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An Eberhard-Like Theorem for Pentagons and Heptagons.
- Source :
-
Discrete & Computational Geometry . Dec2010, Vol. 44 Issue 4, p931-945. 15p. - Publication Year :
- 2010
-
Abstract
- Eberhard proved that for every sequence ( p), 3≤ k≤ r, k≠6, of nonnegative integers satisfying Euler's formula ∑(6− k) p=12, there are infinitely many values p such that there exists a simple convex polyhedron having precisely p faces of size k for every k≥3, where p=0 if k> r. In this paper we prove a similar statement when nonnegative integers p are given for 3≤ k≤ r, except for k=5 and k=7 (but including p). We prove that there are infinitely many values p, p such that there exists a simple convex polyhedron having precisely p faces of size k for every k≥3. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width. Our proof suggests a general method for obtaining results of this kind. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 01795376
- Volume :
- 44
- Issue :
- 4
- Database :
- Academic Search Index
- Journal :
- Discrete & Computational Geometry
- Publication Type :
- Academic Journal
- Accession number :
- 54326376
- Full Text :
- https://doi.org/10.1007/s00454-010-9264-1