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A polyhedral model for Cartan's Hypersurface in S 4
Author(s) -
Brehm Ulrich,
Kühnel Wolfgang
Publication year - 1986
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300013863
Subject(s) - mathematics , hypersurface , quotient , combinatorics , euclidean geometry , regular polygon , embedding , pure mathematics , surface (topology) , polyhedron , boundary (topology) , icosahedral symmetry , geometry , mathematical analysis , artificial intelligence , computer science
E. Cartan's famous isoparametric hypersurface in S 4 with three distinct constant principal curvatures is geometrically a parallel hypersurface of the Veronese surface, and topologically it is an 8‐fold quotient of the 3‐sphere. In the present paper we describe a polyhedral analogue with only 15 vertices. Combinatorially this is an 8‐fold quotient of the boundary complex of the 600‐cell, and geometrically it is a quite regular subcomplex of a certain almost convex simplicial 4‐sphere in E 5 . The euclidean symmetry group of this embedding is isomorphic to the icosahedral group A 5 acting transitively on the 15 vertices.