Models of Locally Regular Heptagonal Dodecahedra

Year: 2010 Authors: David I. McCooey

Core claim

Five of the six combinatorially distinct trivalent heptagonal dodecahedra admit realizable non-self-intersecting models with aesthetically pleasing symmetry.

Topics

polyhedral realizations, genus-2 polyhedra, combinatorial structures, symmetry, physical fabrication

Domains

polyhedral geometry, topology, combinatorics, symmetry groups, mathematical art, 3D modeling, stereolithography, paper modeling

Methods

algorithmic search, combinatorial classification, symmetry selection, model fabrication

Media

stereolithography, paper models, stereo pairs, nets

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

Models of Locally Regular Heptagonal Dodecahedra

David I. McCooey

Pixelux Entertainment, Inc.

Normandy Park, WA 98166, USA

E-mail: dmccooey@mac.com

Abstract

We present models of polyhedra built from 12 heptagons meeting three per vertex. Unlike the analogous case with 12 pentagons, where there is a single unique combinatorial structure, there are six combinatorially distinct ways to combine 12 heptagons, meeting three per vertex, into a (possibly self-intersecting) polyhedron. We identified realizable (non-self-intersecting) examples for five of the six possible structures, and fabricated physical models of them. They all necessarily have genus 2 (topologically equivalent to a 2-holed donut), and they appear in a variety of aesthetically pleasing symmetries. These models demonstrate a form of art emerging from mathematics.

Six years ago, J.M. Wills [4] proved that a trivalent heptagonal dodecahedron could be realized, and more recently [5], he constructed an explicit example. His example was combinatorially different from the one used in his earlier proof, revealing the evident fact that there were at least two combinatorially distinct ways to combine 12 heptagons, meeting three per vertex, into a polyhedron.

Figure 1: The six combinatorially distinct structures for a trivalent heptagonal dodecahedron.

McCooey

It turns out that there are exactly six combinatorially distinct ways to combine 12 heptagons, meeting three per vertex, into a (possibly self-intersecting) polyhedron [1]. Maps for these six structures are shown in Figure 1. See [3] for a thorough description. In this short paper, we present polyhedral realizations without self-intersections for five of these maps $({\mathbf{M}}_1,{\mathbf{M}}_2,{\mathbf{M}}_3,{\mathbf{M}}_5,$ and ${\mathbf{M}}_6)$ . They were found using the same algorithm that was used for finding realizations of the all-heptagon Klein map [2], except that the symmetry of the expected polyhedron in each case was chosen to be consistent with the associated map’s automorphism group [3]. The polyhedron described by Wills in his 2004 existence proof is a realization of ${\mathbf{M}}_1$ , and the one he constructed explicitly in 2008 is a realization of ${\mathbf{M}}_2$ . Any realization of ${\mathbf{M}}_4$ , if it exists at all, has only the trivial symmetry [3]. Figures 2 through 7 show stereo pairs and nets for these polyhedra, and Figure 8 shows some physical models. Examples were selected with qualities such as 90-degree angles, parallel faces, and balanced edge lengths.

Figure 2: Two stereo pairs and a net for a polyhedral realization of $M_1$ with S4 symmetry.

Figure 3: Two stereo pairs and a net for a polyhedral realization of $M_2$ with C3 symmetry.

Models of Locally Regular Heptagonal Dodecahedra

Figure 4: Two stereo pairs and a net for a polyhedral realization of $M_2$ with C3 symmetry.

Figure 5: Two stereo pairs and a net for a polyhedral realization of $M_3$ with D2 symmetry.

Figure 6: Two stereo pairs and a net for a polyhedral realization of $M_3$ with D3 symmetry.

McCooey

Figure 7: Two stereo pairs and a net for a polyhedral realization of $M_h$ with D2 symmetry.

We fabricated stereolithography models at scales of roughly 3.5 inches in diameter, and paper models at scales of roughly 7 inches in diameter. Besides being significant as the simplest possible locally regular genus-2 polyhedra, these models also serve as useful representations of some simple point group symmetries, not to mention their aesthetic appeal as art objects. The ${\mathbf{M}}_1$ polyhedron, in particular, is particularly interesting for its 4-fold rotoreflection (S4) symmetry, which is not often encountered.

Figure 8: Stereolithography models $\left([M_2,M_2,M_1]\right]$ $[M_3,M_3,M_h]$ and paper models $(M_1,M_3,M_h)$ .

References

[1] B. Datta and A. K. Upadhyay, Degree-regular triangulations of the double-torus, Forum Mathematicum, 18, pp. 1011-1025. 2006. [2] D. I. McCooey, A non-self-intersecting polyhedral realization of the all-heptagon Klein map, Symmetry: Culture and Science, 20(1-4), pp. 247-268. 2009. [3] D. I. McCooey, Locally Regular Heptagonal Dodecahedra, to appear in Symmetry: Culture and Science - special issue on Tessellations, 2010. [4] J. M. Wills, Equivelar Polyhedra, The Coxeter Legacy (C. Davis and E. Ellers eds. 2006), [May 12-16 2004], Providence, RI: American Mathematical Society, pp. 121-128. 2004. [5] J. M. Wills, A locally regular heptagon-dodecahedron embedded in 3-space, European Journal of Combinatorics, 29(8), pp. 1952-1955. 2008.