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- W3103844692 abstract "Regular polytopes, the generalization of the five Platonic solids in 3 space dimensions, exist in arbitrary dimension $ngeq-1$; now in {rm dim}. 2, 3 and 4 there are emph{extra} polytopes, while in general dimensions only the hyper-tetrahedron, the hyper-cube and its dual hyper-octahedron exist. We attribute these peculiarites and exceptions to special properties of the orthogonal groups in these dimensions: the $mathrm{SO}(2)=mathrm{U}(1)$ group being (abelian and) emph{divisible}, is related to the existence of arbitrarily-sided plane regular polygons, and the emph{splitting} of the Lie algebra of the $mathrm{O}(4)$ group will be seen responsible for the Schl{a}fli special polytopes in 4-dim., two of which percolate down to three. In spite of {rm dim}. 8 being also special (Cartan's emph{triality}), we argue why there are no emph{extra} polytopes, while it has other consequences: in particular the existence of the three emph{division algebras} over the reals $mathbb{R}$: complex $mathbb{C}$, quaternions $mathbb{H}$ and octonions $mathbb{O}$ is seen also as another feature of the special properties of corresponding orthogonal groups, and of the spheres of dimension 0,1,3 and 7." @default.
- W3103844692 created "2020-11-23" @default.
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- W3103844692 date "2013-04-01" @default.
- W3103844692 modified "2023-09-25" @default.
- W3103844692 title "On Regular Polytopes" @default.
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- W3103844692 doi "https://doi.org/10.1016/s0034-4877(13)60026-9" @default.
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