2 Sentences With "manifoldnesses"

How to use manifoldnesses in a sentence? Find typical usage patterns (collocations)/phrases/context for "manifoldnesses" and check conjugation/comparative form for "manifoldnesses". Mastering all the usages of "manifoldnesses" from sentence examples published by news publications.

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Open Court Publishing: Chicago in 1901. Under the heading On Physiological as Distinguished from Geometrical Space Mach states that "Both spaces are threefold manifoldnesses" but the former is "...neither constituted everywhere and in all directions alike, nor infinite in extent, nor unbounded." A notable attempt at a rigorous formulation was made in 1947 by Rudolf Luneburg, who preceded his essay on mathematical analysis of visionLuneburg, R.K. (1947). Mathematical Analysis of Binocular Vision.
As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an n-fach ausgedehnte Mannigfaltigkeit (n times extended manifoldness or n-dimensional manifoldness) as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a Mannigfaltigkeit evolved into what is today formalized as a manifold. Riemannian manifolds and Riemann surfaces are named after Bernhard Riemann.