Construction of non-Wythoffian perfect \(4\)-polytopes
Two convex polytopes \(P,Q\subset\mathbb R^d\) are symmetrically equivalent if there is an isometry \(\phi\) of \(\mathbb R^d\) and a combinatorial isomorphism \(\psi:P\to Q\) such that for each face \(F<P\) and each isometry \(\sigma\) fixing \(P\) we have that \(\sigma(F)=\psi^{-1}\phi^{-1}\sigma\phi\psi(F)\). A polytope is perfect if each symmetrically equivalent polytope is similar. Perfect polytopes generalize regular polytopes, and further examples arise as special cases of Wythoff's construction, that is, as the convex hull of a point orbit under a finite reflection group. The author constructs \(4\)-dimensional polytopes which cannot be obtained by Wythoff's method.
scientific article; zbMATH DE number 1894857