Toroids1

Stewart Toroid realisations of the Szilassi Toroid and beyond ...

Stewart in 1980 showed that it is also possible to colour a number of Stewart Toroids to produce the equivalent of the Szilassi toroid in that all faces of the same colour are contiguous and each of the seven colours is contiguous with the other six.

The image above is of the Stewart Toroid K₃ / Q₃T₃ coloured in such a fashion.

On pages 200 and 201 of 'Adventures Among the Toroids', Stewart presents, under 'Map Colouring Problems', two interesting toroids closely related to this topic. He presents edge adjacent colourings for genus 2 and genus 3 toroids, but did not consider the possibility of planar representations. It is interesting to see that the reduced face sets are heterogenous.

"Map on a p=2 toroid T₅ / 2R₅(D₅)S₅Q₅ requiring eight colours" (link to OFF file). The reduced face set here consists of four 7-gons and four 8-gons. There are two doubled edges in two independent pairs of 8-gons and both have an edge distance of 2.

"Map on a p=3 toroid K₅ / 3R₅(E₅)S₅Q₅ requiring nine colours" (link to OFF file). The reduced face set here consists of three 8-gons and six 9-gons. There are three doubled edges, in three independent pairs of 9-gons and all have an edge distance of 2.

This arrangement could be extended to toroids of genus 4 (two 9-gons and eight 10-gons with four doubled edges) and genus 5 (one 10-gon and ten 11-gons with five doubled edges). Whether any of these would correspond to a planar toroid is an open question.

The OFF files for the genus 2 and 3 cases above were generated from stel files present on Alex Doskey's site at www.doskey.com/polyhedra (now sadly unavailable).

The Mizhaev genus 3 toroid

In attempting to produce an equivalent to Mizhaev's genus 3 toroid (above), some further topological considerations had to be taken into account.

1. Each 9-gonal face shares two distinct edges with two other faces.
2. Traversing the faces through these doubled edges results in a cycle which passes through all 8 faces.
3. For a pair of doubled edges, the polygons sharing that edge each have a specific number of edges between them (the 'edge distance').

Examination of the Mizhaev toroid reveals that the edge distances around the cycle are as follows [2, 2/3, 2, 2/3, 2, 2/3, 2, 2/3] where an entry of '2' means that both faces have an edge distance of 2, an entry of '2/3' means that one face has an edge distance of 2 and the other face an edge distance of 3. The Mizhaev toroid could then be described as Cycle: [8], Distances [2, 2/3, 2, 2/3, 2, 2/3, 2, 2/3], or in short form C[8] D[2, 2/3].
Using the Stewart Toroid K₅ / 4J₉₂(E₅), I was able to generate a colouring that precisely matched that of the Mizhaev toroid including the edge distance cycle above. (link to OFF file). From the net of the Stewart Toroid I then generated an indexed face set of eight enneagons by removing intermediate vertices where only two colours are present and then reindexing. Analysis of this reduced face set showed it to have a one-to-one correspondence (bijection) with the face set of the Mizhaev Toroid.

The Röst-Vigh Genus 3 Toroid

The equivalent description of the Röst-Vigh Genus 3 Toroid would be Cycle: [4, 4], Distances [2, 2, 2, 2], [2, 2, 2, 2] or in short form C[4, 4] D[2].

Again I was able to to generate a colouring that precisely matched this using the same method as was described above for the Mizhaev Toroid (link to OFF file)

The Mizhaev Genus 8 toroid

This has 8 independent tunnels. No Stewart toroid is known with more than six independent tunnels.

With each face having four collinear edges, which are not all shared with a single other face, the cycle notation breaks down. There appear to be three cycles of 4 and two cycles of 6 faces existing simultaneously. All edge distances between collinear edges are 2.

Are there more examples?

It is also possible to generate a number of colourings of K₅ / 4J₉₂(E₅) that meet the edge neighbourly criterion but have different cycles and distance arrays from the above. I have focused here on the more symmetric symbols as they would result in more symmetric planar toroids. The list of cases lists obtained is not complete but reflects a broad but non-exhaustive search. See notation below for details of the distance array strings:

- For the cycle [8] case C[8] D[2/3] and C[8] D[3, 2/3] were generated along with other less regular arrays.
- For the cycle [4,4] case C[4, 4] D[3] and C[4, 4] D[2, 3] were among those that can be obtained.
- Solutions with a cycle of [3,5] can be obtained with various distance arrays, none regular.
- If we allow faces to share three collinear edges they must necessarily form a cycle of 2. It was possible to form solutions of the form C [2, 2, 2, 2] D[2] ; C [2, 2, 4] D[2] ; and C [2, 6] D[2],
- The only case where no candidates have been generated is cycle [2, 3, 3].

It was also possible to generate solutions where edge distances of 1 occurred. While these are possible in the Stewart Toroid realisations these cannot be formed into toroids with planar faces as the collinear edges and the one edge between them would form one degenerate edge. This demonstrates that a solution to the Stewart Toroid is not necessarily realisable as a planar toroid.

Dismissing the edge distance 1 examples, a total of 15 configurations have been generated to date in addition to the Mizhaev and Röst-Vigh solutions. Whether some of these other cycle arrays and distance length arrays can be realised as toroids with planar faces remains an open question. As the Mizhaev and Röst-Vigh toroids have only been discovered very recently then perhaps there are more.

What about the genus 4 case?

The next toroid in this sequence would be an as yet undiscovered genus 4 toroid with nine 10-gonal faces and nine doubled edges.

For the genus 3 case, the two known solutions may be used as a guide toward the expected symmetry of any further solutions. With 8 faces an octahedral/tetrahedral basis appears favoured with holes making a tetrahedral core with four openings. The four copies of J₉₂ in the Stewart Toroid above were deliberately placed in a tetrahedral symmetry to reflect this. It should be noted though that the genus 1 Szilassi toroid only has C2 (cyclic 2) symmetry so the discovery of two genus 3 cases with higher symmetries could be an artefact of the search methods used.

The genus 4 case is less straight-forward. There is no guidance from known results. Nine faces does not give an obvious symmetry beyond C3 (cyclic 3). If the holes lead to a central core then 5 of them are required. Two configurations were explored
1. K₅ / 5J₉₂(E₅). (link to OFF file) The overall symmetry was given precedence. Three copies of J₉₂ were placed in a C3 symmetry and the remaining two at the poles, somewhat like the five faces of a triangular prism.
2. K₅ / 5Q₅S₅(E₅). (link to OFF file) A symmetric arrangement of the holes was given precedence. The five copies of Q₅S₅ were placed in a C5 (cyclic 5) symmetry.

A similar search was then conducted to the above, again excluding edge distance 1 cases, the lists obtained are not complete but reflect a broad but non-exhaustive search:

The case 1 run has resulted in more than 290 topologically distinct candidates. The potentially symmetric cases are:

- C[9] D[2/3]
- C[3, 3, 3] D[2/3]
- C[3, 6] D[2] [2, 3]

Other interesting looking cases include:

-C[2, 7] D[2] ; C[2, 2, 5] D[2] and C[2, 2, 2, 3] D[2].
- C[2, 3, 4] D[2] [3] [2, 3]
-C[9] D[2⁸, 3]
-C[9] D[2, 2/3⁸]

The case 2 run also resulted in more than 140 topologically distinct candidates. As expected, there is no candidate topology showing any high degree of symmetry, the most interesting looking being:

-- C[2, 3, 4] D[2] [3] [2, 3]
- C[3, 3, 3],D[3] [3, 2/3, 2/3] [3, 2/3, 2/3]
- C[3, 6] D[3] [2, 3, 2, 3, 2, 4]
- C[9] D[2³, 2/3⁶]
-C[9] D[2, 2/3⁸] (again)

Whether the differing lists of cases obtained is a reflection of the differing symmetries of the Johnson Solids, or whether it is merely an artefact of incomplete lists is an open question. The difficulty in generating case 2 examples in comparison to case 1 examples implies the former.

Notation

Each face normally shares two doubled edges with two other faces.    Following the path of these doubled edges leads to cycles of polygons. In some cases one cycle covers the entire polyhedron. In other cases there is more than one distinct cycle. The list of face cycles is denoted by C[X, Y, Z] where X, Y and Z denote the cycle lengths. It is also possible for a polygon to share three edges with one other, in which case the cycle length in necessarily 2 given the class of polyhedra under consideration.

Where two faces share a doubled edge, it becomes necessary to count the number of edges in each polygon in between those edges. This is termed the 'edge distance'. It need not have the same value for the two polygons. The edge distance for each cycle is denoted by the string D[L, M, N ...] where there is one entry for each polygon in the cycle. Where the edge distance between the polygons is different, this is denoted by P/Q where P and Q are the respective edge distances (ordered such that P < Q).   For cycles of length 2, where the polygons have three common edges, the distance is denoted by [M;N] where M is the edge distance between common edges one and two, and N is the edge distance between common edges two and three.

The edge distance strings are alwasys sequenced to their lexigraphically lowest value without losing the cyclic order.

For brevity, the edge distances can be shortened where this does not introduce any uncertainty. D[2, 2, 2, 2] can be denoted as D[2] as can D[2; 2]. D[2, 3, 2, 3] can be denoted as D[2, 3]. Further compression is possible if all edge distances are the same so C[2, 4] D[2;2] [2, 2, 2, 2] can be compressed to C[2, 4] D[2] [2] and further to C[2, 4] D[2].

Repeat values can be denoted with exponents D[2, 3, 3, 3] can be represented as D[2, 3³].   Notation such as D[2, 2/3³] should be read as D[2, 2/3, 2/3, 2/3]. The C arrays are generally short enough so as to not need compacting.

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