple line arrangement may be represented geometrically as a collection of rhombi, one per vertex of the arrangement, with sides perpendicular to the lines that meet at that vertex. These rhombi may be joined together to form a tiling of a convex polygon in the case of an arrangement of finitely many lines, or of the entire plane in the case of a locally finite arrangement with infinitely many lines. investigated special cases of this construction in which the line arrangement consists of k sets of equally spaced parallel lines.
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The Hadwiger–Nelson problem asks how many colors are needed to color the points of the Euclidean plane in such a way that each pair of points at unit distance from each other are assigned different colors. That is, it asks for the chromatic number of the infinite graph whose vertices are all the points in the plane and whose edges are all pairs of points at unit distance. The Moser spindle requires four colors in any graph coloring: in any three-coloring of one of the two rhombi from which it is formed, the two acute-angled vertices of the rhombi would necessarily have the same color as each other. But if the shared vertex of the two rhombi has the same color as the two opposite acute-angled vertices, then these two vertices have the same color as each other, violating the requirement that the edge connecting them have differently-colored endpoints.
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A regular polygon is a zonogon if and only if it has an even number of sides. Thus, the square, regular hexagon, and regular octagon are all zonogons. The four-sided zonogons are the square, the rectangles, the rhombi, and the parallelograms.
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Pritchard (1994), p. 255 The gameboard has an overall hexagonal shape and comprises 72 rhombi in three alternating colors. Each player commands a full set of standard chess pieces. The game was first published in Chess Spectrum Newsletter 2 by the inventor.
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The face angles of these rhombi are approximately 70.528° and 109.471°. The thirty slim rhombic faces have face vertex angles of 41.810° and 138.189°; the diagonals are in ratio of 1 to φ2. It is also called a rhombic enenicontahedron in Lloyd Kahn's Domebook 2.
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Octagonal zonogon Tessellation by irregular hexagonal zonogons Regular octagon tiled by squares and rhombi In geometry, a zonogon is a centrally symmetric convex polygon. Equivalently, it is a convex polygon whose sides can be grouped into parallel pairs with equal lengths and opposite orientations.
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This polyhedron could be constructed by taking a tall uniform hexagonal prism, and making 3 angled cuts on the top and bottom. The trapezoids represent what remains of the original prism sides, and the 6 rhombi a result of the top and bottom cuts.
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A shape with the same exterior appearance as the dodecadodecahedron can be constructed by folding up these nets: 250px 12 pentagrams and 20 rhombic clusters are necessary. However, this construction replaces the crossing pentagonal faces of the dodecadodecahedron with non-crossing sets of rhombi, so it does not produce the same internal structure.
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A skewed grid is a tessellation of parallelograms or parallelepipeds. (If the unit lengths are all equal, it is a tessellation of rhombi or rhombohedra.) A curvilinear grid or structured grid is a grid with the same combinatorial structure as a regular grid, in which the cells are quadrilaterals or [general] cuboids, rather than rectangles or rectangular cuboids.
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Botai horses were primarily ancestors of Przewalski's horses, and contributed 2.7% ancestry to modern domestic horses. Thus, modern horses may have been domesticated in other centers of origin. The pottery of the culture had simple shapes, most examples being gray in color and unglazed. The decorations are geometric, including hatched triangles and rhombi as well as step motifs.
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The sword of Goujian is in length, including an hilt; the blade is wide at its base. The sword weighs . In addition to the repeating dark rhombi pattern on both sides of the blade, there are decorations of blue crystals and turquoise. The grip of the sword is bound by silk, while the pommel is composed of eleven concentric circles.
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In geometry, the elongated dodecahedron,Coxeter (1973) p.257 extended rhombic dodecahedron, rhombo-hexagonal dodecahedronWilliamson (1979) p169 or hexarhombic dodecahedronFedorov's five parallelohedra in R³ is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular depending on the shape of the rhombi. It can be seen as constructed from a rhombic dodecahedron elongated by a square prism.
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Examples of tangential quadrilaterals are the kites, which include the rhombi, which in turn include the squares. The kites are exactly the tangential quadrilaterals that are also orthodiagonal.. A right kite is a kite with a circumcircle. If a quadrilateral is both tangential and cyclic, it is called a bicentric quadrilateral, and if it is both tangential and a trapezoid, it is called a tangential trapezoid.
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If the number of sides n is odd, then for any given set of sidelengths a_1, \dots , a_n satisfying the existence criterion above there is only one tangential polygon. But if n is even there are an infinitude of them.. For example, in the quadrilateral case where all sides are equal we can have a rhombus with any value of the acute angles, and all rhombi are tangential to an incircle.
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Kites are examples of ex-tangential quadrilaterals. Parallelograms (which include squares, rhombi, and rectangles) can be considered ex- tangential quadrilaterals with infinite exradius since they satisfy the characterizations in the next section, but the excircle cannot be tangent to both pairs of extensions of opposite sides (since they are parallel). Convex quadrilaterals whose side lengths form an arithmetic progression are always ex-tangential as they satisfy the characterization below for adjacent side lengths.
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It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:The Regular Polyhedra (of index two), David A. Richter 250px Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.
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Honeycomb section containing transition from worker to drone (larger) cells – here bees make irregular and five-cornered cells (marked with red dots). Western honeybees and honeycomb In 1965, László Fejes Tóth discovered the trihedral pyramidal shape (which is composed of three rhombi) used by the honeybee is not the theoretically optimal three- dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or about one part per 2850) more efficient.
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It is topologically equivalent to a quotient space of the hyperbolic order-5 square tiling, by distorting the rhombi into squares. As such, it is topologically a regular polyhedron of index two:The Regular Polyhedra (of index two), David A. Richter 250px Note that the order-5 square tiling is dual to the order-4 pentagonal tiling, and a quotient space of the order-4 pentagonal tiling is topologically equivalent to the dual of the medial rhombic triacontahedron, the dodecadodecahedron.
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A Fokker D.VII shows a four-color Lozenge-Tarnung (lozenge camouflage) During the First World War, the Germans developed Lozenge-Tarnung (lozenge camouflage). This camouflage was made up of colored polygons of four or five colors. The repeating patterns often used irregular four-, five- and six-sided polygons, but some contained regular rhombi or hexagons. Because painting such a pattern was very time consuming, and the paint added considerably to the weight of the aircraft, the pattern was printed on fabric.
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There are several different variations of Penrose tilings with different tile shapes. The original form of Penrose tiling used tiles of four different shapes, but this was later reduced to only two shapes: either two different rhombi, or two different quadrilaterals called kites and darts. The Penrose tilings are obtained by constraining the ways in which these shapes are allowed to fit together. This may be done in several different ways, including matching rules, substitution tiling or finite subdivision rules, cut and project schemes, and coverings.
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Regarding zoomorpic items, there are caprids, bovids, dogs, "ostrich-like" animals (big birds) and schematic linear quadrupeds. Geometric signs show T-shaped figures, vertical parallel lines, horizontal zigzags, vertical parallel zigzags, branch-like or tree-like figures, chessboard patterns, rhombi, horizontal stair-like patterns, crossed networks, honeycomb networks and crossed circles. Few rayed circle figures, mainly the two unica of the so-called calendar scene, likely represent a sun depiction. Taking count of some associated figures, it is possible to recognize dancing, hunting, and mating scenes.
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In geometry, the ten-of-diamonds decahedron is a space-filling polyhedron with 10 faces, 2 opposite rhombi with orthogonal major axes, connected by 8 identical isosceles triangle faces. Although it is convex, it is not a Johnson solid because its faces are not composed entirely of regular polygons. Michael Goldberg named it after a playing card, as a 10-faced polyhedron with two opposite rhombic (diamond-shaped) faces. He catalogued it in a 1982 paper as 10-II, the second in a list of 26 known space-filling decahedra.
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The concept of similarity extends to polygons with more than three sides. Given any two similar polygons, corresponding sides taken in the same sequence (even if clockwise for one polygon and counterclockwise for the other) are proportional and corresponding angles taken in the same sequence are equal in measure. However, proportionality of corresponding sides is not by itself sufficient to prove similarity for polygons beyond triangles (otherwise, for example, all rhombi would be similar). Likewise, equality of all angles in sequence is not sufficient to guarantee similarity (otherwise all rectangles would be similar).
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It can also be seen as a nonuniform truncated icosahedron with pyramids augmented to the pentagonal and hexagonal faces with heights adjusted until the dihedral angles are zero, and the two pyramid type side edges are equal length. This construction is expressed in the Conway polyhedron notation jtI with join operator j. Without the equal edge constraint, the wide rhombi are kites if limited only by the icosahedral symmetry. joined truncated icosahedron The sixty broad rhombic faces in the rhombic enneacontahedron are identical to those in the rhombic dodecahedron, with diagonals in a ratio of 1 to the square root of 2.
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Central nave The church and the monastery occupy a plateau at the top of a hill, at some 540 meters above sea level, behind a large square. The exterior appearance of the church is characterized by the juxtaposition of black (basalt) and light white (sandstone) stones, as typical of Pisan-Lucchese medieval churches. The façade is divided into four horizontal sectors: the three lower sectors (that of the portal and those above it) feature blind arcades and loggias decorated by rhombi, another typical decoration of contemporary Tuscan religious buildings. On the first step of the portal is the inscription Mariane maistro, likely the name ("Master Marianus") of the mason who directed the construction.
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Click here for an animated version. The Herschel graph is planar and 3-vertex- connected, so it follows by Steinitz's theorem that it is a polyhedral graph: there exists a convex polyhedron (an enneahedron) having the Herschel graph as its skeleton.. This polyhedron has nine quadrilaterals for faces, which can be chosen to form three rhombi and six kites. Its dual polyhedron is a rectified triangular prism, formed as the convex hull of the midpoints of the edges of a triangular prism. This polyhedron has the property that its faces cannot be numbered in such a way that consecutive numbers appear on adjacent faces, and such that the first and last number are also on adjacent faces.
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Westminster School master William Camden cultivated the artistic genius of Ben Jonson. The Scottish poet William Drummond of Hawthornden was friend and confidant to Jonson. In midlife, Jonson claimed that his paternal grandfather, who 'served King Henry 8 and was a gentleman', was a member of the extended Johnston family of Annandale in the Dumfries and Galloway, a genealogy that is attested by the three spindles (rhombi) in the Jonson family coat of arms: one spindle is a diamond-shaped heraldic device used by the Johnston family. Jonson's father lost his property, was imprisoned, and suffered forfeiture under Queen Mary; having become a clergyman upon his release, he died a month before his son's birth.
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As a quilting pattern it also has many other names including cubework, heavenly stairs, and Pandora's box. It has been suggested that the tumbling blocks quilt pattern was used as a signal in the Underground Railroad: when slaves saw it hung on a fence, they were to box up their belongings and escape. See Quilts of the Underground Railroad.. In these decorative applications, the rhombi may appear in multiple colors, but are typically given three levels of shading, brightest for the rhombs with horizontal long diagonals and darker for the rhombs with the other two orientations, to enhance their appearance of three-dimensionality. There is a single known instance of implicit rhombille and trihexagonal tiling in English heraldry – in the Geal/e arms.
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There are an infinite number of uniform tilings of the hyperbolic plane by kites, the simplest of which is the deltoidal triheptagonal tiling. Kites and darts in which the two isosceles triangles forming the kite have apex angles of 2π/5 and 4π/5 represent one of two sets of essential tiles in the Penrose tiling, an aperiodic tiling of the plane discovered by mathematical physicist Roger Penrose. Face-transitive self- tesselation of the sphere, Euclidean plane, and hyperbolic plane with kites occurs as uniform duals: for Coxeter group [p,q], with any set of p,q between 3 and infinity, as this table partially shows up to q=6. When p=q, the kites become rhombi; when p=q=4, they become squares.
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A rectangle (equiangular quadrilateral) with integer side lengths may be tiled by unit squares, and an equiangular hexagon with integer side lengths may be tiled by unit equilateral triangles. Some but not all equilateral dodecagons may be tiled by a combination of unit squares and equilateral triangles; the rest may be tiled by these two shapes together with rhombi with 30 and 150 degree angles. A cyclic polygon is equiangular if and only if the alternate sides are equal (that is, sides 1, 3, 5, ... are equal and sides 2, 4, ... are equal). Thus if n is odd, a cyclic polygon is equiangular if and only if it is regular.De Villiers, Michael, "Equiangular cyclic and equilateral circumscribed polygons", Mathematical Gazette 95, March 2011, 102-107.
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It has been claimed by some that the glass panes in the Louvre Pyramid number exactly 666, "the number of the beast", often associated with Satan. Dominique Stezepfandt's book François Mitterrand, Grand Architecte de l'Univers declares that "the pyramid is dedicated to a power described as the Beast in the Book of Revelation (...) The entire structure is based on the number 6." The story of the 666 panes originated in the 1980s, when the official brochure published during construction did indeed cite this number (even twice, though a few pages earlier the total number of panes was given as 672 instead). The number 666 was also mentioned in various newspapers. The Louvre museum, however, states that the finished pyramid contains 673 glass panes (603 rhombi and 70 triangles).
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A definition intended to describe a given set of individuals fails if its description of matching individuals is incongruous: too broad (excessively loose with parameters) or too narrow (excessively strict with parameters). For example, "a shape with four sides of equal length" is not a sufficient definition for "square", because squares are not the only shapes that can have four sides of equal length; rhombi do as well. Likewise, defining a "rectangle" as "a shape with four perpendicular sides of equal length" is inappropriate because it is too narrow, as it describes only squares while excluding all other kinds of rectangles, thus being a plainly incorrect definition. If a cow were defined as an animal with horns, this would be overly broad (including goats, for example), while if a cow were defined as a black-and-white quadruped, this would be both overly narrow (excluding: all-black, all-white, all-brown and white-brown cows, for example) and overly broad (including Dalmatians, for example).
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In the same year, due to the difference in views in the development of the products, the relationship between Vicentelli and Plastwood comes to an end. This created the introduction of Geomag SA in January 2003, a Swiss company with its headquarters in Ticino, with the signing of the agreement between Vicentelli and Geomag SA allowing for the production of the patented construction toy line. At the end of July Geomag SA began the production and marketing campaigns of the products, including the development of panel (triangular platforms, rhombi, squares and pentagonal semi-transparent coloured polycarbonate that can be used to fill the gaps of the structures bars, with decorative function and support) which demonstrated a second success (representing 35% of the global market) and particularly became an important product line for Geomag as it differentiated the company from its imitators. This development is also a result from a patent licensed from Vicentelli.
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It is impossible here to give any full description of these different systems; but a brief account may be given of the principles underlying them. Most of them are based upon a decimal system, doubtless owing to the habit of counting on the fingers. In some cases the symbols are simple and obvious, as in the Cretan script, where circles (or rhombi), dots and lines are used for hundreds, tens and units, each being repeated as often as necessary; and a similar system for the lower denominations is used at Epidaurus in the 4th century BC. In Athens the usual system was to indicate each denomination by its initial, M for Μύριοι (10,000), X for χίλιοι (1,000), H for εκατόν (100), Δ for δέκα (10), π for πεντε (5) and I for units. The other Greek system followed that derived from the Phoenicians, using the letters of the alphabet in their conventional order from one to nine, 10 to 90 and 100 to 900; in this arrangement obsolete letters were retained in their original places so as to give the requisite number of 27 symbols.
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The majority of the design consists of squares, rhombi, and isosceles triangles all arranged along a diagonal axis with the empty upper right corner corresponding to the center of the muqarnas vault. The angles of each element are in multiples of 45° with few exceptions. Scholars and archeologist of Islamic architecture have made numerous attempts to reconstruct the designs of the muqarnas plate found at Takht-e Soleyman to then understand what the muqarnas might have looked like in the palace. These scholars have based their designs in historical and cultural context by referencing the writings by Islamic mathematician and astronomer Ghiyath al-Din al-Kashi (in which he describes the use, design, and construction of muqarnas), various structures of the same Ilkhanid time period that contain muqarnas (the Great Mosque of Natanz and the tomb of Shaykh Abd al-Samad al-Isfahani of Natanz), previous interpretations of the muqarnas plate (chiefly of experts Ulrich Harb and Mohammad-Ali Jalal Yaghan), as well as the muqarnas traditions used today directly inherited through the traditions of the Ilkhanid period (as with the muqarnas workshops of Fez, Morocco).
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