Abstract
We construct a symmetric polyhedron of genus 4 in R 3 with 11 vertices. This shows that for given genus g the minimal numbers of vertices of combinatorial manifolds and of polyhedra coincide in the first previously unknown case g=4 also. We show that our polyhedron has the maximal symmetry for the given genus and minimal number of vertices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barnette, D., ‘Polyhedral Maps on 2-Manifolds’ in Convexity and Related Combinatorial Geometry, Proc. Second University of Oklahoma Conf., (D.C. Kay, and M. Breen, eds).
Bokowski, J., ‘Aspects of Computational Synthetic Geometry, II: Combinatorial Complexes and their Geometric Realization — an Algorithmic Approach’, Proc. Computer-Aided Geometric Reasoning, INRA, Antibes (France), 1987.
Bokowski, J., ‘On the Geometric Flat Embedding of Abstract Complexes with Symmetries’, Preprint 1076, Darmstadt, 1987.
Bokowski, J., and U. Brehm, ‘A New Polyhedron of Genus 3 with 10 Vertices’, Colloquia Math. Soc. János Bolyai, Siofok, 1985.
Bokowski, J. and Eggert, A., ‘All Realizations of Möbius' Torus with 7 Vertices’, Preprint 1009, Darmstadt, 1986.
Bokowski, J., Richter, J. and Sturmfels, B, ‘Nonrealizability Proofs in Computational Geometry’, Discrete Comput. Geom. (to appear).
Bokowski, J. and Sturmfels, B., ‘On the Coordinatization of Oriented Matroids’, Discrete Comput. Geom. 1 (1986), 293–306.
Brehm, U., ‘Polyeder mit zehn Ecken vom Geschlecht drei’, Geom. Dedicata 11 (1981), 119–124.
Brehm, U., ‘A Nonpolyhedral Triangulated Möbius Strip’, Proc. Amer. Math. Soc. 89 (1983), 519–522.
Brehm, U., ‘A Maximally Symmetric Polyhedron of Genus 3 with 10 Vertices’, Mathematika 34 (1987), 237–242.
Brehm, U., ‘Maximally Symmetric Polyhedral Realizations of Dyck's Regular Map’, Mathematika, 34 (1987), 229–236.
Császár, A., ‘A Polyhedron without Diagonals’, Acta Sci. Math. Szeged. 13 (1949) 140–142.
Dress, A., Dreiding, A. and Haegi, H., ‘Chirotops-a New Approach to Oriented Matroids’, Manuscript, Bielefeld/Zürich, 1984.
Dress, A., ‘Chirotops and Oriented Matroids’, Bayreuther Math. Schriften 21 (1986), 14–68.
Duke, R. A., ‘Geometric Embedding of Complexes’, Amer. Math. Monthly 77 (1970), 597–603.
Grünbaum, B., Convex Polytopes, Interscience, London, 1967.
Jungerman, M. and Ringel, G., ‘Minimal Triangulations on Orientable Surfaces’, Acta Math. 145 (1980), 121–154.
Sturmfels, B., ‘Aspects of Computational Synthetic Geometry, I: Algorithmic Coordinatization of Matroids’, Proc. Computer-Aided Geometric Reasoning, INRIA, Antibes (France), 1987.
Sturmfels, B., ‘Computational Synthetic Geometry’, PhD Dissertation, University of Washington, Seattle, 1987.
Szilassi, L., ‘Regular Toroids’, Structural Topology 13 (1986), 69–80.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bokowski, J., Brehm, U. A polyhedron of genus 4 with minimal number of vertices and maximal symmetry. Geom Dedicata 29, 53–64 (1989). https://doi.org/10.1007/BF00147470
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00147470