Polyhedral Embeddings of Triangular Regular Maps of Genus g, 2 ⩽ g ⩽ 14, and Neighborly Spatial Polyhedra ^†

Abstract

This article provides a survey of polyhedral embeddings of triangular regular maps of genus g, $2⩽g⩽14$, and of neighborly spatial polyhedra. An old conjecture of Grünbaum from 1967, although disproved in 2000, lies behind this investigation. We discuss all duals of these polyhedra as well, whereby we accept, e.g., the Szilassi torus with its non-convex faces to be a dual of the Möbius torus. A numerical optimization approach by the second author for finding such embeddings was first applied to finding (unsuccessfully) a dual polyhedron of one of the 59 closed oriented surfaces with the complete graph of 12 vertices as their edge graph. The same method has been successfully applied for finding polyhedral embeddings of triangular regular maps of genus g, $2⩽g⩽14$. The effectiveness of the new method has led to ten additional new polyhedral embeddings of triangular regular maps and their duals. There do exist symmetrical polyhedral embeddings of all triangular regular maps with genus g, $2⩽g⩽14$, except in a single undecided case of genus 13. Among these results, there are three new Leonardo polyhedra, each with 156 vertices, 546 edges, and 364 triangular faces, based on the Hurwitz triplet of genus 14 with Conder notation R14.1, R14.2, and R14.3.

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], p. 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 counterexample was only first given in 2000 [3], and additional counterexamples were 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. However, for polyhedra with a higher genus, there are instances where the dual embeddings require non-convex faces. A famous example for such an embedding is Szilassi’s polyhedron with a dual face lattice compared with the seven-vertex torus of Möbius. We do allow non-convex faces when we use duality in this paper.

The use of the second author’s numerical optimization approach has led to all but one embedding 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. Triangular regular maps of genus g, $2⩽g⩽14$, and their duals with polyhedral embeddings. Embeddings marked with a * were previously known; all others are new.

Conder Notation	Genus	Schläfli Type	${\mathit{f}}_0$	${\mathit{f}}_1$	${\mathit{f}}_2$	Map Author	Comb. Sym.	Embedding Symmetries	Dual Embedding	Figure
R3.1	3	${\{3,7\}}_8$	24	84	56	Klein	336 PSL(2,7) × $C_2$	T *, D3	T *, D3	Section 5.1
R3.2	3	${\{3,8\}}_6$	12	48	32	Dyck	192	D3 *, D2, S2		Section 5.3
R5.1	5	${\{3,8\}}_{12}$	24	96	64	Fricke, Klein	384	O *, S2	D2, C3	Section 5.4
R6.1	6	${\{3,10\}}_6$	15	75	50	Coxeter, Moser	300	C3, C2, C1 *		Section 5.6
R7.1	7	${\{3,7\}}_{18}$	72	252	168	Hurwitz, Macbeath	1008 PSL(2,8) × $C_2$	D3, D2, S2, C1 *	D3, D2	Section 5.7
R8.1	8	${\{3,8\}}_8$	42	168	112		672 PSL(3,2) ⋊ $C_2$	D4, D3, S4, S2		Section 5.9
R8.2	8	${\{3,8\}}_{14}$	42	168	112		672 PSL(3,2) ⋊ $C_2$	D4, D3, S4		Section 5.10
R10.1	10	${\{3,9\}}_{12}$	36	162	108		648	D2, S4		Section 5.11
R10.2	10	${\{3,12\}}_6$	18	108	72		432	C2		Section 5.12
R13.1	13	${\{3,10\}}_{30}$	36	180	120		720 $A_5\times S_3$	C3, C2		Section 5.13
R13.2	13	${\{3,12\}}_{12}$	24	144	96		576			Section 5.14
R14.1	14	${\{3,7\}}_{12}$	156	546	364		2184 PSL(2,13)	T, D3, S4	T	Section 5.15
R14.2	14	${\{3,7\}}_{26}$	156	546	364		2184 PSL(2,13)	T	T	Section 5.17
R14.3	14	${\{3,7\}}_{14}$	156	546	364		2184 PSL(2,13)	T, D3, S4	T	Section 5.19

. For only five of them, polyhedral embeddings were known. For the remaining cases, we now have 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 an error in a previous paper [6].

The geometrical symmetry of three new polyhedral embeddings based on the Hurwitz triplet of genus 14 are chiral tetrahedral, telling us that they are all solutions to a long-standing question of Jörg M. Wills that was posed again in [7,8]. These polyhedral embeddings are new Leonardo polyhedra, each with 156 vertices, 546 edges, and 364 triangular faces, based on the regular maps with Conder notation R14.1, R14.2, and R14.3.

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.

Afterward, we describe the algorithm of the second author.

2. Polyhedral Embeddings of Triangulated Orientable Regular Maps with Genus g, $\mathbf{2}⩽\mathit{g}⩽\mathbf{14}$, 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. Geometric symmetries are listed with Schoenflies notation. All triangular polyhedral embeddings in Table 1 are new for genus g, $g⩾8$. The embeddings of genera 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.

The Appendix A contains coordinates for the cases with the highest symmetry order. Additional embeddings by the second author, including all available symmetries and regular maps of higher genus, can be interactively viewed and downloaded at http://codeparade.net/embeddings (accessed on 20 March 2025).

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

In

Table 2. Neighborly spatial polyhedra according to complete graph embeddings on 2-manifolds and pseudo-manifolds.

Graph	Genus	${\mathit{f}}_0$	${\mathit{f}}_1$	${\mathit{f}}_2$	Number of Embeddings	Combinatorial Polyhedra, Articles	Geometrical Embeddings, Articles
$K_4$	0	4	6	4	1
$K_7$	1	7	21	14	4	[9]	[10,11]
$K_7$ dual	1	14	21	7	1		[12,13]
$K_9$	–	9	36	24	16	[14]	[14]
$K_{10}$	–	10	45	30	4	[15]	[15]
$K_{12}$	6	12	66	44	none	[16]	[3,4]
$K_{15}$	11	15	105	70	unknown	[17]

, 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 (accessed on 20 March 2025).

The investigation of the second author to study the dual cases, which have been investigated by Bokowski, Guedes de Oliveira, and Schewe, did not find additional embeddings in this area.

4. Polyhedral Embeddings as an Optimization Problem

Finding a polyhedral 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 three-dimensional vertices and triangulation with no additional constraints. This greatly simplifies the optimization. By contrast, faces with polygons of more than three 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 a vertex adjacency list. Since the edge graph is cubic, the vertices of the dual are simply the intersection of the three 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 if a regular map is embeddable, it will also be embeddable with at least one non-trivial geometric symmetry.

Possible geometric symmetries can be inferred from the automorphisms of the regular map. Some examples are in

Table 3. Geometric symmetries from permutation groups.

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, S4
(a, b, c, d) (e, f)	S4
(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 two 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.

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

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

The general algorithm is listed in Algorithm 1.

Algorithm 1 Primary search for embeddings

Input: List of index triplets representing the triangles T  1: Procedure SearchEmbeddings (T, $iters$, $clusters$, $\sigma$ = 1.0, $\beta$ = 0.997, $\gamma$ = 1.25)  2: for i in $clusters$ do  3:    $V_i$ ← ApplySymmetry(RandomVertices())  4:    $p_i$ ← Penalty(T,Vi)  5: end for  6: for j in $iters$ do  7:    $m$ ← ArgMinimum(p)  8:    for i in $clusters$ do  9:      if $p_i>p_m\ast \gamma$ then 10:         $r=$ RandomIndex() 11:         $V_i\leftarrow V_r$ 12:         $p_i\leftarrow p_r$ 13:      end if 14:      $V_{new}\leftarrow \beta \ast (V_i+\mathrm{R}\mathrm{A}\mathrm{N}\mathrm{D}\mathrm{O}\mathrm{M}\mathrm{N}\mathrm{O}\mathrm{I}\mathrm{S}\mathrm{E}\left(\sigma \right))$ 15:      $V_{new}\leftarrow \mathrm{A}\mathrm{P}\mathrm{P}\mathrm{L}\mathrm{Y}\mathrm{S}\mathrm{Y}\mathrm{M}\mathrm{M}\mathrm{E}\mathrm{T}\mathrm{R}\mathrm{Y}\left(V_{new}\right)$ 16:      $p_{new}\leftarrow \mathrm{P}\mathrm{E}\mathrm{N}\mathrm{A}\mathrm{L}\mathrm{T}\mathrm{Y}(T,V_{new})$ 17:      if $p_{new}⩽p_i \mathbf{or} (i\ne m \mathbf{and} p_{new}⩽p_m\ast \gamma )$ 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, the best results are Kepler–Poinsot-like with a low intersection count. R13.2 has been realized with as few as 64 edge intersections with S4 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 $\{3,7\}$ [18]. Its abstract symmetry group has an order of 336. References are [19,20]. Polyhedral embeddings are shown in Figure 2.

The corresponding polyhedron 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 [21]. For additional articles about regular Leonardo polyhedra, see [7,8,22,23,24,25]. We also provide a section describing new Leonardo polyhedra in Section 6. Figure 3 shows the regular map R3.1 on a hyperbolic disk.

5.2. The Dual Case R3.1′

An embedding of the dual polyhedron of R3.1 with chiral tetrahedral symmetry is due to D. McCooey [26]. Polyhedral embeddings are shown in Figure 4.

5.3. Case R3.2

The regular map R3.2 is due to W. Dyck [27,28] with Schläfli type ${\{3,8\}}_6$ and genus 3. The combinatorial symmetry group has order 192. Polyhedral embeddings are shown in Figure 5.

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. For more details about these embeddings, see Figure 6, Figure 7, Figure 8 and Figure 9.

5.4. Case R5.1

The regular map R5.1, due to Klein and Fricke [33], has type ${\{3,8\}}_{12}$ and genus 5 with symmetry group of order 384. Polyhedral embeddings are shown in Figure 10.

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

5.5. The Dual Case R5.1′

The embedding of the dual is a new result of the second author in this paper. Polyhedral embeddings are shown in Figure 11.

5.6. Case R6.1

The regular map R6.1, due to Coxeter and Moser, has type ${\{3,10\}}_6$ and genus 6 with symmetry group of order 300. Polyhedral embeddings are shown in Figure 12.

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.

5.7. Case R7.1

The regular map R7.1 is due to Hurwitz and has type ${\{3,7\}}_{18}$ and genus 7. It is the second element of the infinite Hurwitz sequence with types $\{3,7\}$, also denoted as a Macbeath surface, as she rediscovered it, with symmetry group of order 1008. Polyhedral embeddings are shown in Figure 13. A first embedding without geometrical symmetries was found in 2018 by Bokowski and Cuntz [35]. The symmetry of this new embedding has order 6, although we find in [6] the claim that such a symmetry is not possible. In other words, this example tells us that an error in that paper has been detected. For more details about these embeddings, see Figure 14, Figure 15 and Figure 16.

5.8. The Dual Case R7.1′

Polyhedral embeddings are shown in Figure 17. Additional internal structures are isolated in Figure 18.

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. This regular map has 42 vertices and 112 faces. Polyhedral embeddings are shown in Figure 19.

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. Polyhedral embeddings are shown in Figure 20.

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. Polyhedral embeddings are shown in Figure 21.

5.12. Case R10.2

The regular map R10.2 has type ${\{3,12\}}_6$ and genus 10 with symmetry group of order 432. Polyhedral embeddings are shown in Figure 22. A combinatorial description of the map is illustrated in Figure 23.

5.13. Case R13.1

The regular map R13.1 has type ${\{3,10\}}_{30}$ and genus 13 with symmetry group of order 720. Polyhedral embeddings are shown in Figure 24.

5.14. Case R13.2

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

After extensive search, we found that it is the only triangular regular map of genus g, $2⩽g⩽14$, that is not likely to be embeddable in ${\mathbb{R}}^3$.

5.15. Case R14.1

The regular map R14.1, due to Hurwitz, has type ${\{3,7\}}_{12}$ and genus 14 with symmetry group of order 2184. Polyhedral embeddings are shown in Figure 25.

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. The polyhedral embeddings of the three regular maps from the Hurwitz triplet are all new examples of regular Leonardo polyhedra due to their chiral tetrahedral geometric symmetry.

5.16. The Dual Case R14.1′

Polyhedral embeddings are shown in Figure 26.

5.17. Case R14.2

The regular map R14.2, due to Hurwitz, has type {3,7}₂₆ and genus 14 with symmetry group of order 2184. Polyhedral embeddings are shown in Figure 27.

5.18. The Dual Case R14.2′

Polyhedral embeddings are shown in Figure 28.

5.19. Case R14.3

The regular map R14.3, due to Hurwitz, has type {3,7}₁₄ and genus 14 with symmetry group of order 2184. Polyhedral embeddings are shown in Figure 29.

5.20. The Dual Case R14.3′

Polyhedral embeddings are shown in Figure 30.

6. Three New Leonardo Polyhedra and Understandable Spatial Representation

So far, only six Leonardo Polyhedra were known. You can find them in [7,8] and in the references cited there. Two of them have triangular faces, so they are in Table 1 with Conder notation R3.1 and R5.1.

In Figure 31, we can see a combinatorial description and a colored explanation of the Leonardo polyhedron of R3.1 in a window of the Blender software. We have studied all new polyhedra with Blender. If the reader is interested in viewing the polyhedra of Table 1, it is possible with some basic explanations of Blender to see them rotating on the screen. You need only to download the free Blender software and the corresponding .obj files that are provided at http://codeparade.net/embeddings (accessed on 20 March 2025).

The R3.1 example based on Klein’s quadric with the polyhedral embedding of Schulte and Wills and the second known triangular Leonardo polyhedron of Grünbaum based on the Fricke–Klein regular map R5.1 have geometries that are relatively easy to describe.

For the second example, you start with six small squares at the midpoints of the faces of a cube. All small squares are rotated symmetrically by a small angle. Now, connect these six small squares with triangles in two ways such that it forms two topological spheres. When the small squares are then removed, you obtain the closed triangular 2-manifold of R5.1. See the description of this Leonardo polyhedron in Figure 32.

Can we explain simply the shape of these three amazing new Leonardo polyhedra? They all have 156 vertices, 546 edges, and 364 triangular faces! The inner and outer parts of the regular map R14.2 can be seen in Figure 33; however, the Leonardo polyhedron is the union of all seven of these parts. We encourage the reader to view and rotate these Leonardo polyhedra in three dimensions to fully admire them. The regular maps with maximal spatial geometric symmetries as generalizations of Platonic solids now provide at least nine Leonardo polyhedra, i.e., polyhedral embeddings that have again the rotational geometric symmetry of a Platonic solid.

7. 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 = 2), where a polyhedron need not be convex or have convex faces 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 topological embeddings have played a significant role in results like the map color theorem [17].

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].

Computing combinatorially possible embeddings is a second step before we discuss possible polyhedral embeddings.

We find in this section the solution to a long-standing conjecture of Branco Grünbaum, which is that there does exist a cell decomposition of a triangulated orientable surface that cannot have an embedding in $R^3$.

7.1. The Tetrahedron

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

7.2. The Seven Vertex Torus of Möbius

For the seven vertex torus of Möbius in his collected works [9], Császár in [10] 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 [11] found via oriented matroid techniques additional three symmetrical polyhedral embeddings of this seven vertex torus of Möbius. One such embedding is shown in Figure 34.

In this video, we see another embedding as a YouTube video: https://youtu.be/LGUyT6xfTFs (accessed on 20 March 2025). We also find all four symmetric embeddings of this neighborly seven-vertex torus in [37]. The best version might be a video from 1986 that can be downloaded from http://science-to-touch.com/ForJB/MoebiusTorus.mov (accessed on 20 March 2025).

7.3. 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 [14]. These 59 surfaces can be drawn topologically on this ceramic model in Figure 35. It is a shape that is topologically a sphere with six handles.

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 12 vertex neighborly 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.

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

We have additional spatial polyhedra without diagonals with nine vertices in [14]. In Figure 36, we see an example from this paper.

We have additional spatial polyhedra without diagonals with 10 vertices in [15]. 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.

8. The Dual Case of the Former Section

8.1. The Tetrahedron

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

8.2. Szilassi’s Polyhedron

The combinatorial property of Szilassi’s polyhedron is dual to the seven-vertex torus of Möbius with non-convex faces and can be seen in Figure 37. That this embedding of Szilassi is essentially unique was shown via oriented matroid techniques in [13].

8.3. 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 were 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 38. The two orthogonal projections along the x-, y-, and z-axes are shown in the four columns. The lower part shows an exploded view of all 12 eleven-gons.

9. 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 the non-embeddable cases of the 59 neighborly polyhedra with 12 vertices. 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 carried out 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 by 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 find a dual embedding in Figure 38. Another topological embedding for the example with the highest symmetry has been constructed by Carlo Séquin and Ling Xiao; see Figure 39 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.