Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-24T13:03:35.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Maximally symmetric polyhedral realizations of Dyck's regular map

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

Ulrich Brehm
Ulrich Brehm
Affiliation:
FB 3-Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 1000 Berlin 12, West-Germany.
Get access

Abstract

We construct realizations of Dyck's regular map of genus three as polyhedra in ℝ3. One of these has one axis of symmetry of order three and three axes of symmetry of order two. The other polyhedra have three axes of symmetry. We show that a polyhedron realizing Dyck's regular map cannot have a symmetry group of order larger than six. Thus the symmetry groups of our realizations are maximal.

MSC classification

Secondary: 52A37: Other problems of combinatorial convexity
Type
Research Article
Information
Mathematika , Volume 34 , Issue 2 , December 1987 , pp. 229 - 236
Copyright
Copyright © University College London 1987

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

1.Bokowski, J.. A geometric realization for Dyck's regular map does exist without self-intersections. Discrete Comput. Geometry. In print.Google Scholar
2.Brehm, U.. A maximally symmetric polyhedron of genus 3 with 10 vertices. Mathematika, 34 (1987). 237–242.CrossRefGoogle Scholar
3.Dyck, W.. Über Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer Riemannscher Flächen. Math. Ann., 17 (1880), 473–508.CrossRefGoogle Scholar
4.Dyck, W.. Notiz über eine reguläre Riemannsche Fläche vom Geschlecht 3 und die zugehörige Normalkurve 4. Ordnung. Math. Ann., 17 (1880), 510–516.CrossRefGoogle Scholar
5.Dyck, W.. Gruppentheoretische Studien. Math. Ann., 20 (1882), 1–45.CrossRefGoogle Scholar
6.McMullen, P.Schulz, Ch. and Wills, J. M.. Polyhedral 2-manifolds in E3 with unusually large genus. Israel J Math., 46 (1983), 127–144.CrossRefGoogle Scholar
7.McMullen, P.Schulte, E. and Wills, J. M.. Infinite series of combinatorially regular polyhedra in three-space. Geometriae Dedicata. To appear.Google Scholar
8.Schulte, E. and Wills, J. M.. A polyhedral realization of Felix Klein's map {3, 7}8 on a Riemannian manifold of genus 3. J. London Math. Soc., 32 (1985), 539'547.CrossRefGoogle Scholar
9.Schulte, E. and Wills, J. M.. Geometric realizations for Dyck's regular map on a surface of genus 3. Discrete Comput. Geometry, 1 (1986), 141–153.CrossRefGoogle Scholar
10.Schulte, E. and Wills, J. M.. On Coxeter's regular skew polyedra. Discrete Math., 60 (1986), 253–262.CrossRefGoogle Scholar
11.Schulte, E. and Wills, J. M.. Combinatorially regular polyhedra in three-space. To appear in: Symmetry—a collection of essays, ed. Hofmann, K. H. and Wille, R. (Heldermann-Verlag, Berlin).Google Scholar