### 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 so, 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. hyperbolic tesselations using finite (spherical polytopial) tiles only
2. hyperbolic tesselations which additionally (to class 1) include lower dimensional euclidean tesselations as infinite tiles
3. 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 on the other hand virtually any Dynkin symbol, which does not classify as spherical or euclidean, would be hyperbolic. This makes clear that this hyperbolic case is beyond any complete listing. Only the first of the above 3 classes makes sense of deeper detailed investigations, and will be handled below. As their fundamental domains clearly will be compact, this attribute then is used alike for the symmetry groups of that class and the therefrom derived tesselations. – Onto the other 2 classes just few general remarks are provided in the sequel.

```----
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 explicitely in that listing for the general Schwarz triangle oPoQoR*a (providing cases and their general incidence matrices; even so 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.)

linear ones loop ones o3o7o o3o8o ... o4o5o Just to provide some examplifying symmetries ... ```x3o7o o3x7o o3o7x x3x7o x3o7x o3x7x x3x7x s3s7s ``` ```x3o8o o3x8o o3o8x x3x8o x3o8x o3x8x x3x8x o3o8s x3o8s o3x8s x3x8s s3s8o s3s8x s3s8s ``` ```x4o5o o4x5o o4o5x x4x5o x4o5x o4x5x x4x5x s4o5o s4x5o s4o5x s4x5x o4s5s x4s5s s4s5s ``` ```x5o5o o5x5o x5x5o x5o5x x5x5x s5s5s ``` ```x3o3o4*a o3x3o4*a x3x3o4*a x3o3x4*a x3x3x4*a s3s3s4*a ``` ```x3o4o5*a o3x4o5*a o3o4x5*a x3x4o5*a x3o4x5*a o3x4x5*a x3x4x5*a s3s4s5*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 x7o7/2o

#### Non-Compact 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 will be understood to describe an horocyclic tile.

linear ones loop ones
oPo∞o o∞o∞o oPoQo∞*a oPo∞o∞*a o∞o∞o∞*a
```x3o∞o
o3x∞o
o3o∞x
x3x∞o
x3o∞x
o3x∞x
x3x∞x

s3s∞s

...
```
```x∞o∞o
o∞x∞o
x∞x∞o
x∞o∞x
x∞x∞x

s∞s∞s
```
```x3o3o∞*a
o3x3o∞*a
x3x3o∞*a
x3o3x∞*a
x3x3x∞*a

s3s3s∞*a

...
```
```x3o∞o∞*a
o3o∞x∞*a
x3x∞o∞*a
x3o∞x∞*a
x3x∞x∞*a

s3s∞s∞*a

...
```
```x∞o∞o∞*a
x∞x∞o∞*a
x∞x∞x∞*a

s∞s∞s∞*a
```

```----
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
o3x5o3o
x3x5o3o
x3o5x3o
x3o5o3x
o3x5x3o
x3x5x3o
x3x5o3x
x3x5x3x
```
```x4o3o5o
o4x3o5o
o4o3x5o
o4o3o5x
x4x3o5o
x4o3x5o
x4o3o5x
o4x3x5o
o4x3o5x
o4o3x5x
x4x3x5o
x4x3o5x
x4o3x5x
o4x3x5x
x4x3x5x

s4o3o5o
```
```x5o3o5o
o5x3o5o
x5x3o5o
x5o3x5o
x5o3o5x
o5x3x5o
x5x3x5o
x5x3o5x
x5x3x5x
```
```x3o3o *b5o
o3x3o *b5o
o3o3o *b5x
x3x3o *b5o
x3o3x *b5o
x3o3o *b5x
o3x3o *b5x
x3x3x *b5o
x3x3o *b5x
x3x3x *b5x
```
loop ones
o3o3o3o4*a o3o4o3o4*a o3o3o3o5*a o3o4o3o5*a o3o5o3o5*a
```x3o3o3o4*a
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
```

Trying to extend the class with linear Dynkin diagrams into non-convex realms, i.e. asking for regular star-honeycombs (within class 1), would come out to be hopeless either. The actual choice of any Kepler-Poinsot polyhedron (as well for cell as for vertex figure) produces spherical curvatures only.

On the other hand to class 1 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 (even so those could be seen as being pseudo elements thereof). More generally the laminates. This class of honeycombs is obtained from members of class 3 (of 3D hyperbolic honeycombs), which do not contain Euclidean elements (for this purpose). Therefrom the contained 2D hyperbolic tilings just are replaced by mirrors, and thus no longer count as elements of the derived honeycomb.

#### Non-Compact 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++
loop ones
D2++
2-loop ones
more A2++
simplexial ones
more A2++
```o3o6o3o

o3o4o4o
o4o4o4o

o3o3o6o
o4o3o6o
o5o3o6o
o6o3o6o
```
```o3o3o *b6o

o4o4o *b3o
o4o4o *b4o
```
```o3o3o3o3*b
o4o3o3o3*b
o5o3o3o3*b
o6o3o3o3*b
```
```o3o3o3o6*a
o3o4o3o6*a
o3o5o3o6*a
o3o6o3o6*a

o3o3o4o4*a
o3o4o4o4*a
o4o4o4o4*a
```
```o3o3o3o3*a3*c
```
```o3o3o3o3*a3*c *b3*d
```

More generally, this linear class would require for oPoQoRo 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)
cos2(π/hP,Q) = cos2(π/P) + cos2(π/Q)
```

Besides lots of others, for instance all the prisms of any 2D hyperbolic tiling surely would belong to the 3rd class.

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

#### Compact Tetracombs

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

o3o3o3o5o (convex) o4o3o3o5o (convex) o5o3o3o5o (convex) o3o3o5o5/2o (µ=5) o3o5o5/2o5o (µ=10) o3o3o *b3o5o (convex) o3o3o3o3o4*a (convex) linear ones ```x3o3o3o5o o3x3o3o5o o3o3x3o5o o3o3o3x5o o3o3o3o5x ... ``` ```x4o3o3o5o o4x3o3o5o o4o3x3o5o o4o3o3x5o o4o3o3o5x ... ``` ```x5o3o3o5o o5x3o3o5o o5o3x3o5o ... ``` ```x3o3o5o5/2o o3x3o5o5/2o o3o3x5o5/2o o3o3o5x5/2o o3o3o5o5/2x ... ``` ```x3o5o5/2o5o o3x5o5/2o5o o3o5x5/2o5o o3o5o5/2x5o o3o5o5/2o5x ... ``` others ```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.

#### Non-Compact Tetracombs

The potential irreducible convex cases within class 2 are

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
```

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

Just providing irreducible convex symmetries. In class 1 there is none. In class 2 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
```

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

Just providing irreducible convex symmetries. In class 1 there is none. In class 2 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 there is none. In class 2 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 there is none. In class 2 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 there is none. In class 2 we have only

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

### General Fundamental Domains (up)

There is a further main issue to be regarded within hyperbolic geometry: In hyperbolic geometry even the compact fundamental domains are no longer restricted to simplices. Such more general fundamental domain polytopes commonly are called Coxeter domains (or: Coxeter polytopes), at least if those are elementary (i.e. the produced symmetry is convex). Accordingly, the dihedral angles (of the pairs of facets Fi and Fj) of Coxeter polytopes are bound to be ≤ π/2 (convexity), and those submultiplicative numbers (to π) have to be integral (finite order, elementary domain).

The other way round the known main results here state that any such polytope, with dihedral angles αi,j ≤ π/2, must be simple, i.e. its vertex figures remain simplexial. That any such polytope furthermore, provided its αi,j are of the form π / mi,j with integral mi,j, does generate a Coxeter group which acts properly on this space by means of reflections at its facets. Thereby that given polytope then will be its fundamental domain. And the entries of the Coxeter matrix (mi,j)i,j of this group are derivable as follows:

```mi,j = 1 ,         if i = j ;  else:
mi,j = π / αi,j ,  if Fi ∩ Fj ≠ Ø
mi,j = ∞ ,         if Fi ∩ Fj = Ø
```

Sure, the set of such polytopes is infinite. Even its classification is a still on-going task.

```----
2D  More general Tilings
----
```

The picture shows such a more general uniform hyperbolic tiling with a tetragonal domain: there are squares (red), hexagons (blue), octagons (magenta), and decagons (golden) around any vertex.

• In fact, in 2D virtually any cyclical arrangement of even-numbered polygons around a vertex would give rise to an uniform tiling (generalising the omnitruncates): their vertex figure thus is [2p,2q,2r,...,2s,2t].
• Further, in those tilings any single edge type might be reduced to zero (generalising the truncates and rhombates), and the vertex figure would become [p,2q,2r,...,2s,t,2s,...,2r,2q].
• A second to be reduced edge then has to be adjacent (thereby reducing some face type completely), the vertex figure thereby would become [(q,2r,...,2s,t,2s,...,2r,q)p] (this generalises the regular and quasiregular tilings).
• And finally, as one started with all even-numbered tiles only, a vertex alternation could apply, thus leading to the corresponding snub tiling. It should be noted here, that the additional snubbing face depends on the starting cyclical arrangement. Esp. it is no longer bound to remain a triangle only.

Dynkin diagrams for such more general hyperbolics are not defined, neither for the symmetry groups themselves, nor for the therefrom derived tesselations. Moreover, as the Wythoff caleidoscopic construction essentially is based on those diagrams (resp. on the information decoded therein), this "more general case" also could be called non-Wythoffian.

In view of the above cited theorem however, one might attempt to extend that diagrammal description for instance by using a virtual simplexial description, i.e. the number of nodes being the number of facets of the Coxeter polytope, and the links are marked according to those mi,j (i≠j). – The diagram of the provided picture then accordingly could read x3x∞x4x∞*a5*c. (In fact, the nodes at positions b,a,c,d in this sequence cyclically provide the faces around any vertex: hexagon (mb,a = 3), decagon (ma,c = 5), octagon (mc,d = 4), and (closing back) square (md,b = 2). I.e. node a represents the mirrors midway orthogonal to the hexagon/decagon edges, b correspondingly to the square/hexagon edges, c to the octagon/decagon edges, and d to the square/octagon edges. But there are no tiles (polygons) to be spanned by the remaining pairs of mirrors (mb,c = md,a = ∞).)

The main problem here is, as already for the Coxeter matrix above, that hyperbolic space does not have parallels (as for Euclidean space). I.e. 2 non-intersecting straight lines might be limit tangential ("intersection" at an ideal point) or even there they won't. But that ∞ symbol would not distinguish between those 2 rather different possibilities. For instance ...x∞o... or ...x∞x... would describe on the one hand, in the limit tangential case, a true horocyclical (aze) tile, i.e. both "ends" approach the same ideal point (this is the way, it could occur within simplicial domains only), but in the other case those "ends" would approach 2 different ideal points. (Thus we would need better some transinfinity sign or even transinfinite numbers here.) – In fact, both paths of edges, either running from an ideal point into finite reach and back to the same point, or running from one ideal point into finite reach and then towards some other ideal point (i.e. those pseudo tile paths) both would have a count of ∞ consecutive edges. This is why that ∞ symbol deserves its right. But only those edge sequences, running back to the original ideal point, give rise to true tiles (aze), the other ones never can. – Even so there is a small exception. When this edge sequence within Poicaré disk display would come orthogonally onto the circle of infinity, this serves like a hyperbolic boundary "tile" (in the sense of class 3), at which either a reflection or a glide-reflection reproduces the one side at the other, thereby describing a laminat.

#### Degrees of freedom

For hyperbolic fundamental triangles, the size of sides is already fixed by the choice of the angles. But this does not hold true for other fundamental domains. In fact, for a fundamental n-gon there are n-3 degrees of freedom, usable for alterable side lengths. – This is just the same as for the single such reflection group of euclidean space: the one with a rectangular fundamental domain, where alike the relative scale of side length can be varied.