Abstract
Five polyhedra in ordinary three-dimensional space have attracted special interest since ancient times because of their regularity: the tetrahedron, the cube (hexahedron), the octahedron, the icosahedron, and the dodecahedron. Platon describes them in the dialogue “Timaios” and thus they were called Platonic solids. We present them quite informally in §1. in order to answer the question why there are not more regular solids, the general concepts of convex polytopes and their regularity are introduced. We do this for arbitrary finite dimensions and show that in all dimension > 2 the possibilities for regular solids are very restricted. From dimension five on only the analogs of the tetrahedron, the cube, the octahedron survive. Further study of the six possibilities in four dimensions is postponed to chapter II, §3. In this chapter I we restrict attention again to three dimensions from §5 on.
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© 1986 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig
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Lamotke, K. (1986). Regular Solids and Finite Rotation Groups. In: Regular Solids and Isolated Singularities. Advanced Lectures in Mathematics. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-322-91767-6_1
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DOI: https://doi.org/10.1007/978-3-322-91767-6_1
Publisher Name: Vieweg+Teubner Verlag, Wiesbaden
Print ISBN: 978-3-528-08958-0
Online ISBN: 978-3-322-91767-6
eBook Packages: Springer Book Archive