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Corresponding Polyhedra in the Three Spaces of Constant Curvature

Published online by Cambridge University Press:  20 November 2018

Ernst Roeser
Ernst Roeser*
Affiliation:
Mainz, Germany
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The five Platonic solids can be drawn in elliptic or hyperbolic space just as well as in Euclidean space. Their numerical properties are, of course, the same in all three. So are the various angles subtended at the centre. But the face-angles and dihedral angles are greater in elliptic space, smaller in hyperbolic. It is a special feature of the non-Euclidean spaces that we cannot change the size of a solid without changing its shape.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 5 , 1953 , pp. 40 - 45
Copyright
Copyright © Canadian Mathematical Society 1953

References

1. Coxeter, H. S. M., The functions of Schldfii and Lobatschefsky, Quarterly J. Math., 6 (1935), 13–29.Google Scholar
2. Coxeter, H. S. M., Regular Polytopes (London, 1948; New York, 1949).Google Scholar
3. Coxeter, H. S. M. and Whitrow, G. J., World-structure and non-Euclidean honeycombs, Proc. Royal Soc, A, 201 (1950), 417–437.Google Scholar
4. Roeser, Ernst, Über die nichteuklidischen regular en Polyeder, S.B. Heidelberger Akad. Wiss., Math.-Nat. Kl. (1932), 1–10.Google Scholar
5. Roeser, Ernst, Die regularen Polygone der drei Raumformen, Dtsch. Math., S (1938), 288–301.Google Scholar
6. Schlegel, Victor, Théorie der homogen zusammengesetzten Raumgebilde, Verh. K. Leopold.-Carol. D. Akad. Naturf., U (1883), 343–459.Google Scholar
7. Sommerville, D. M. Y., Non-Euclidean Geometry (London, 1914; Chicago, 1919).Google Scholar
8. Sommerville, D. M. Y., The regular divisions of space of n dimensions and their metrical constants, R.C. Circ. mat. Palermo, 48 (1924), 9–22.Google Scholar