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A polyhedral model for Cartan's Hypersurface in S4

Published online by Cambridge University Press:  26 February 2010

Ulrich Brehm  and
Wolfgang Kühnel
Ulrich Brehm
Affiliation:
Fachbereich Mathematik der Techn., Universität Berlin, Str. des 17. Juni 136, D-1000 Berlin 12, Federal Republic of Germany.
Wolfgang Kühnel
Affiliation:
Fachbereich Mathematik der Universität Duisburg, Lotharstr. 65, D-4100 Duisburg 1, Federal Republic of Germany.
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Abstract

E. Cartan's famous isoparametric hypersurface in S4 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 E5. The euclidean symmetry group of this embedding is isomorphic to the icosahedral group A5 acting transitively on the 15 vertices.

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Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 55 - 61
Copyright
Copyright © University College London 1986

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References

1.Bertini, E.. Geometria proiettiva degli iperspazi (Pisa, 1907). German translation: Einführung in die projektive Geometrie mehrdimensionaler Räume (Wien, 1924).Google Scholar
2.Brehm, U. and Kühnel, W.. A 5-neighborly triangulation of the quaternionic projective plane with 15 vertices. In preparation.Google Scholar
3.Cartan, E.. Sur des families remarquables d'hypersurfaces isoparamétriques dans les espaces sphériques, Math. Zeit., 45 (1939), 335–367.Google Scholar
4.Cecil, T. and Ryan, P.. Tight spherical embeddings. Proc. Conf. Global Diff. Geom. and Global Analysis, Berlin 1979 (Berlin, 1981), 94–104. Lecture Notes in Mathematics, 838.Google Scholar
5.Coxeter, H. S. M.. Regular polytopes, 2nd edition (Macmillan, New York, 1969).Google Scholar
6.Coxeter, H. S. M.. Twisted honeycombs. Regional Conf. Ser. Math., Nr. 4 (Amer. Math. Soc., 1970).Google Scholar
7.Coxeter, H. S. M. and Moser, W. O. S.. Generators and Relations for Discrete Groups, 4th ed. (Springer, Berlin-Heidelberg-New York, 1980). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 14.Google Scholar
8.Kühnel, W. and Banchoff, T. F.. The 9-vertex complex projective plane. The Math. Intelligencer, vol. 5, issue 3 (1983), 11–22.Google Scholar
9.Kuiper, N. H.. On convex maps. Meuiv Archief voor Wiskunde, 10 (1962), 147–164.Google Scholar
10.Kuiper, N. H.. Geometry in Total Absolute Curvature Theory. Perspectives in Mathematics, Anniversary of Oberwolfach (Birkhäuser, Basel-Boston-Stuttgart, 1984), 377–392.Google Scholar
11.Mannoury, G., Surfaces—images. Nieuw Archief voor Wiskunde, 4 (1900), 122–129.Google Scholar
12.Miyaoka, R.. Taut embeddings and Dupin hypersurfaces. Proc. Conf. Diff. Geom. of Submanifolds, Kyoto 1984 (Springer, Berlin, 1984). Lecture Notes in Mathematics, 1090.Google Scholar
13.Pinkall, U.. Dupinsche Hyperflachen in E 4. Manuscripta Math., 51 (1985), 89–119.Google Scholar
14.Steinitz, E.. Konfigurationen der projektiven Geometrie. Enz. d. Math. Wiss. Ill A, B, 5a (1910), Nr. 14.Google Scholar