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Hyperbolic Tesselations



Tesselations based on simplicial domains

Both, spherical space tesselations (aka polytopes) and Euclidean space tesselations, base their reflection symmetry groups on simplicial fundamental domains. In 2D spherical geometry those are known as Schwarz triangles, in 3D they are called Goursat tetrahedra. In Euclidean space this restriction was relaxed in so far as parallel mirrors would be allowed, i.e. ones which don't intersect anymore. The submultiplicative number of the dihedral angle (i.e. the link mark) in such cases was set to ∞. This results in an offshore vertex of the former simplicial fundamental domain, drifted far away to infinity. But that's all what can happen. Now taking over this view onto hyperbolic space, already gives lots of stuff to deal with. – Even though, it should be noted, that this still is not the end of the story, there are other possibilities too.

Besides from the shape and the size of the fundamental domain, for instance its extend to infinity ("cusps"), also the extend of the tiles can be considered. For Euclidean tilings, honeycombs, etc. there are 2 classes, one using only finite tiles (polytopes of spherical geometry) but building up complexes which nonetheless fill all of Euclidean space, or alternatively those other ones, which use additionally infinitely extended tiles, i.e. euclidean tilings, honeycombs, etc. from one dimension less as building blocks within the next dimension. In hyperbolic space, there even is one case more:

  1. compact:   hyperbolic tesselations using finite (spherical polytopial) tiles only
  2. paracompact:   hyperbolic tesselations which additionally (to class 1) include lower dimensional euclidean tesselations as infinite tiles
  3. hypercompact:   hyperbolic tesselations which additionally (to either class 1 or 2) include lower dimensional hyperbolic constituends

Finally, right from their definition, Dynkin symbols, both for the mere symmetry groups, and for the described polytopes or tesselations, are essentially based on that simplicial restriction and thus not versatile outside that very scope. (But attempts for more general Coxeter domains can be made too.)

A further class are laminates. These are built from infinite regions of a tesselation, which are made from compact tiles only. Those regions further are bounded by some infinite (pseudo) facetings, which serve as reflection, glide-reflection, etc. As such laminates belong to class 1. – Note that these facets might become real ones within class 3: There such laminar regions could occur between bollotiles (tiles of hyperbolic structure themselves). Clearly we should restrict here to cases, where these bollotiles do have exactly the same curvature as the whole tesselation. Because then those behave like hemi-facets of spherical space: One could consider the external, infinitely multiple self-blends of those tesselations, which would just blend out these bollotiles. That is, these serve as mirrors, each reflecting the laminar region. The such derived laminate accordingly is called the lamina-truncate of the un-blended structure. Both these notions were introduced by W.Krieger.

Just as for spherical geometry, the hyperbolic one has also a non-vanishing uniform curvature. Accordingly a circumradius here too is well-defined. Only that this quantity would provide purely imaginary values for hyperbolics. In fact, the formulas for radius derivation, based on a given Wythoffian Dynkin diagram (being implemented within the spreadsheet, which is provided at the download page), would work here exactly the same.



---- 2D Tilings (up) ----

In this dimension any Dynkin symbol of type oPoQo would be hyperbolic, whenever 1/P + 1/Q < 1/2 (or equivalently (P-2)(Q-2) > 4). (In fact ">" within the first formula would qualify spherical, and "=" would qualify euclidean.) For convex cases, i.e. integral line mark numbers, the single euclidean solution for a loop Dynkin symbol is o3o3o3*a; no spherical does exist. Anything beyond thus qualifies as a symmetry group of hyperbolic space. As long as finite line mark numbers are used only, for this dimension we remain within the above mentioned first class. – But even the general hyperbolic case for oPoQoR*a can be formalized by 1/P + 1/Q + 1/R < 1 for any rational P,Q,R (each >1), thereby extending the above formula to cases with R<>2 as well.

With respect to the node markings we will have exactly the same cases as given explicitly in that listing for the general Schwarz triangle oPoQoR*a (providing cases and their general incidence matrices; although there, in addition for each of those general incidence matrix cases, so far only links to spherical and euclidean space representants are provided).


Compact Tilings

(A nice applet for visualization of 2D hyperbolic tilings (as well as euclidean ones) is tyler. In case make sure to check "hyperbolic". In fact it was designed to work beyond triangular domains as well.)

Just to provide some examplifying symmetries ...
linear ones
o3o7o o3o8o ... o4o5o o4o6o ... o4o8o ...
x3o7o - hetrat
o3x7o - thet
o3o7x - heat
x3x7o - thetrat
x3o7x - sirthet
o3x7x - theat
x3x7x - grothet
x3o8o - otrat
o3x8o - toct
o3o8x - ocat
x3x8o - totrat
x3o8x - srotoct
o3x8x - tocat
x3x8x - grotoct
 
x4o5o - pesquat
o4x5o - tepet
o4o5x - peat
x4x5o - topesquat
x4o5x - srotepet
o4x5x - topeat
x4x5x - grotepet
x4o6o - hisquat
o4x6o - tehat
o4o6x - shexat
x4x6o - thisquat
x4o6x - srotehat
o4x6x - toshexat
x4x6x - grotehat
 
x4o8o - osquat
o4x8o - teoct
o4o8x - socat
x4x8o - ocat
x4o8x - sroteoct
o4x8x - tosocat
x4x8x - groteoct
 
s3s7s - snathet
o3o8s
x3o8s
o3x8s - toct
x3x8s - totrat

s3s8o
s3s8x - srotoct

s3s8s
 
s4o5o - pepat
s4x5o - tepet
s4o5x - topepat
s4x5x - topeat

o4s5s
x4s5s - srotepet

s4s5s
s4o6o - hihexat
s4o6x - thihexat
o4s6o
x4s6x - srotehat
o4o6s
x4o6s
...

s4s6o
s4s6x - srotehat
s4o6s
s4o6s'
o4s6s
x4s6s - srotehat
...

s4s6s
 
s4o8o - ococat
o4s8o
o4o8s - osquat
s4s8o
s4o8s
o4s8s
s4s8s
...
 
o5o5o ... o5o10o ... o6o6o ... o8o8o ...
x5o5o - pepat
o5x5o - peat
x5x5o - topepat
x5o5x - tepet
x5x5x - topeat
 
x5o10o - depat
o5x10o - pidect
o5o10x - pedecat
x5x10o - decat
x5o10x - sropdect
o5x10x - topdecat
x5x10x - gropdect
 
x6o6o - hihexat
o6x6o - shexat
x6x6o - thihexat
x6o6x - tehat
x6x6x - toshexat
 
x8o8o - ococat
o8x8o - socat
x8x8o - tococat
x8o8x - teoct
x8x8x - tosocat
 
s5s5s
 
o5o10s - depat
s5s10o
s5s10s
...
 
s6o6o
o6s6o
x6s6o
s6s6o
s6o6s
s6s6s
...
 
s8o8o
o8s8o - osquat
s8s8o
s8o8s
s8s8s
...
 
loop ones  
o3o3o4*a ... o3o4o4*a o3o4o5*a ... o4o4o4*a ...
x3o3o4*a
o3x3o4*a - otrat
x3x3o4*a
x3o3x4*a - toct
x3x3x4*a - totrat
 
x3o4o4*a
o3o4x4*a - hisquat
x3x4o4*a - tehat
x3o4x4*a
x3x4x4*a - thisquat
x3o4o5*a
o3x4o5*a
o3o4x5*a
x3x4o5*a
x3o4x5*a
o3x4x5*a
x3x4x5*a
 
x4o4o4*a - osquat
x4x4o4*a - teoct
x4x4x4*a - ocat
  
s3s3s4*a
 
o3o4s4*a - hihexat
x3o4s4*a
s3s4o4*a
s3s4s4*a
...
s3s4s5*a
 
s4o4o4*a - ococat
s4s4o4*a
s4s4s4*a
...
 

In contrast to the situation of euclidean space tilings for both the spherical and hyperbolical tilings the size of the tiles is fixed by the absolute geometry of the filled manifold, i.e. its curvature, and the to be used vertex figure. For instance, let P0 be a vertex of xPoQo, let P1 be the center of an adjacent edge, and P2 the center of an adjacent face (in fact a xPo), then the distances φ = P0P1, χ = P0P2, and ψ = P1P2 depend on the absolute geometry of oPoQo via

cosh(φ) = cos(π/P) / sin(π/Q)
cosh(χ) = cot(π/P) · cot(π/Q)
cosh(ψ) = cos(π/Q) / sin(π/P)

Therefore xPoQo itself can be described as a tiling with edge length 2φ, having Q P-gons at each vertex, and the P-gons will have a circumradius of χ and an inradius of ψ.

The only regular star-tesselations have the symmetries o-P-o-P/2-o, here P being an odd integer greater than 5. All those star-tesselations would have density 3. (The case P = 5 already describes the spherical space tesselation or polyhedron sissid respectively gad.) In fact, x-P/2-o-P-o are derived as stellations of xPo3o. Dually, the edge-skeletons of x-P-o-P/2-o and of x3oPo are the same.

x7/2o7o - sheat x7o7/2o - gheat

Paracompact Tilings

As there is just a single 1D euclidean space tiling, aze, the only wythoffian tilings of hyperbolic plane, which use euclidean tiles in addition to polygons, are based on the reflection groups oPoQoR*a, where still 1/P + 1/Q + 1/R < 1, but at least one of those link marks being infinite. Here aze then will be understood to describe an horocyclic tile (a.k.a. apeirogon).

linear ones loop ones
oPo∞o o∞o∞o oPoQo∞*a oPo∞o∞*a o∞o∞o∞*a
x3o∞o - aztrat
o3x∞o - tazt
o3o∞x - azat
x3x∞o - taztrat
x3o∞x - srotazt
o3x∞x - tazat
x3x∞x - grotazt

x4o∞o - asquat
o4x∞o - tezt
o4o∞x - squazat
x4x∞o - tasquat
x4o∞x - srotezt
o4x∞x - tosquazat
x4x∞x - grotezt

o6o∞x - hazat
...
x∞o∞o - azazat
o∞x∞o - squazat
x∞x∞o - azat
x∞o∞x - tezt
x∞x∞x - tosquazat
x3o3o∞*a
o3x3o∞*a - aztrat
x3x3o∞*a
x3o3x∞*a - tazt
x3x3x∞*a - taztrat

...
x4o4x∞*a - tezt
x4x4x∞*a - tasquat

...
x3o∞o∞*a
o3o∞x∞*a - hazat
x3x∞o∞*a
x3o∞x∞*a
x3x∞x∞*a

...
x∞o∞o∞*a - azazat
x∞x∞o∞*a - squazat
x∞x∞x∞*a - azat
o3o∞s
s3s∞o
s3s∞s

o4s∞s
...
s∞s∞s
s3s3s∞*a

...
s3s∞s∞*a

...
s∞s∞s∞*a

(For tilings with more general fundamental domains cf. Coxeter domains.)



---- 3D Honeycombs (up) ----

Compact Honeycombs

Here the restriction to finite tiles is much more effective, at least if being considered with respect to non-product honeycombs. For convex cases (integral line mark numbers) we only have the following 9 irreducible symmetry groups, resp. the therefrom derived listed Wythoffian hyperbolic honeycombs.

linear ones tri-dental ones
o3o5o3o o4o3o5o o5o3o5o o3o3o *b5o
x3o5o3o - ikhon
o3x5o3o - rih
x3x5o3o - tih
x3o5x3o - srih
x3o5o3x - spiddih
o3x5x3o - dih
x3x5x3o - grih
x3x5o3x - prih
x3x5x3x - gipiddih
x4o3o5o - pechon
o4x3o5o - ripech
o4o3x5o - riddoh
o4o3o5x - doehon
x4x3o5o - tipech
x4o3x5o - sripech
x4o3o5x - sidpicdoh
o4x3x5o - ciddoh
o4x3o5x - sriddoh
o4o3x5x - tiddoh
x4x3x5o - gripech
x4x3o5x - priddoh
x4o3x5x - pripech
o4x3x5x - griddoh
x4x3x5x - gidpicdoh
x5o3o5o - pedhon
o5x3o5o - ripped
x5x3o5o - tipped
x5o3x5o - sripped
x5o3o5x - spidded
o5x3x5o - diddoh
x5x3x5o - gripped
x5x3o5x - pripped
x5x3x5x - gipidded
x3o3o *b5o - apech
o3x3o *b5o - riddoh
o3o3o *b5x - doehon
x3x3o *b5o - tapech
x3o3x *b5o - ripech
x3o3o *b5x - birapech
o3x3o *b5x - tiddoh
x3x3x *b5o - ciddoh
x3x3o *b5x - bitapech
x3x3x *b5x - griddoh
s3s5s3s - snih *) 
...
s4o3o5o - apech
s4o3x5o - tapech
s4o3o5x - birapech
s4o3x5x - bitapech
...
o4x3o5β
...
β5o3x5o
...
...
loop ones
o3o3o3o4*a o3o4o3o4*a o3o3o3o5*a o3o4o3o5*a o3o5o3o5*a
x3o3o3o4*a - gadtatdic
o3x3o3o4*a
x3x3o3o4*a
x3o3x3o4*a
x3o3o3x4*a
o3x3x3o4*a
x3x3x3o4*a
x3x3o3x4*a
x3x3x3x4*a
x3o4o3o4*a
x3x4o3o4*a
x3o4x3o4*a
x3o4o3x4*a
x3x4x3o4*a
x3x4x3x4*a
x3o3o3o5*a
o3x3o3o5*a
x3x3o3o5*a
x3o3x3o5*a
x3o3o3x5*a
o3x3x3o5*a
x3x3x3o5*a
x3x3o3x5*a
x3x3x3x5*a
x3o4o3o5*a
o3x4o3o5*a
x3x4o3o5*a
x3o4x3o5*a
x3o4o3x5*a
o3x4x3o5*a
x3x4x3o5*a
x3x4o3x5*a
x3x4x3x5*a
x3o5o3o5*a
x3x5o3o5*a
x3o5x3o5*a
x3o5o3x5*a
x3x5x3o5*a
x3x5x3x5*a
...
...
...
...
...

*) These figures occur only as alternations. An all unit edged representation does not exist.

Trying to extend the class with linear Dynkin diagrams into non-convex realms, i.e. asking for compact regular star-honeycombs, would come out to be hopeless either. In fact, the actual choice of any Kepler-Poinsot polyhedron (as well for cell as for vertex figure) produces spherical curvatures only. – But this would not bother the other (non-linear) types of Dynkin diagram structures (nor non-compact linears)!

So far just some random star examples...
loop ones loop'n'tail ones
o5/2o5o3o5*a o5o3o3o5/2*b
x5/2o5o3o5*a
o5/2o5x3o5*a
x5/2o5x3o5*a
x5/2o5o3x5*a
o5/2o5x3x5*a
x5/2o5x3x5*a
x5o3o3o5/2*b - ditdih
o5x3o3o5/2*b
o5o3x3o5/2*b
o5o3o3x5/2*b
x5x3o3o5/2*b
x5o3x3o5/2*b
x5o3o3x5/2*b
o5x3x3o5/2*b
o5x3o3x5/2*b - [Grünbaumian]
o5o3x3x5/2*b
x5x3x3o5/2*b
x5x3o3x5/2*b - [Grünbaumian]
x5o3x3x5/2*b
o5x3x3x5/2*b - [Grünbaumian]
x5x3x3x5/2*b - [Grünbaumian]

Also to class 1 would belong additionally all the honeycomb products of any 2D hyperbolic tiling with (an appropriate hyperbolic space version of) aze. This is due to the fact that in this product neither of the full-dimensional elements themselves (considered as bodies) remain true elements of the product (although those could be seen as being pseudo elements thereof).

More generally the laminates belong here. The only known uniform lamina-truncates (cf. definition) are lamina-trunc( x4x3o8o ) and lamina-trunc( o8o4x *b3x ).

Already in 1997 W. Krieger found an infinite series of uniform, pyritohedral vertex figured honeycombs belonging here as well (each featuring 8 cubes and 6 p-gonal prisms per vertex). Other singular cases would be spd{3,5,3} (with 2 does, 6 paps, and 6 ikes per vertex) and pd{3,5,3} (with 4 does and 12 paps per vertex), as detailed under subsymmetric diminishings.

In early 2021 a compact non-Wythoffian though uniform hyperbolic was found by a guy calling himself "Grand Antiprism", the octsnich, having 2 octs and 8 snics per vertex.


Paracompact Honeycombs

Irreducible 3D hyperbolic reflectional symmetry groups within class 2, with finite integral link marks only, would group into the following classes, which would include euclidean tilings in addition to spherical space tiles. Those noncompact hyperbolic groups can be considered over-extended forms, like the affine groups, adding a second node in sequence to the first added node, with letter names marked up by a '++' superscript.

linear ones
C2++ & G2++
tri-dental ones
B2++
loop-n-tail ones
some A2++
o3o6o3o

o3o4o4o
o4o4o4o
o3o3o6o
o4o3o6o
o5o3o6o
o6o3o6o
o3o3o *b6o

o4o4o *b3o
o4o4o *b4o
o3o3o3o3*b
o4o3o3o3*b
o5o3o3o3*b
o6o3o3o3*b
x3o6o3o - trah
o3x6o3o - ritrah
x3x6o3o - hexah
x3o6x3o - sritrah
x3o6o3x - spidditrah
o3x6x3o - ditrah
x3x6x3o - gritrah
x3x6o3x - pritrah
x3x6x3x - gipidditrah

x3o4o4o - octh
o3x4o4o - rocth
o3o4x4o - risquah
o3o4o4x - squah
x3x4o4o - tocth
x3o4x4o - srocth
x3o4o4x - sidposquah
o3x4x4o - osquah
o3x4o4x - srisquah
o3o4x4x - tisquah
x3x4x4o - grocth
x3x4o4x - prisquah
x3o4x4x - procth
o3x4x4x - grisquah
x3x4x4x - gidposquah

x4o4o4o - sisquah
o4x4o4o - squah
x4x4o4o - tissish
x4o4x4o - risquah
x4o4o4x - spiddish
o4x4x4o - dish
x4x4x4o - tisquah
x4x4o4x - prissish
x4x4x4x - gipiddish
x3o3o6o - thon
o3x3o6o - rath
o3o3x6o - rihexah
o3o3o6x - hexah
...

x4o3o6o - hachon
o4x3o6o - rihach
o4o3x6o - rishexah
o4o3o6x - shexah
x4o3o6x - sidpichexah
o4x3x6o - chexah
...

x5o3o6o - hedhon
o5o3o6x - phexah
...

x6o3o6o - hihexah
o6x3o6o - rihihexah
o6x3x6o - hexah
...
x3o3o *b6o - ahach
o3x3o *b6o - tachach
x3o3o *b6x - birachach
x3x3o *b6x - bitachach
...

x4o4o *b3o
o4x4o *b3o
o4o4o *b3x
x4x4o *b3o
x4o4x *b3o - risquah
x4o4o *b3x
o4x4o *b3x - tocth
x4x4x *b3o
x4x4o *b3x
x4o4x *b3x
x4x4x *b3x

x4o4o *b4o - sisquah
o4x4o *b4o - squah
x4x4o *b4o - tissish
x4o4x *b4o - squah
x4x4x *b4o
x4o4x *b4x - risquah
x4x4x *b4x
x3o3o3o3*b - thon
o3x3o3o3*b - rath
o3o3x3o3*b - ahexah
x3x3o3o3*b
x3o3x3o3*b - birahexah
o3x3x3o3*b - tahexah
o3o3x3x3*b
x3x3x3o3*b - bitahexah
x3o3x3x3*b
o3x3x3x3*b
x3x3x3x3*b

x4o3o3o3*b - hachon
o4x3o3o3*b
o4o3x3o3*b - ashexah
x4x3o3o3*b
x4o3x3o3*b - birashexah
o4x3x3o3*b - tashexah
o4o3x3x3*b
x4x3x3o3*b - bitashexah
x4o3x3x3*b
o4x3x3x3*b
x4x3x3x3*b

x5o3o3o3*b
o5x3o3o3*b
o5o3x3o3*b - aphexah
x5x3o3o3*b
x5o3x3o3*b - biraphexah
o5x3x3o3*b - taphexah
o5o3x3x3*b
x5x3x3o3*b - bitaphexah
x5o3x3x3*b
o5x3x3x3*b
x5x3x3x3*b

x6o3o3o3*b - hihexah
o6x3o3o3*b - rihihexah
o6o3x3o3*b - trah
x6x3o3o3*b
x6o3x3o3*b
o6x3x3o3*b - ritrah
o6o3x3x3*b
x6x3x3o3*b
x6o3x3x3*b
o6x3x3x3*b - hexah
x6x3x3x3*b
s3s6o3o - ahexah
s3s6o3x
s3s6s3s - snatrah *) 
...

o3o4o4s
o3o4s4o
x3o4s4o
s3s4o4o **)
x3x4o4s
s3s4o4x **)
s3s4o4s' *)
...

s4o4o4o - sisquah
o4s4o4o
s4o4s4o
s4o4o4s
...
o3o3o6s - ahexah
...

s4o3o6o - ahach
o4o3o6s - ashexah
s4o3o6x - birachach
x4o3o6s - birashexah
s4o3o6s' - quishexah
...

o5o3o6s - aphexah
...

o6s3s6o - ahexah
...
x3o3o *b6s - quishexah
...

s4o4s *b3o
x4s4o *b3s *)
...

s4o4o *b4o - sisquah
o4s4o *b4o
s4o4s *b4o
...
s4o3o3o3*b
s4o3x3o3*b - quishexah
...

o6s3s3s3*b - ahexah
...
loop ones
D2++
2-loop ones
more A2++
simplicial ones
more A2++
o3o3o3o6*a
o3o4o3o6*a
o3o5o3o6*a
o3o6o3o6*a
o3o3o4o4*a
o3o4o4o4*a
o4o4o4o4*a
o3o3o3o3*a3*c
o3o3o3o3*a3*c *b3*d
...

...

...

x3o6o3o6*a
x3x6o3o6*a - shexah
...
o3o3o4x4*a
...

x3o4o4o4*a
...
o3o4x4o4*a
...

x4o4o4o4*a - sisquah
x4x4o4o4*a
x4o4x4o4*a - squah
x4x4x4o4*a
x4x4x4x4*a
o3x3o3o3*a3*c - ahach
x3x3o3o3*a3*c - quishexah
...
x3o3o3o3*a3*c *b3*d - trah
x3x3o3o3*a3*c *b3*d - rihihexah
x3x3x3o3*a3*c *b3*d - ritrah
x3x3x3x3*a3*c *b3*d - hexah
s3s6o3o6*a - ashexah
...
o3o3o4s4*a
...

o3o4s4o4*a
...

s4o4o4o4*a - sisquah
s4o4s4o4*a
...
...
s3s3s3s3*a3*c *b3*d - ahexah
...

*) These figures occur only as alternations. An all unit edged representation does not exist.
**) Although being rescalable to equal edge lengths, those figures are only scaliform.


E.g. the linear diagrams oPoQoRo, in order to be at most paracompact, in general would require to bow under both, (P-2)(Q-2) ≤ 4 and (Q-2)(R-2) ≤ 4. Further, those numbers again can be used to derive the according geometry: Any xPoQoRo consists of xPoQo-cells only, those having edges of length 2φ, an circumradius of χ, and an inradius of ψ, where

cosh(φ) = cos(π/P) sin(π/R) / sin(π/hQ,R)
cosh(ψ) = sin(π/P) cos(π/R) / sin(π/hP,Q)
cosh(χ) = cos(π/P) cos(π/Q) cos(π/R) / sin(π/hP,Q) sin(π/hQ,R)
with:
cos2(π/hP,Q) = cos2(π/P) + cos2(π/Q)

(The last equation clearly evaluates into hP,2 = Ph2,Q = Qh3,3 = 4h3,4 = h4,3 = 6h3,5 = h5,3 = 10h3,6 = h4,4 = h6,3 = ∞. Geometrically this number is related to the Petrie polygon of each of the corresponding regular polyhedra or tilings, i.e. their largest regular shadow polygon.)


Hypercompact Honeycombs

As any neither compact nor paracompact hyperbolic honeycomb would be hypercompact, those clearly have infinite count. So there cannot be a complete listing, not even tentatively. Only a few, randomly selected examples might follow here.

Just some random examples...
linear ones tri-dental ones loop-n-tail ones loop ones 2-loop ones simplicial ones prisms
o3o4o8x
o3o4o8s
o3o5o10x
o3o5o10s

o4x4oPo (for P>4)
o4s4oPo (for P>4)

x4oPo4x (for P>4)
s4oPo4s (for P>4)

x4x3o8o
x4x3x8o
x8o4o *b3o
o8o4x *b3x

x5o5o *b5/2o (non-convex)
o3o4x4o4*b
o3o5x5o5*b
 
o4x4o4x4*aP*c (for P>2)
o4s4o4s4*aP*c (for P>2)
 
x xPoQo (for any hyperbolic xPoQo)
x oPxQo (for any hyperbolic oPxQo)
x xPxQo (for any hyperbolic xPxQo)
x xPoQx (for any hyperbolic xPoQx)
x xPxQx (for any hyperbolic xPxQx)

x x3x8o

(For honeycombs with more general fundamental domains cf. Coxeter domains.)



---- 4D Tetracombs (up) ----

Compact Tetracombs

Dwelling within class 1 only, is equally restrictive here. Potential irreducible symmetries are:

linear ones
o3o3o3o5o (convex) o4o3o3o5o (convex) o5o3o3o5o (convex) o3o3o5o5/2o (µ=5) o3o5o5/2o5o (µ=10)
x3o3o3o5o - pennit
o3x3o3o5o
o3o3x3o5o
o3o3o3x5o
o3o3o3o5x - hitte
...
x4o3o3o5o - pitest
o4x3o3o5o
o4o3x3o5o
o4o3o3x5o
o4o3o3o5x - shitte
...
x5o3o3o5o - phitte
o5x3o3o5o
o5o3x3o5o
...
x3o3o5o5/2o
o3x3o5o5/2o
o3o3x5o5/2o
o3o3o5x5/2o
o3o3o5o5/2x
...
x3o5o5/2o5o
o3x5o5/2o5o
o3o5x5/2o5o
o3o5o5/2x5o
o3o5o5/2o5x
...
others
o3o3o *b3o5o (convex) o3o3o3o3o4*a (convex) o5o3o3o3/2o3*c (µ=2) o3o3o5o5o3/2*c (µ=4) o3o3o5o *b3/2o3*c (µ=3) o3o3/2o3o *b5o5*c (µ=6) ...
x3o3o *b3o5o
o3x3o *b3o5o
o3o3o *b3x5o
o3o3o *b3o5x
...
x3o3o3o3o4*a
o3x3o3o3o4*a
o3o3x3o3o4*a
...
...
...
...
...
...

Only the convex symmetries are exhausted within the table. This already is enough to show that here there are exactly 5 convex regulars and 4 regular star-tetracombs within class 1.

Beyond 4D, there will be no irreducible symmetry within class 1 anymore.

Within this dimension as well lamina-truncate uniforms are known, contit and odipt, which btw. are the only known non-regular dual pair, both of which are uniform.


Paracompact Tetracombs

The potential irreducible convex cases within class 2 are provided by the following groups. Those provide 2 more regular figures.

linear ones tridental ones cross ones loop-n-tail ones loop ones 2-loop ones
o3o4o3o4o
o3o3o *b4o3o
o3o4o *b3o3o
o3o4o *b3o4o
o3o4o *b3o *b3o
o3o3o3o3o3*b
o4o3o3o3o3*b
o3o3o4o3o4*a
o3o3o3o3*a3o3*c
x3o4o3o4o
o3x4o3o4o
o3o4x3o4o
o3o4o3x4o
o3o4o3o4x - chont
...
x3x4o3o4o
...

...
s3s4o3o4o
...
...
...
...
...
...

Hypercompact Tetracombs

As any neither compact nor paracompact hyperbolic tetracomb would be hypercompact, those clearly have infinite count. So there cannot be a complete listing, not even tentatively. Only a few, randomly selected examples might follow here.

Just some random examples...
linear ones
o3x4x3o8o
x3x4x3x8o

Additionally a still hypercompact lamina-truncate uniform is known here, the lamina-truncate( x3x4x3x8o ).



---- 5D Pentacombs (up) ----

Just providing irreducible convex symmetries. In class 1 (compact ones) there is none. In class 2 (paracompact ones) we have only

linear ones tridental ones cross ones pentadental ones loop-n-tail ones loop ones
o3o3o3o4o3o
o3o3o4o3o3o
o3o4o3o3o4o
o3o3o *b3o4o3o
o3o3o3o4o *c3o
o4o3o3o4o *c3o
o3o3o *b3o *b3o3o
o3o3o *b3o *b3o4o
o3o3o *b3o *b3o *b3o
o3o3o3o3o3o3*b
o3o3o3o3o3o4*a
o3o3o4o3o3o4*a
x3o3o3o4o3o
o3o3o3o4o3x
...
o3o3o3o4s3s
s3s4o3o4o3o
         


---- 6D Hexacombs (up) ----

Just providing irreducible convex symmetries. In class 1 (compact ones) there is none. In class 2 (paracompact ones) we have only

tridental ones bi-tridental ones loop-n-tail ones
o3o3o3o3o4o *c3o
o3o3o3o3o *b3o *c3o
o3o3o3o3o3o3o3*b


---- 7D Heptacombs (up) ----

Just providing irreducible convex symmetries. In class 1 (compact ones) there is none. In class 2 (paracompact ones) we have only

tridental ones bi-tridental ones loop-n-tail ones
o3o3o3o3o3o *c3o3o
o3o3o3o3o3o4o *c3o
o3o3o3o3o3o *b3o *d3o
o3o3o3o3o3o3o3o3*b


---- 8D Octacombs (up) ----

Just providing irreducible convex symmetries. In class 1 (compact ones) there is none. In class 2 (paracompact ones) we have only

tridental ones bi-tridental ones loop-n-tail ones
o3o3o3o3o3o3o3o *c3o
o3o3o3o3o3o3o4o *c3o
o3o3o3o3o3o3o *b3o *e3o
o3o3o3o3o3o3o3o3o3*b


---- 9D Enneacombs (up) ----

Just providing irreducible convex symmetries. In class 1 (compact ones) there is none. In class 2 (paracompact ones) we have only

tridental ones bi-tridental ones
o3o3o3o3o3o3o3o3o *c3o
o3o3o3o3o3o3o3o4o *c3o
o3o3o3o3o3o3o3o *b3o *f3o


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