The special cuts of the $600$-cell

Mathieu Dutour Sikiri\'c and Wendy Myrvold

Abstract: A polytope is called regular-faced if each of its facets is a regular polytope. The $4$-dimensional regular-faced polytopes were determined by G. Blind and R. Blind [Bl], [R1], [R2]. Regarding this classification, the class of such polytopes not completely known is the one which consists of polytopes obtained by removing some set of non-adjacent vertices (an independent set) of the $600$-cell. These independent sets are enumerated up to isomorphism, and we show that the number of polytopes in this last class is $314\;248\;344$.

[Bl] Blind, G.; Blind, R.: {\em Die konvexen Polytope im $\RR^4$, bei denen alle Facetten reguläre Tetraeder sind}. Monatsh. Math. {\bf 89} (1980), 87--93. [R1] Blind, R.: {\em Konvexe Polytope mit regulären Facetten im $R\sp{n}$ $(n\geq 4)$}. Contributions to Geometry (Proc. Geom. Sympos., Siegen, 1978), 248--254, Birkhäuser, Basel-Boston, Mass., 1979. [R2] Blind, R.: {\em Konvexe Polytope mit kongruenten regulären $(n-1)$-Seiten im $R\sp{n}$ $(n\geq 4)$}. Comment. Math. Helv. {\bf 54}(2) (1979), 304--308.

Keywords: grand antiprism, regular-faced polytope, regular polytope, semiregular polytope, $600$-cell, (snub) $24$-cell, symmetry group

Classification (MSC2000): 52B11, 52B15

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