Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T16:36:33.608Z Has data issue: false hasContentIssue false
  • English
  • Français

A Short Proof of Euler’s Relation for Convex Polytopes

Published online by Cambridge University Press:  20 November 2018

Jim Lawrence
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The purpose of this paper is to present a short, self-contained proof of Euler’s relation. The ingredients of this proof are (i) the principle of inclusion and exclusion of combinatorics and (ii) the Euler characteristic; a development of the Euler characteristic is included.

Keywords

52A25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 40 , Issue 4 , 01 December 1997 , pp. 471 - 474
Copyright
Copyright © Canadian Mathematical Society 1997

References

1. Bruggesser, H. and Mani, P., Shellable decompositions of cells and spheres. Math. Scand., 29 (1972), 197–205.Google Scholar
2. Cordovil, R., Las Vergnas, M., and Mandel, A., Euler's relation, Möbius functions and matroid identities. Geom. Dedicata, 12 (1982), 147–162.Google Scholar
3. Folkman, J. and Lawrence, J. F., Oriented matroids. J. Combin. Theory Ser. B, 25 (1978), 299–236.Google Scholar
4. Groemer, H., On the Euler characteristic in spaces with a separability property. Math. Ann., 211 (1974), 315–321.Google Scholar
5. Grünbaum, B., Convex Polytopes. Interscience, 1967.Google Scholar
6. Hadwiger, H., Eulers Characteristik und kombinatorische Geometrie. J. Reine Angew. Math., 194 (1955), 101–110.Google Scholar
7. Klee, V., The Euler characteristic in combinatorial geometry. Am. Math. Monthly, 70 (1963), 119–127.Google Scholar
8. McMullen, P. and Schneider, R., Valuations on convex bodies. Convexity and its Applications (P.M. Gruber and J. M. Wills, eds.), Birkhauser, Basel, 1983, 170–247.Google Scholar
9. Mandel, A., Topology of oriented matroids. Ph. D. thesis, University of Waterloo, 1981.Google Scholar
10. Nef, W., Zur einfürung der Eulerschen charakteristik. Monatshefte für Mathematik, 92 (1981), 41–46.Google Scholar
11. Rota, G. -C., On the combinatorics of the Euler characteristic. Studies in Pure Mathematics (L. Mirsky, ed.), Academic Press, London, 1971, 221–233.Google Scholar
12. Tverberg, H., How to cut a convex polytope into simplices. Geom. Dedicata, 3 (1974), 239–240.Google Scholar
13. Zaslavsky, T., Counting the faces of cut-up spaces. Bulletin of the A. M. S., 81 (1975), 916–918.Google Scholar