HyperRogue is a turn-based roguelike game that most notably takes place on the hyperbolic plane, which introduces a few gameplay mechanics unique to that type of space. It uses a board that is the truncated order-7 triangular tiling, a tiling of the hyperbolic plane composed of hexagons and heptagons. There is also a land in the game, the Warped Coast, that changes the board from its truncated form to its rectified form, known as the triheptagonal tiling, composed of triangles and heptagons in a rectangular vertex figure.
As a game taking place in a weird space, one can wonder: what would it be like if it were higher dimensional? More specifically, what higher dimensional tesselations have the properties HyperRogue's tiling has?
Definition
HyperRogue-like tesselations can be primarily defined as convex uniform compact hyperbolic tesselations with a simplicial vertex figure. This will be known as the
Another requirement for HyperRogue-like tesselations is that they must have a
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Note that this article will only concern itself with finding tesselations within Wythoffian symmetries, i.e., Coxeter symmetries with simplicial fundamental domains, also known as the Lannér symmetries. This is mainly because we have not discovered all compact hyperbolic Coxeter symmetries have been discovered, which is not the case with Wythoffian symmetries.[1] I also do not know how to interpret the non-simplicial fundamental domain diagrams.
Two dimensional
For completeness, we will also analyze what other tilings are HyperRogue-like. It turns out any regular hyperbolic tilings works for this; with the default tesselation being its truncation and the warped variant being its rectification. This is consistent with what HyperRogue uses, with the base tiling being the order-7 triangular tiling {3,7}.
Three dimensional
There are nine symmetries we can look into for valid honeycombs: the three regular symmetries, the one branched diagram symmetry and the five cyclic symmetries.
Regular & branched symmetries
We'll first look into the regular symmetries, being the symmetries of the order-5 cubic honeycomb, the order-4 dodecahedral honeycomb, both [5,3,4], the order-5 dodecahedral honeycomb [5,3,5] and the icosahedral honeycomb [3,5,3]. The one branched diagram symmetry, [5,31,1], which is derived from [5,3,4] by its halving, does not have any valid default honeycombs.
Accounting for the aesthetic preference, the valid default honeycombs of those symmetries are their bitruncates (cubidodecahedral honeycomb, disdodecahedral honeycomb & disicosahedral honeycomb), the omnitruncates of [5,3,5] and [3,5,3] (great prismatodisdodecahedral honeycomb & great prismatodisicosahedral honeycomb) and the truncated icosahedral honeycomb.
As for their warped variants, only the two omnitruncates admit valid warped variants, being the runcinate of their symmetry group (small prismatodisdodecahedral honeycomb & small prismatodisicosahedral honeycomb). In general, the expansion (in this case, runcination) of a polytope is always 2-colorable.
Cyclic symmetries
Despite being more difficult to comprehend, the cyclic symmetries are simpler to analyze. They are the ones with cyclic Coxeter diagrams: [(4,3,3,3)], [(5,3,3,3)], [(4,3,4,3)] and [(5,3,4,3)]. The only valid default honeycomb within these symmetries are their omnitruncates. As for their warped variants, the possibilities are the two of their cyclotruncates (two ringed nodes that are adjacent) of each symmetry that preserve the symmetry of its diagram.
Four dimensional
The diagram shapes of compact symmetries are similar to those in 3D; there are five symmetries we can look into for valid fourcombs, the three regular symmetries, the one branched diagram symmetry and the one cyclic symmetry.
Regular & branched symmetries
It turns out that due to the increase in the facet type limit, there are many valid default fourcombs that fit the aesthetic preference. But there's a trick we can use to additionally trim them by the existence of a valid warped variant: the only 2-colorable fourcomb within these symmetries is the sterication (expansion) of that symmetry, and due to the aesthetic requirements, only the self-dual symmetries have an expansion with three teron types. Thus, the only valid default fourcomb is the great prismatodishecatonicosachoral fourcomb, the omnitruncate of [5,3,3,3,5], with its warped variant being the small prismatodishecatonicosachoral fourcomb.
And just like the 3D branched symmetry, the one 4D branched symmetry, [5,3,31,1], does not yield any valid default fourcombs.
Cyclic symmetry
There is one compact hyperbolic cyclic symmetry in 4D, [(4,3,3,3,3)]. Unlike the 3D cyclic symmetries, which have twice as many warped variants for the four symmetries, the one 4D cyclic symmetry has only one warped variant. The default tesselation is, again, the omnitruncate: x4x3x3x3x3*a, and the warped variant is, again, a cyclotruncate: x4x3o3o3o3*a.
Five dimensional and beyond
Past 4D, there are no compact hyperbolic Wythoffian symmetries. There actually turn out to be compact hyperbolic Coxeter symmetries past this point, with provably none existing past 29D. The highest dimensional symmetries found are two 7D ones and one 8D symmetry, discovered by V. O. Bugaenko.[2][3]
Full list
Three dimensions
Images of these can be found on the Wikipedia page Uniform honeycombs in hyperbolic space.
| Default tesselation | Warped variant | ||
|---|---|---|---|
| Coxeter diagram | Name | Coxeter diagram | Name |
| x3x5x3x | Great prismatodisicosahedral honeycomb | x3o5o3x | Small prismatodisicosahedral honeycomb |
| x5x3x5x | Great prismatodisdodecahedral honeycomb | x5o3o5x | Small prismatodisdodecahedral honeycomb |
| x4x3x3x3*a | Cyclomnitruncated cyclotetracubihedral honeycomb | x4x3o3o3*a | Cyclotruncated cyclocubitetrahedral honeycomb |
| o4o3x3x3*a | Cyclotruncated cyclotetroctahedral honeycomb | ||
| x5x3x3x3*a | Cyclomnitruncated cyclotetradodecahedral | x5x3o3o3*a | Cyclotruncated cyclododecatetrahedral honeycomb |
| o5o3x3x3*a | Cyclotruncated cyclotetricosahedral honeycomb | ||
| x4x3x4x3*a | Cyclomnitruncated cyclocubic honeycomb | x4x3o4o3*a | Cyclotruncated cyclocuboctahedral honeycomb |
| o4x3x4o3*a | Cyclotruncated cycloctacubic honeycomb | ||
| x5x3x4x3*a | Cyclomnitruncated cyclocubidodecahedral honeycomb | x5x3o4o3*a | Cyclotruncated cyclododecoctahedral honeycomb |
| o5o3x4x3*a | Cyclotruncated cyclicosicubic honeycomb | ||
| x5x3x5x3*a | Cyclomnitruncated cyclododecahedral honeycomb | x5x3o5o3*a | Cyclotruncated cyclododecicosahedral honeycomb |
| o5x3x5o3*a | Cyclotruncated cyclicosidodecahedral honeycomb | ||
Four dimensions
| Default tesselation | Warped variant | ||
|---|---|---|---|
| Coxeter diagram | Name | Coxeter diagram | Name |
| x5x3x3x5x | Great prismatodishecatonicosachoral fourcomb | x5o3o3o5x | Small prismatodishecatonicosachoral fourcomb |
| x4x3o3o3o3*a | Cyclomnitruncated cyclopentahexadecicositetrachoral fourcomb | x4x3x3x3x3*a | Cyclotruncated cyclotesseractipenticositetrachoral fourcomb |
References
- ↑ Felikson, Anna. Lannér diagrams: Coxeter diagrams of compact hyperbolic simplices.
- ↑ Felikson, Anna. Both known examples of compact Coxeter 7-polytopes (due to Bugaenko).
- ↑ Felikson, Anna. The unique known compact polytope in H8.