Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-04T21:14:21.518Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-Schlegelian polyhedral map on the torus

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Amos Altshuler  and
Ulrich Brehm
Amos Altshuler
Affiliation:
Ben Gurion University of the Negeov, Beer-Sheva, Israel.
Ulrich Brehm
Affiliation:
Mathematisches Institut, Albert-Ludwigs-Universität, Freiburg, West Germany.
Get access

Abstract

We describe a toroidal polyhedral map which can be geometrically realized in R3 but not via a Schlegel diagram of a convex 4-polytope. Moreover, this map is not isomorphic to a subcomplex of the boundary complex of any convex polytope.

MSC classification

Secondary: 52A99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 83 - 88
Copyright
Copyright © University College London 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

A1.Altshuler, A.. Polyhedral realization in R 3 of triangulations of the torus and 2-manifolds in cyclic 4-polytopes. Discrete Math., 1 (1971), 211–238.CrossRefGoogle Scholar
A2.Altshuler, A.. Manifolds in stacked 4-polytopes. J. Comb. Th., 10 (1971), 198–239.CrossRefGoogle Scholar
BA.Brehm, U. and Altshuler, A.. Weakly neighborly polyhedral maps on the torus. To appear.Google Scholar
BSW.Betke, U., Schulze, Ch. and Wills, J. M.. Bänder und Möbiusbänder in Konvexen Polytopen. Abh. Math. Sem. Hamburg, 44 (1975), 249–262.CrossRefGoogle Scholar
C.Császár, A.. A polyhedron without diagonals. Acta Sci. Math. Szeged, 13 (1949–50), 140–142.Google Scholar
G.Grünbaum, B.. Convex Polytopes (Wiley, New York, 1967).Google Scholar
M.Möbius, A. F.. Gesammelte Werke, Vol. 2 (Leipzig, 1886).Google Scholar
S.Simutis, J.. Geometric Realizations of Toroidal Maps (Ph.D. Thesis, 1977, University of California, Davis).Google Scholar
SR.Steinitz, E. and Rademacher, H.. Vorlesungen über die Theorie der Polyeder (Berlin, 1934).Google Scholar