Open Access
Regular Surfaces and Regular Maps
Faniry Razafindrazaka and Konrad Polthier
- 01 Jan 2014
- pp 225-234
TL;DR: In this article, the symmetry group of a regular map is used to visualize the regular surface formed around the tubular neighborhood of the regular map, which can be interactively modified and used as a target shape for other regular maps.
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Abstract: A regular surface is a closed genus g surface defined as the tubular neighbourhood of the edge graph of a regular map A regular map is a family of disc type polygons glued together to form a 2-manifold which is flag transitive The visualization of this highly symmetric surface is an intriguing and challenging problem Unlike regular maps, regular surfaces can always be visualized as 3D embeddings In this paper, we introduce an algorithm to visualize the regular surface formed around the tubular neighborhood of a regular map Our algorithm takes as input the symmetry group of a regular map and outputs a 3D realization of its regular surface This surface can be interactively modified and used as a target shape for other regular maps As a result, we find new realizations of regular maps ranging from genus 9 to 85
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Citations
Turning Hild’s Sculptures into Single-Sided Surfaces
Carlo H. Séquin
- 25 Jan 2019
TL;DR: In this article, the authors explore ways in which similar-looking shapes may be created that are single-sided and highlight some differences in their two approaches to create some complex 2-manifolds that are clearly different from Hild's repertoire.
7
Hurwitz's regular map (3, 7) of genus 7: A polyhedral realization
Jürgen Bokowski,Michael Cuntz +1 more
- 10 Nov 2017
TL;DR: For the next possible genus of a Hurwitz surface, i.e., for the genus 7 case with 72 vertices, a polyhedral realization without self-intersections is provided and a topological representation for which a corresponding model is shown.
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Generators and relations for discrete groups
H. S. M. Coxeter,W. O. J. Moser +1 more
- 01 Jan 1957
TL;DR: In this article, the authors present a systematic enumeration of cosets based on the following groups: cyclic, Dicyclic and Metacyclic groups, Graphs, Maps, Cayley Diagrams, Hyperbolic Tessellations and Fundamental Groups.
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Symmetric tiling of closed surfaces: visualization of regular maps
Jarke J. van Wijk
- 27 Jul 2009
TL;DR: In this paper, the authors present a method for the generation of space models of regular maps for genus 2 and higher, based on a generalization of the method for tori.
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My Search for Symmetrical Embeddings of Regular Maps
Carlo H. Séquin
- 01 Jan 2010
TL;DR: The symmetrical patterns discovered could be further modified to create Escher-like tilings on low-genus handle bodies in space models of regular maps embedded in surfaces of genus 2 to 5.
Patterns on the Genus-3 Klein Quartic
Carlo H. Séquin
- 01 Jan 2006
TL;DR: In this article, projections of Klein's quartic surface of genus 3 into 3D space are used as canvases on which they present regular tessellations, Escher tilings, knotand graph embedding problems, Hamiltonian cycles, Petrie polygons and equatorial weaves derived from them.