Locally and Globally Regular Toroids with Less than 16 Hexagonal Faces

Year: 2010 Authors: Lajos Szilassi

Core claim

All combinatorial types of locally regular toroids with F ≤ 15 are exhibited, with conditions for global regularity and realizations generated via Euler3D.

Topics

equivelar toroids, hexagonal faces, polyhedral realizations, sculptural geometry

Domains

polyhedral combinatorics, topology, symmetry groups, toroidal geometry, sculpture, mathematical art, visual design

Methods

dual construction, combinatorial classification, Euler3D modeling

Media

conference CD-ROM, polyhedral models, sculpture drawings

Paper text

The text below is the locally extracted OCR/Markdown version of the paper. Raw PDF files remain local and are not published here.

Bridges 2010: Mathematics, Music, Art, Architecture, Culture

Locally and Globally Regular Toroids with Less than 16 Hexagonal Faces

Lajos Szilassi

University of Szeged, Hungary

E-mail: szilassi@jgytf.u-szeged.hu

Abstract

In 2008 [1], we presented three classes of equivelar toroids. We have shown examples of al combinatorial types of toroids for face number $F\le 12$ . The data which uniquely determine these polyhedra are described in [2]. As a continuation of [1], here we show further examples for all combinatorial types of toroids with $F\le 15$ . These constructions represent outstanding mathematical achievements, and they also have significance in the art as sculptures having pleasant visual appearance. Such sculptures will be shown at the exhibition of the conference.

A polyhedron is called a toroid, if it is a torus in topological sense. A toroid is locally regular, if the number of edges meeting at each vertex is the same, and all of its faces are polygons with equal number of sides. A polyhedron satisfying these two conditions is usually called now an equivelar polyhedron (Brehm and Wills, 1993); here we use the term locally regular in the particular case of toroidal polyhedra. It is a requirement that the polygons must not be self-intersecting, and the polyhedron must not have overarching faces, i.e. two faces must not have common vertices more than two, and if they have two common vertices, then they have common edges as well.

A toroid is called globally regular (or, simply, regular), if the group of its combinatorial automorphisms is transitive on the flags. (A flag is a triple consisting of a vertex, an edge and a face, which are mutually incident). The F9 B and F12 D are globally regular toroids.

We attach to our paper on the conference CD the maps of all the locally regular toroids with face

number $F\le 15$ ; we give necessary condition for the global regularity of these maps; finally, we give a one or more polyhedral realizations of these maps. These realization are given via the software Euler3D [3]; the corresponding files contain also their numerical data.

Here in the Appendix we present some drawings of maps belonging to toroids with hexagonal faces, for $13\le \mathrm{F}\le 15$ .

Finally, we remark that these polyhedra were obtained, using duality, from the polyhedra of type $\{3,6\}$ , which can easily be constructed by the method described in [1,2]. Constructing polyhedra of type $\{3,6\}$ with larger and larger number of faces is more and more easier. The contrary is true in the case of their duals, i.e. polyhedra of type $\{6,3\}$ . Constructing the latter is more and more difficult, and they are less and less interesting mathematically. On the other hand, from aesthetic - artistic - point of view they may still be interesting.

A sculpture (F 7 toroid) in the Fermat’s birthplace, Beaumont de Lomagne France

Conclusion: It is a lucky circumstance, when an abstract sculpture exhibit a significant mathematical achievement as well, such as in our case.

References

[1] L. Szilassi, Some Regular Toroids, Proceedings of the $11^{th}$ Bridges Conference, Leeuwarden, 2008, pp. 459-460. [2] L. Szilassi, Locally Regular Toroids with Hexagonal Faces Symmetry: Culture and Science Vol. 20. Nos.1-4, 269-295, 2009 [3] Proceedings of the $11^{th}$ Bridges Conference, Leeuwarden, 2008, CD-ROM /Extras/Some_Regular_Toroids/

Szilassi

F13 A

APPENDIX F13 B1

F13 B2

F14 A1

F14 A2

F14 B

F15 A1

F15 A2

F15 B

F15 C

F15 D1

F15 D2

F15D3

512