On Symmetrical Equivelar Polyhedra of Type {3, 7} and Embeddings of Regular Maps

Abstract

A regular map is an abstract generalization of a Platonic solid. It describes a group, a topological cell decomposition of a 2-manifold of type $\{p, q\}$ with only p-gons, such that q of them meet at each vertex in a circular manner, and we have maximal combinatorial symmetry, expressed by the flag transitivity of the symmetry group. On the one hand, we have articles on topological surface embeddings of regular maps by F. Razafindrazaka and K. Polthier, C. Séquin, and J. J. van Wijk.On the other hand, we have articles with polyhedral embeddings of regular maps by J. Bokowski and M. Cuntz, A. Boole Stott, U. Brehm, H. S. M. Coxeter, B. Grünbaum, E. Schulte, and J. M. Wills. The main concern of this partial survey article is to emphasize that all these articles should be seen as contributing to the common body of knowledge in the area of regular map embeddings. This article additionally provides a method for finding symmetrical equivelar polyhedral embeddings of type $\{3, 7\}$ based on symmetrical graph embeddings on convex surfaces.

1. Preliminary Remarks

This article can be viewed as a contribution to computational synthetic geometry that deals with methods for realizing abstract geometric objects in concrete vector spaces [1]. In this book, we find the following example on page 132.

INPUT: Combinatorial or geometric condition, Felix Klein’s regular map ${\{3, 7\}}_8$.

OUTPUT: Coordinates or a realizability proof, polyhedral realization of Felix Klein’s map [2].

We see the Platonic solids with a labeling of their vertices in Figure 1. This allows us to study and to describe their symmetries via permutations of their vertices.

Regular maps are generalizations of Platonic solids with their Schläfli symbols $\{p, q\}$, $\{3, 3\}$ for the tetrahedron, $\{4, 3\}$ for the cube, $\{3, 4\}$ for the octahedron, $\{5, 3\}$ for the dodecahedron, and $\{3, 5\}$ for the icosahedron. We have only p-gons as facets, q of them cyclically at each vertex.

A regular map is an abstract cell decomposition of a closed connected topological 2-manifold with only topological p-gons such that q of them meet in a cyclic manner at each vertex. We require (global) dihedral symmetry of the p-gons and a rotational symmetry at each vertex such that the whole combinatorial structure of the topological 2-manifold does not change.

For the above example of Felix Klein’s regular map, we provide two combinatorial descriptions for the input that might lead to a better understanding of the concept of a regular map. In Figure 2 and Figure 3, we see two essentially different combinatorial symmetries for this example. The resulting combinatorial structure, the regular map, can be viewed as either a symmetry group or an abstract topological 2-manifold. The existence of a geometrical symmetry for a spatial embedding of the abstract closed connected topological 2-manifold is, in general, completely open.

Each regular map can be considered as an INPUT for a problem in computational synthetic geometry. Regular maps provide a special problem class in this area. We have about 3000 generated examples of regular maps that are available via [3,4].

This article contains a survey of articles in which we have a corresponding OUTPUT of such problems either as a spatial topological embedding or as a spatial polyhedral realization. These two possible embedding results should be seen in context.

One example concerns the regular R5.1 map in the list of M. Conder [3]. A spatial polyhedral embedding was known as a result of Grünbaum [5] (compare [6]), while J.J.van Wijk [7] presented their spatial topological embedding without citing this foregoing result.

2. Historical Remarks

The regular map of Felix Klein from 1879 that we have used as a first example is also known in the literature as Felix Klein’s quartic (see [8]). It is the first regular map of an infinite sequence of regular maps of Adolf Hurwitz of type $\{3, 7\}$ [9]. Other early discoveries of regular maps, e.g., symmetry groups of Riemann surfaces or symmetry groups of regular polyhedra in dimension 4, are those of Walther von Dyck [10,11], Alicia Boole Stott [12], Felix Klein and Robert Fricke [13], and Harold Scott MacDonald Coxeter [14]. In those times, the notion regular map was not used. Within a recent article [15] about Felix Klein, seen by three of his friends, David Hilbert, Adolf Hurwitz, and Herrmann Minkowski, we can read some of this. Nowadays, we have the lists of regular maps by M. Conder and P. Dobcsányi in [3,4]. For comments about historical origins, we have the book of H.S.W. Coxeter and W. O. J. Moser [16]. There is useful survey article about polyhedral embeddings for maps of small genus by E. Schulte and J. M. Wills in [17]. It gives precise definitions of some of the terms used in the current article and it describes additional maps.

3. The Known Spatial Polyhedral Realizations of Regular Maps

We have six spatial polyhedral realizations of regular maps with at least the rotational symmetry of one of the Platonic solids. These have been named regular Leonardo polyhedra. The reason for this name is shown in Figure 4. Here, we see Leonardo da Vinci’s drawing in which he has replaced the edges of the dodecahedron with struts. This was a first polyhedron with a genus g, $g\ge 2$. When you do the opposite and you replace the struts of the polyhedron on the right picture with edges, you obtain the Schlegel digram of the 24-cell in dimension 4 that Alicia Boole Stott found in 1880 when she was 18. You can see this model rotating in a YouTube video at https://www.youtube.com/watch?v=3n_hyMpbNL8 (accessed on 8 April 2024).

The remaining five Leonardo polyhedra can be seen together with this example in two recent articles [18,19].

The authors of these regular maps have been mentioned in the last section. The authors of the polyhedral realizations or polyhedral embeddings are A. Boole Stott [12], H.S.M. Coxeter [14], E. Schulte and J. M. Wills [2], and B. Grünbaum [5].

For the example that we have used at the beginning for explaining the regular map of Felix Klein, we have a YouTube video of its realization of Schulte and Wills that shows not only the geometric symmetry of this polyhedron but also an additional symmetry of order 2 that allows us to turn the polyhedron inside out without changing the polyhedron: https://www.youtube.com/watch?v=GP_dDMCRSFc (accessed on 21 November 2021).

Here, we have to mention an additional polyhedral embedding of the dual regular map compared with what we have discussed so far for Klein’s quartic. When thinking of the Platonic solids with its convex facets, you require facets for a polyhedral embedding of a regular map convex as well. However, the corresponding polyhedral embedding of D.I. McCooey [20], with non-convex pentagons, is a very interesting result together with several equivelar (=local regular) polyhedra of type $\{7, 3\}$.

The polyhedral realization of H. S. M. Coxeter’s Regular Leonardo Polyhedron $\{8, 4|3\}$ can be seen in the following YouTube video: https://www.youtube.com/watch?v=o-VSIqtmGNM (accessed on 8 April 2024).

This provides the background for the YouTube video for this volume Symmetries in Combinatorial Structures. https://www.youtube.com/watch?v=AailDBdIPVY&t=15s (accessed on 22 November 2023).

The regular Leonardo Polyhedron of Coxeter’s regular map $\{8, 4|3\}$ (see [19]) was based on a corresponding 2-manifold in $R^4$. For an attempt to explain a corresponding embedding via a Schlegel diagram, I have used the following picture in Figure 5. It shows 14 truncated cubes that were adjacent to a further truncated cube C in dimension 4. When we rotate these 14 cubes by using their common 2-faces of C as hinges so that they all lie in the hyperplane of the cube C, we obtain the model in Figure 5, left part. The video https://youtu.be/AailDBdIPVY (accessed on 21 November 2023) shows the corresponding projection of the Schlegel diagram in which the 14 truncated cubes are glued at the central truncated cube. The picture on the right contains an image from the video.

So far, there are only two additional regular maps known for which we have a spatial polyhedral realization. For the regular map of Walther von Dyck [10,11], the conjecture that there does not exist such a spatial polyhedral realization was disproved by J. Bokowski in [21]. Afterwards, U. Brehm found an even better polyhedral realization with the best possible geometrical symmetries [22]. You can see this model in Figure 6 and in a YouTube video: https://www.youtube.com/watch?v=HahcRHIc-YY&t=2s (accessed on 21 November 2023).

The last and most involved example of a spatial polyhedral realization of a regular map is based on the second example of the infinite list of regular maps of type $\{3, 7\}$ of Hurwitz. A symmetric description of this regular map can be seen in Figure 7. A polyhedral realization model was made by J. Bokowski and M. Cuntz [23]. You can see this model in two YouTube videos: https://www.youtube.com/watch?v=4Bh_6Zl9ILc (accessed on 20 June 2020). https://www.youtube.com/watch?v=VhmtQ9zhF8Q&t=89s (accessed on 2 June 2020).

Figure 8 shows a picture from the first of these two YouTube videos.

An investigation of the possible geometrical symmetries of this polyhedron was carried out by J. Bokowski and G. Gévay [24]. So far, we have not mentioned any methods for finding polyhedral embeddings. For this last example, the theory of oriented matroids has played a decisive role, like in many other problem cases of computational synthetic geometry.

When you insist on no intersections for a polyhedral realization of a regular map, most symmetries remain hidden in general. On the other hand, we have the polyhedra by Kepler and Poinsot with self-intersections. Can we construct a similar polyhedron of the genus 7 Hurwitz surface ${\{3, 7\}}_{18}$ of this type with symmetries? Yes. There is such a Kepler–Poinsot-type Polyhedron due to J. Bokowski [25]. You can see this in Figure 9 and in the following YouTube video: https://www.youtube.com/watch?v=dVAl9EdKSmM&t=5s (accessed on 2 December 2020).

4. The Known Spatial Topological Embeddings

We have topological surface embeddings of regular maps of C. Séquin [26,27], J. J, van Wijk [7,28], and F. Razafindrazaka and K. Polthier [29,30], with excellent pictures. Most of these pictures are available online. If you type in “Jarke J. van Wijk regular maps” on the Internet or you go to the homepages of Carlo Sequin and Konrad Polthier, you will find them. A study of these articles is informative about regular maps, and you see mathematics being at the borderline of art.

5. Symmetrical Non-Regular Equivelar Polyhedra of Type {3, 7}

When we do not require polyhedra with the highest possible combinatorial symmetry, i.e, as a regular map, we can still find polyhedra with only p-gons, such that, at each vertex, we have a circular sequence with q of them meeting there. These polyhedra are called equivelar polyhedra. When they do not have the combinatorial symmetry of a regular map, we also use the notion of non-regular equivelar polyhedra. In this section, we describe a construction method that leads to symmetrical non-regular equivelar polyhedra of type $\{3, 7\}$ when we have a symmetrical three-valent graph embedding on a convex surface. The method can be seen easily when we describe the building block at the vertices of such a graph. When the graph has the rotational symmetry of one of the Platonic solids, we obtain a non-regular Leonardo polyhedron.

The example with a tetrahedral symmetry in Figure 10 has 72 vertices, 252 edges, and 168 triangles.

Euler’s formula tells us its genus: it is seven. The same properties are shared by the regular map of genus 7 of type ${\{3, 7\}}_{18}$ of Hurwitz, for which there does exist a polyhedral embedding [23], although no symmetrical embedding is known so far [24].

When I tried to construct an equivelar polyhedron of type $\{3, 7\}$ with a symmetry that we know from the underlying Archimedian solid in Figure 11, I realized later that this construction can be completed whenever a three-valent graph is given as a one-skeleton of a convex polyhedron.

In Figure 12, you see the decisive triangles of the building blocks for all equivelar polyhedra of type $\{3, 7\}$ in this article.

In Figure 13 you see two pictures of an example with a rotational cubical symmetry.

In Figure 14 you see the more involved case with a rotational dodecahedral symmetry.

6. Conclusions

We have compared two types of spatial embeddings for regular maps: general topological surface embeddings and polyhedral embeddings. There are interactions between these outcomes. Initial results suggest that, in many cases, the existence of a corresponding polyhedral embedding is highly unlikely. It would be valuable to obtain proofs demonstrating the non-existence of a polyhedral embedding in many of these cases.

When we begin with a polyhedral embedding of a regular map, there are methods to construct other topological embeddings, such as through the subdivisions of the polyhedron (see Figure 8). In the search for additional regular Leonardo polyhedra, one could first examine which of the possible candidate regular maps has a subgroup that is known to be a rotational symmetry of a Platonic solid.

In the search for additional non-regular equivelar polyhedra, we have discovered new examples of non-regular Leonardo polyhedra compared with [31]. The construction technique somewhat resembles what we have observed from Alicia Boole Stott and from constructions for finding topological surface embeddings. I believe that there are additional construction methods available, similar to the one illustrated in Figure 15. I recommend a further investigation to be carried out in this direction.