Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 49, No. 1, pp. 269275 (2008) 

The special cuts of the $600$cellMathieu Dutour Sikiri\'c and Wendy MyrvoldRudjer Bo\u skovi\'c Institute, Bijenicka 54, 10000 Zagreb, Croatia, email: Mathieu.Dutour@ens.fr. Department of Computer Science, University of Victoria, P.O. Box 3055, Stn CSC, Victoria, B.C. Canada V8W 3P6, email: wendym@cs.uvic.caAbstract: A polytope is called regularfaced if each of its facets is a regular polytope. The $4$dimensional regularfaced 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 nonadjacent 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), 8793. [R1] Blind, R.: {\em Konvexe Polytope mit regulären Facetten im $R\sp{n}$ $(n\geq 4)$}. Contributions to Geometry (Proc. Geom. Sympos., Siegen, 1978), 248254, Birkhäuser, BaselBoston, Mass., 1979. [R2] Blind, R.: {\em Konvexe Polytope mit kongruenten regulären $(n1)$Seiten im $R\sp{n}$ $(n\geq 4)$}. Comment. Math. Helv. {\bf 54}(2) (1979), 304308. Keywords: grand antiprism, regularfaced polytope, regular polytope, semiregular polytope, $600$cell, (snub) $24$cell, symmetry group Classification (MSC2000): 52B11, 52B15 Full text of the article:
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