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Visualization of regular maps : the chase continues Citation for published version (APA): Wijk, van, J. J. (2014). Visualization of regular maps : the chase continues. IEEE Transactions on Visualization and Computer Graphics, 20(12), 2614-2623. https://doi.org/10.1109/TVCG.2014.2352952 DOI: 10.1109/TVCG.2014.2352952 Document status and date: Published: 01/01/2014 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 24. Sep. 2021 2614 IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, VOL. 20, NO. 12, DECEMBER 2014 Visualization of Regular Maps: The Chase Continues Jarke J. van Wijk, Member, IEEE Abstract—A regular map is a symmetric tiling of a closed surface, in the sense that all faces, vertices, and edges are topologically indistinguishable. Platonic solids are prime examples, but also for surfaces with higher genus such regular maps exist. We present a new method to visualize regular maps. Space models are produced by matching regular maps with target shapes in the hyperbolic plane. The approach is an extension of our earlier work. Here a wider variety of target shapes is considered, obtained by duplicating spherical and toroidal regular maps, merging triangles, punching holes, and gluing the edges. The method produces about 45 new examples, including the genus 7 Hurwitz surface. Index Terms—regular maps, tiling, tessellation, surface topology, mathematical visualization 1INTRODUCTION In graph theory, a map is a crossing-free embedding of a graph on a combination of heuristics, insights, and a variety of media, ranging a surface. A regular map is a map that is vertex-, face-, and edge- from sketches to computer models to 3D printing. An alternative is to transitive, i.e., each vertex, face, and edge is topologically indistin- use an automatic approach. In 2009 we translated the problem into au- guishable from the other elements of the same type. Classic examples tomatically finding matching patterns in hyperbolic space. About 50 of regular maps are the Platonic solids, which are regular maps on new solutions resulted, but for many regular maps it is still unknown spheres. But also for surfaces with higher genus g such regular maps what their space models could look like. can be found. The genus is informally equal to the number of holes The choice of the target surface is important, both in manual and in the shape. The Euler number χ = V − E + F, with V the num- automatic approaches. Typically a smooth closed surface is used, cov- ber of vertices, E the number of edges, and F the number of faces is ered with a face-transitive map. Decorating each face with the same an invariant for the genus of the surface, and for orientable surfaces pattern should hopefully lead to a depiction of a regular map. Sequin´ χ = 2 − 2g. used a tetrus, the shape that results when replacing the edges of a tetra- Regular maps on tori, genus 1 surfaces, can be found easily. As hedron by tubes, to visualize regular maps of genus 3. We extended an example, folding a chessboard by matching opposite sides gives this approach, and used tubified regular maps as target surfaces, where a torus decorated with a regular map. For genus 2 and higher, the quarter tubes were used as target faces. problem to produce visualizations of regular maps is much more com- In this paper we present a generalization of our earlier approach. We plex. It is known what regular maps exist. Conder [4] has enumerated use a richer set of target surfaces, obtained by duplicating space mod- all reflexible regular maps on orientable surfaces up to genus 101, and els of regular maps of genus 0 and 1, merging triangles, systematically found 6104 different cases. Each map is described here via its symme- punching holes, and gluing the edges. This leads to a face-transitive try group. In graphics terminology, the structure of the faces, edges, map, consisting of topologically identical faces, configured in a cer- and vertices is given, but no possible geometric realization, i.e.,no tain pattern, and which may have vertices of different valences around possible space models are given. their perimeter. Also, regular maps can be simplified to face-transitive It is a fascinating problem to produce space models of regular maps: maps. This approach is more generic than the earlier approach, which surfaces embedded in 3D, decorated with colored faces, edges, and was limited to the use of tubified regular maps as target shapes. vertices depicting regular maps. Regular maps are not an esoteric, Like in the earlier approach, we use a two-step matching process to abstract concept, they can be described in a very concrete and geo- find new solutions. First, simplified regular maps and target surfaces metric way. For instance, the Hurwitz genus 7 surface consists of 168 are matched, to find combinations of these with similar faces and sim- triangles, at each vertex 7 triangles meet, and each Petrie-polygon (a ilar patterns; next, exact solutions are sought for by aligning points of closed zig-zag path over the edges) has 18 edges. But, what could the target surface with points of the regular map in hyperbolic space. A such a shape look like? The papers of Carlo Sequin´ [20, 21, 22]give novelty here is the smoothing of meshes in hyperbolic space, leading an interesting and entertaining account on how to attack this problem; to edges with less kinks. we argued earlier [24] that this class of puzzles is close to perfect. Compared to the earlier approach, the number of regular maps that Also, the scope of this topic is large. Regular maps relate to many can be visualized almost doubles, and all examples given in this paper different branches of mathematics [14]. Symmetry is described by are new. Specifically interesting is the first visualization of the Hurwitz algebra, shapes by geometry, and regular maps can be studied from the genus 7 surface, also known as the MacBeath surface. point of view of surface topology, combinatorial group theory, graph In the next section we give a short introduction on concepts used, theory, hyperbolic geometry, and algebraic geometry. Finally, there is and discuss related work. Next, we give a global overview of our ap- an artistic interest, as these highly symmetric objects lead to complex proach in Section 3, followed by a detailed description of the various and fascinating shapes. steps in Sections 4 to 7. Results are presented and discussed in Sec- Roughly, there are two main approaches to come up with solutions tion 8, followed by conclusions in Section 9. for the visualization of regular maps. The work of Sequin´ is the key ex- ample of the manual approach. He has produced a steadily growing set 2BACKGROUND of space models of regular maps by attacking them one by one, using As mentioned, regular maps are located at the intersection of differ- ent branches of mathematics, and to understand and process regular • Jarke J. van Wijk is with Eindhoven University of Technology. E-mail: maps, some background is needed. We assume basic knowledge of [email protected]. group theory and hyperbolic geometry. We used Beardon [2], Ka- Manuscript received 31 Mar. 2014; accepted 1 Aug. 2014. D ate of tok [12], and Anderson [1] for understanding tilings, Fuchsian groups, publication11Aug. 2014; date of current version 9 Nov. 2014. and hyperbolic geometry; Firby and Gardiner [9] for surface topology; For information on obtaining reprints of this article, please send Coxeter [6], Coxeter and Moser [7] and Conder and Dobcsanyi´ [5] for e-mail to: [email protected]. regular maps. Here we describe some basic concepts and terminology, Digital Object Identifier 10.1109/TVCG.2014.2352952 mainly by examples, that are used in the following sections. 1077-2626 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. VAN WIJK: VISUALIZATION OF REGULAR MAPS: THE CHASE CONTINUES 2615 R3 R4 2 2 2 M M R R S R 1 aba R2S N O b ab R I a bc S 2 N c O RS 2 ca S RS Fig.
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