Polyhedral Embeddings of Triangular Regular Maps of Genus g, 1 < g < 15, and Neighborly Spatial Polyhedra

1. Introduction

This article provides a contribution to the field of Computational Synthetic Geometry, which explores methods for realizing abstract geometric objects in concrete vector spaces, [1]. Our focus lies in the construction and analysis of triangular regular maps of genus g, $2⩽g⩽14$ and of neighborly spatial polyhedra and of all these embeddings. These spatial polyhedra are all related to an old conjecture of Branko Grünbaum from 1967 [2], page 253, who conjectured that all oriented simplicial 2-manifolds have polyhedral embeddings in $R^3$. This conjecture was not believed by many experts, however, a proof of a counter example was given only first in 2000, [3], and additional counter-examples provided by Lars Schewe in his Ph.D. thesis, see his publication in 2010 [4]. When we study convex polytopes with their convex faces, we have the polar dual polytopes with convex faces again. When we have polyhedra with higher genus there are polyhedra for which we have in the dual face lattice embeddings with non-convex faces. A famous example for such an embedding is Szilassi’s polyhedron with a dual face lattice compared with the 7 vertex torus of Möbius. We do allow non-convex faces when we use duality in this paper.

The investigation of the second author to study the dual cases, of what has been investigated by Bokowski, Guedes de Oliveira, and Schewe, could not find questionable embeddings in this area. The use of his numerical optimization approach has led to all but one embeddings of triangular regular maps of genus g, $2⩽g⩽14$. We recall that a regular map generalizes in a topological manner what we know from Platonic solids. A regular map is a decomposition of a two-dimensional manifold into topological disks such that every flag (an incident vertex-edge-face triple) can be transformed into any other flag by a combinatorial symmetry of the decomposition. We use the result of M. Conder [5] and his notation for triangular oriented regular maps of genus g, $2⩽g⩽14$ with multiplicity 1. These 14 regular maps are listed in Table 1. For only five of them polyhedral embeddings were known. For the remaining cases, we have now eight new polyhedral embeddings of regular maps and for two former polyhedral embeddings, the symmetry properties have been improved. This has even led to detecting a fault in a previous paper, [6].

In the next two sections we provide tables with the list of triangular regular maps and neighborly polyhedra that were tested according to Grünbaum’s conjecture.

Afterwards we describe the algorithm of the second author.

2. Polyhedral Embeddings of Triangulated Orientable Regular Maps with Genus g, $2⩽g⩽14$ and Some of Their Duals

We have listed in Table 1 all polyhedral embeddings of triangulated orientable regular maps with genus g, $2⩽g⩽14$ and some of their duals. All triangular polyhedral embeddings in Table 1 are new for genus g, $g⩾8$. Geometric symmetries are listed with Schoenflies Notation. The embeddings of genus 6 and 7 have higher geometrical symmetries compared to those that were known before. A former embedding of R6.1 of Brehm has never been published. It had no geometric symmetries according to a private communication of J.M.Wills.

3. Polyhedral Embeddings of Neighborly Spatial Polyhedra with Complete Graphs as Their Edge Graph and Their Duals

In Table 2 we have listed neighborly spatial polyhedra according to complete graph embeddings on 2-manifolds and pseudo-manifolds. For complete graphs with 9 and 10 vertices we have polyhedra with pseudo-manifolds as their boundaries: A vertex can have more than one circular sequence of incident faces. For the seven vertex torus of Möbius you can download a video with all four embeddings at http://science-to-touch.com/ForJB/MoebiusTorus.mov.

4. Polyhedral Embeddings as an Optimization Problem

Finding a flat embedding of a simplicial complex without intersections is, in general, a difficult problem. While solutions can be easily verified in polynomial time, there are no efficient algorithms to generate them or prove their non-existence without a full search of the space. For small vertex counts, there has been some success using SAT solvers with oriented matroids [4] however, this can take significant computational resources and becomes intractable for larger complexes.

There are methods that are more efficient at solving NP problems of this sort if we introduce heuristics and non-determinism to our search. As a consequence, we will not be able to guarantee that a solution will be found for a given complex, and questions about which symmetries can be realized will remain open. This compromise is acceptable as we are looking for any embeddable examples where none exist currently.

We choose to focus on triangular regular maps specifically. This is because a polyhedron with all triangle faces can be completely defined by its set of 3-dimensional vertices and triangulation with no additional constraints. This greatly simplifies the optimization. By contrast, faces with polygons of more than 3 vertices need additional constraints to find embeddings since the points may be skew and not all lie on a common plane.

Duals of triangular maps are also equally efficient since these polyhedra can be entirely described by a set of planes and vertex adjacency list. Since the edge graph is cubic, the vertices of the dual are simply the intersection of the 3 planes that share the vertex.

Enforcing a geometric symmetry helps reduce the search space, speed up the computation, and in general seems to produce the best results when a suitable symmetry is used. All embeddings found so far have had some non-trivial geometric symmetry, and there have been no cases where an asymmetric solution has had fewer intersections than a symmetric one for a non-embeddable example. Therefore, we conjecture that all embeddable regular maps can be embedded with at least 1 non-trivial geometric symmetry.

Possible geometric symmetries can be inferred from the automorphisms of the regular map. Examples:

Permutation Group	Possible Geometric Symmetries
(a,b)(c,d)(e,f)	S2, C2, Cs
(a,b)(c,d)(e)(f)	C2, Cs
(a,b,c)(d,e,f)	C3
(a,b,c,d)(e,f,g,h)	C4, D2
(a,b,c,d)(e,f)	D2
(a,b,c,d)(e)(f)	C4

To enforce a geometric symmetry, the first point of a permutation becomes the reference and the other points in the permutation get defined relative to the first one by the given symmetry type.

The solver works in 2 stages; First a large random search is conducted to find low-intersection candidates, then a second stage is used to refine each solution to both reduce the intersections if they are non-zero, and improve the aesthetics of the shape to eliminate things like large scale differences between edges, near-intersections, or very skinny polygons. This stage can also truncate the vertex positions to produce small integer coordinates.

The heuristic used for the large search is simply the number of edge-polygon intersections plus the number of self-crossings of each polygon as illustrated in Figure 1. For triangular maps, self-crossings are always zero, but duals may have crossings.

Figure 1. Left: An edge-polygon intersection. Right: A polygon with a self-crossing.

Figure 1. Left: An edge-polygon intersection. Right: A polygon with a self-crossing.

The second stage heuristic includes the penalty from the first stage plus one minus the minimum of the following metrics:

Length Quality:	${\displaystyle \frac{\mathrm{minimum}\mathrm{side}\mathrm{length}}{\mathrm{maximum}\mathrm{side}\mathrm{length}}}$ for each polygon
Distance Quality:	${\displaystyle \frac{\mathrm{closest}\mathrm{distance}\mathrm{between}\mathrm{non}-\mathrm{neighboring}\mathrm{edges}}{\mathrm{farthest}\mathrm{distance}\mathrm{between}2\mathrm{points}}}$
Angle Quality:	$1-cos\left(\mathrm{smallest}\mathrm{angle}\mathrm{in}\mathrm{any}\mathrm{polygon}\right)$
Plane Quality:	$1-cos\left(\mathrm{smallest}\mathrm{angle}\mathrm{between}\mathrm{neighboring}\mathrm{faces}\right)$

The general algorithm is listed in Algorithm 1

Algorithm 1:Primary search for embeddings

Input:  List of index triplets representing the triangles T  1: ProcedureSearchEmbeddings(T, $iters$, $clusters$, $\sigma$=1.0, $\beta$=0.997, $\gamma$=1.25)  2: fori in $clusters$ do  3:    $V_i\leftarrow ApplySymmetry\left(RandomVertices\left(\right)\right)$  4:    $p_i\leftarrow Penalty(T,V_i)$  5: end for  6: forj in $iters$ do  7:    $m\leftarrow ArgMinimum\left(p\right)$  8:    for i in $clusters$ do  9:      if $p_i>p_m*\gamma$ then  10:         $r=RandomIndex\left(\right)$  11:         $V_i\leftarrow V_r$  12:         $p_i\leftarrow p_r$  13:      end if  14:      $V_{new}\leftarrow \beta *(V_i+RandomNoise\left(\sigma \right))$  15:      $V_{new}\leftarrow ApplySymmetry\left(V_{new}\right)$  16:      $p_{new}\leftarrow Penalty(T,V_{new})$  17:      if $p_{new}⩽p_i\mathbf{or}(i\ne m\mathbf{and}{p}_{new}⩽p_m*\gamma )$ then  18:         $V_i\leftarrow V_{new}$  19:         $p_i\leftarrow p_{new}$  20:      end if  21:    end for  22: end for  23:   return $V_m$ 24: End Procedure

Depending on the complexity of the problem, the optimizer can usually find solutions within only seconds or minutes for the smallest examples such as R3.2, and about a day for the largest ones like R14.2 on a standard desktop computer. Again for dual problems, the algorithm is nearly identical, but each point represents a plane instead, which has the same degrees of freedom. These are generally slower and harder to find since the placement of the planes is more sensitive than the vertex positions.

For any undecided cases, best results are Kepler–Poinsot-like with low intersection count. R13.2 has been realized with as few as 64 edge intersections with D2 symmetry.

5. Polyhedral Embeddings According to Table 1

5.1. Case R3.1

This regular map R3.1 is also called Felix Klein’s quartic of Schläfli type ${\{3,7\}}_8$ and genus 3. It is the first element of the infinite sequence of Hurwitz of type $\{3,7\}$,[16]. Its abstract symmetry group has order 336. References are [17,18].

It is an example of a regular Leonardo polyhedron, i.e., it is a polyhedral embedding of a regular map with a geometrical symmetry group of the rotational symmetry group of a Platonic solid. This polyhedral embedding is due to Schulte and Wills [19]. For additional articles about the six regular Leonardo polyhedra, see [20,21,22,23,24,25].

Figure 2. R3.1 with chiral tetrahedral symmetry

Figure 2. R3.1 with chiral tetrahedral symmetry

Figure 3. The combinatorial input for a polyhedron of Felix Klein’s quartic of type ${\{3,7\}}_8$. Here we see an example of a Petrie polygon.

Figure 3. The combinatorial input for a polyhedron of Felix Klein’s quartic of type ${\{3,7\}}_8$. Here we see an example of a Petrie polygon.

5.2. The Dual Case R3.1’

An embedding of the dual polyhedron of R3.1 is due to D. McCooey, [26].

Figure 4. R3.1’ with chiral tetrahedral symmetry

Figure 4. R3.1’ with chiral tetrahedral symmetry

5.3. Case R3.2

The regular map of R3.2 is due to W. Dyck, [27] [28] of Schläfli type ${\{3,8\}}_6$ and genus 3. The combinatorial symmetry group has order 192.

A first embedding was found by J. Bokowski, [29]. An embedding with better geometrical symmetries, with the dihedral group D3, is due to U. Brehm, [30]. An additional embedding symmetry S2 and an alternative D3 embedding are new and due to the second author.

Figure 5. R3.2 with D3 and S2 symmetry

Figure 5. R3.2 with D3 and S2 symmetry

Figure 6. The triangles of Dyck’s regular map R3.2 of type ${\{3,8\}}_6$ shown with a cyclic symmetry of order 3.

Figure 6. The triangles of Dyck’s regular map R3.2 of type ${\{3,8\}}_6$ shown with a cyclic symmetry of order 3.

Figure 7. Two pictures of Jarke van Wijk’s video with topological embeddings of regular maps, [31,32]. Here we see Dyck’s regular map R3.2’ of type ${\{8,3\}}_6$.

Figure 7. Two pictures of Jarke van Wijk’s video with topological embeddings of regular maps, [31,32]. Here we see Dyck’s regular map R3.2’ of type ${\{8,3\}}_6$.

Figure 8. Here we see Brehm's polyhedral embedding of Dyck's regular map R3.2 of type {3.8}₆. The polyhedron is complete when the red parts cannot be seen. In the front and back parts, eight blue triangles have to be reinserted. The red triangles are the inner sides of the polyhedron. The polyhedron has a geometrical dihedral symmetry D3 of order 6.

Figure 8. Here we see Brehm's polyhedral embedding of Dyck's regular map R3.2 of type {3.8}₆. The polyhedron is complete when the red parts cannot be seen. In the front and back parts, eight blue triangles have to be reinserted. The red triangles are the inner sides of the polyhedron. The polyhedron has a geometrical dihedral symmetry D3 of order 6.

Figure 9. This embedding of Dyck’s regular map R3.2 of type ${\{3,8\}}_6$ has the geometrical dihedral symmetry D3 like Behm’s embedding. However, it is different. For a front triangle and for a triangle of the rear part we see only their boundaries, so that we can imagine the shape of this polyhedron.

Figure 9. This embedding of Dyck’s regular map R3.2 of type ${\{3,8\}}_6$ has the geometrical dihedral symmetry D3 like Behm’s embedding. However, it is different. For a front triangle and for a triangle of the rear part we see only their boundaries, so that we can imagine the shape of this polyhedron.

5.4. Case R5.1

R5.1, due to Klein and Fricke [33] of type ${\{3,8\}}_{12}$ and genus 5, symmetry group of order 384.

The first embedding was found by B. Grünbaum, [34]. This is another example of the six regular Leonardo polyhedra. Later U. Brehm and J. M. Wills independently discovered this embedding again.

Figure 10. R5.1 with chiral octahedral and S2 symmetry

Figure 10. R5.1 with chiral octahedral and S2 symmetry

5.5. The Dual Case R5.1’

The embedding of the dual is a new result of the second author in this paper.

Figure 11. R5.1’ with D2 and C3 symmetry

Figure 11. R5.1’ with D2 and C3 symmetry

5.6. Case R6.1

R6.1, due to Coxeter and Moser of type ${\{3,10\}}_6$ and genus 6, symmetry group of order 300.

According to a private communication by J. M. Wills: A former polyhedral embedding of this regular map without any symmetry was found by U. Brehm. However, It was never published.

Figure 12. R6.1 with C3 and C2 symmetries

Figure 12. R6.1 with C3 and C2 symmetries

5.7. Case R7.1

R7.1, due to Hurwitz of type ${\{3,7\}}_{18}$ and genus 7, the second element of the infinite Hurwitz sequence with types $\{3,7\}$, also denoted as Macbeath surface as she rediscovered it, symmetry group of order 1008. A first embedding without geometrical symmetries was found in 2018, [35]. The symmetry of this new embedding has order 3, although we find in [6] the claim that such a symmetry is not possible. In other words, this example tells us that a fault in that paper has been detected.

Figure 13. R7.1 with C3, C2, and S2 symmetry

Figure 13. R7.1 with C3, C2, and S2 symmetry

Figure 14. We have 72 vertices, 252 edges, and 168 triangles.

Figure 14. We have 72 vertices, 252 edges, and 168 triangles.

Figure 15. We see two orthogonal projections of the order 2 symmetric polyhedral embedding of Hurwitz's regular map {3,7}₁₈ of genus 7 through the axis of symmetry. The 168 triangles are marked as their boundaries.

Figure 15. We see two orthogonal projections of the order 2 symmetric polyhedral embedding of Hurwitz's regular map {3,7}₁₈ of genus 7 through the axis of symmetry. The 168 triangles are marked as their boundaries.

Figure 16. An order 3 symmetrical polyhedral embedding of Hurwitz’s regular map ${\{3,7\}}_{18}$ of genus 7 with triangles progressively removed to show internal structure.

Figure 16. An order 3 symmetrical polyhedral embedding of Hurwitz’s regular map ${\{3,7\}}_{18}$ of genus 7 with triangles progressively removed to show internal structure.

5.8. The Dual Case R7.1’

Figure 17. R7.1’ with C3 and C2 symmetry

Figure 17. R7.1’ with C3 and C2 symmetry

Figure 18. R7.1’ with C3 symmetry. Faces removed to show: central axis, knotted cycle, ring cycle

Figure 18. R7.1’ with C3 symmetry. Faces removed to show: central axis, knotted cycle, ring cycle

5.9. Case R8.1

The regular map R8.1 has Schläfli type ${\{3,8\}}_8$ and genus 8. Its combinatorial symmetry group has order 672. The regular map has 42 vertices, 112 faces.

Figure 19. R8.1 with D2, C4, C3, and S2 symmetry

Figure 19. R8.1 with D2, C4, C3, and S2 symmetry

5.10. Case R8.2

The regular map R8.2 has Schläfli type ${\{3,8\}}_{14}$ and genus 8. Its combinatorial symmetry group has order 672. This regular map has 42 vertices and 112 faces.

Figure 20. R8.2 with D2, C4, and C3 symmetry

Figure 20. R8.2 with D2, C4, and C3 symmetry

5.11. Case R10.1

The regular map R10.1 has Schläfli type ${\{3,9\}}_{12}$ and genus 10. Its combinatorial symmetry group has order 648.

Figure 21. R10.1 with D2 symmetry

Figure 21. R10.1 with D2 symmetry

5.12. Case R10.2

R10.2, of type ${\{3,12\}}_6$ and genus 10, symmetry group of order 432.

Figure 22. R10.2 with C2 symmetry

Figure 22. R10.2 with C2 symmetry

Figure 23. The triangles in the outer circular sequence appear twice. The colored line segments indicate these pairs.

Figure 23. The triangles in the outer circular sequence appear twice. The colored line segments indicate these pairs.

5.13. Case R13.1

R13.1, of type ${\{3,10\}}_{30}$ and genus 13, symmetry group of order 720.

Figure 24. R13.1 with C3 and C2 symmetry

Figure 24. R13.1 with C3 and C2 symmetry

5.14. Case R14.1

R14.1 due to Hurwitz of type ${\{3,7\}}_{12}$ of genus 14, symmetry group of order 2184.

In the theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces (R14.1, R14.2, and R14.3) with the identical automorphism group of the lowest possible genus, here 14.

Figure 25. R14.1 with D2 symmetry

Figure 25. R14.1 with D2 symmetry

5.15. Case R14.2

R14.2 due to Hurwitz of type ${\{3,7\}}_{26}$ of genus 14, symmetry group of order 2184.

Figure 26. R14.2 with C2 symmetry

Figure 26. R14.2 with C2 symmetry

5.16. Case R14.3

R14.3 due to Hurwitz of type ${\{3,7\}}_{14}$ of genus 14, symmetry group of order 2184.

Figure 27. R14.3 with D2 symmetry

Figure 27. R14.3 with D2 symmetry

5.17. Case R13.2

The regular map R13.2 has Schläfli type ${\{3,12\}}_{12}$ and genus 13. Its combinatorial symmetry group has order 576.

It is the only triangular regular map of genus g, $2⩽g⩽14$ that is not likely to be embeddable.

6. Complete Graphs with 4, 7, and 12 Vertices on Closed Oriented 2-Manifolds, No Diagonals

In general, a k-neighborly polytope is a convex polytope where any k or fewer vertices form a face. We concentrate on the spatial case (k=1), where a polyhedron must not be convex and it is considered neighborly if either its edge graph is complete (no diagonals) or each face shares exactly one edge with every other face (face-sharing property). These two properties are linked through duality: The dual of a neighborly polyhedron with a complete edge graph is a neighborly polyhedron with the face-sharing property, and vice versa.

Crucially, we are interested in embeddings of these neighborly polyhedra that are free of self-intersections. This requires us to consider only oriented closed surfaces for our polyhedra, as non-orientable surfaces cannot be embedded in $R^3$. Furthermore, neighborly polyhedra with complete edge graphs necessarily have triangular faces. Therefore, our investigation centers on embeddings of triangular complete graphs on surfaces and their duals. These embeddings have played a significant role in results like the map color theorem, [15].

While we do not assume familiarity with oriented matroids, it is worth noting that all such triangular complete graph embeddings can be obtained by naturally extending the concept of pseudoline arrangements to curve arrangements on surfaces, [36].

Computations of all combinatorially possible embeddings is a second step before we discuss possible polyhedral embeddings.

We find in this section the solution of a long standing conjecture of Branco Grünbaum whether a cell decomposition of a triangulated orientable surface exists that cannot have an embedding in $R^3$.

6.1. The Tetrahedron

The Tetrahedron has the complete graph with 4 vertices as its edge graph. This is the easiest example of a neighborly polyhedron.

6.2. The Seven Vertex Torus of Möbius

For the seven vertex torus of Möbius in his collected works, [7], Császár in [8] was the first to find an embedding for Möbius’s combinatorial description, although he was not aware of this former reference. Here the edge graph has seven vertices. In 1991 Bokowski and Eggert [9] have found via oriented matroid techniques additional three symmetrical polyhedral embeddings of this seven vertex torus of Möbius.

In this video we see another embedding as a YouTube video: https://youtu.be/LGUyT6xfTFs We find all four symmetric embeddings of this neighborly seven vertex torus also in [37]. The best version might be a video from 1986 that can be downloaded from http://science-to-touch.com/ForJB/MoebiusTorus.mov.

Figure 28. A symmetric Möbius torus embedding of Bokowski and Eggert [9] as a 3D-print, https://youtu.be/6GhtRzemOwU

Figure 28. A symmetric Möbius torus embedding of Bokowski and Eggert [9] as a 3D-print, https://youtu.be/6GhtRzemOwU

6.3. The 59 Examples of the Complete Graph with 12 Vertices

The 59 combinatorial different examples of candidates for a triangular embedding with the complete graph with 12 vertices have been published in 1994, [12]. These 59 surfaces can be drawn topologically on this ceramic model in Figure 26. It is a shape that is topologically a sphere with six handles.

Figure 29. A genus 6 surface for all 59 topological embeddings of the complete graph with 12 vertices. Even finding just one such embedding is a challenge.

Figure 29. A genus 6 surface for all 59 topological embeddings of the complete graph with 12 vertices. Even finding just one such embedding is a challenge.

A crucial first proof that an oriented closed triangular 2-manifold does not allow an embedding in 3-space was provided by Bokowski and Guedes de Oliveira in 2000, [3]. The theory of oriented matroids helped decisively: The set of possible oriented matroids that are not forbidden because of edge-face intersections turned out (after long computations) to be the empty set. Several models of the corresponding topological embedding of the corresponding surface have been produced by Bokowski. See: Die Geschichte eines Modells, in [37], p. 88ff.

For the topological embedding of this surface we have the attempt to visualize it in Figure 27.

Figure 30. Here you see the model for the topological shape of an example surface from the 59 surfaces that was used in [3]. Some topological triangles are drawn. The others are clear when you go along adjacent topological edges and take as a third edge the one connecting the other endpoints. A membrane with these three edges forms a triangle of the surface.

Figure 30. Here you see the model for the topological shape of an example surface from the 59 surfaces that was used in [3]. Some topological triangles are drawn. The others are clear when you go along adjacent topological edges and take as a third edge the one connecting the other endpoints. A membrane with these three edges forms a triangle of the surface.

6.4. Neighborly Spatial Pseudo-Manifolds with 9 and 10 Vertices

We have additional spatial polyhedra without diagonals with 9 vertices in [12]. In Figure 28 we see an example of this paper.

Figure 31. A neighborly pseudomanifold with 9 vertices. Computer graphics and model.

Figure 31. A neighborly pseudomanifold with 9 vertices. Computer graphics and model.

We have additional spatial polyhedra without diagonals with 10 vertices in [13]. Among these examples are embeddings of four pinched spheres. The polyhedra in all these cases are orientable neighborly 2-pseudomanifolds. There are several circular sequences of triangles around a vertex.

7. The Dual Case of the Former Section

7.1. The Tetrahedron

The tetrahedron has also the face-sharing property. It is the only dual case with convex faces.

7.2. Szilassi’s Polyhedron

The combinatorial property of Szilassi’s polyhedron is dual to the seven vertex torus of Möbius with non-convex faces.

Figure 32. Szilassi’s polyhedron with seven hexagons and the face-sharing property

Figure 32. Szilassi’s polyhedron with seven hexagons and the face-sharing property

That this embedding of Szilassi is essentially unique, was shown via oriented matroid techniques in [11].

7.3. The 59 Examples of the Complete Graph with 12 Vertices Used for Its 59 Duals

No embeddings of the 59 dual abstract polyhedra with 12 eleven-gons and 44 vertices was found with methods of the second author. Therefore a Kepler-Poinsot version of an embedding with a low number of intersections is of interest. Such a polyhedron has been depicted in Figure 30. The two orthogonal projections along the x- ,y- and z-axis are shown in the four columns. The lower part shows an exploded view of all 12 eleven-gons.

Figure 33. A Kepler-Poinsot model with only 2 crossed polygons. Back, front, side, top, and faces to construct the polyhedron.

Figure 33. A Kepler-Poinsot model with only 2 crossed polygons. Back, front, side, top, and faces to construct the polyhedron.

8. Conclusions

Compared with former progress in finding polyhedral embeddings of regular maps, these new results provide a huge step forward. Finding the maximal order of symmetry is in many cases still open. So far we have proofs for non-embeddable cases when the number of vertices is 12. For the open case R13.2, we have 24 vertices. We cannot hope that this open case is easy to tackle. What can still be done is the non-triangular case of regular maps. Whether we can still find an additional regular Leonardo polyhedron remains an interesting open problem.

The paper of A. Altshuler and U. Brehm [38] has additional neighborly pseudo-manifolds with 11 vertices. An investigation of their polyhedral embeddings is still open.

We do not have a neighborly spatial polyhedron according to a complete graph embedding with 12 vertices on a 2-manifold, however, we have seen the attempt to show such a topological embedding in Figure 27. Another topological embedding for the example with the highest symmetry has been constructed by Carlo Séquin and Ling Xiao, see Figure 31 and the article [39]. When you insert in each topological triangle a vertex and you construct topological eleven-gons around former vertices, you obtain a dual topological embedding.

Figure 34. A topological embedding of the complete graph with 12 vertices on a surface of genus 6 with the highest symmetry by Carlo Séquin and Ling Xiao.

Figure 34. A topological embedding of the complete graph with 12 vertices on a surface of genus 6 with the highest symmetry by Carlo Séquin and Ling Xiao.