Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
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:
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 boundet by some infinite (pseudo) facetings, which serve as reflection, glidereflection, 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 hemifacets of spherical space: One could consider the external, infinitely multiple selfblends 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 laminatruncate of the unblended structure. Both these notions were introduced by W.Krieger.
Just as for spherical geometry, the hyperbolic one has also a nonvanishing uniform curvature. Accordingly a circumradius here too is welldefined. 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 (P2)(Q2) > 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).
(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.)
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 P_{0} be a vertex of xPoQo, let P_{1} be the center of an adjacent edge, and P_{2} the center of an adjacent face (in fact a xPo), then the distances φ = P_{0}P_{1}, χ = P_{0}P_{2}, and ψ = P_{1}P_{2} 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 Pgons at each vertex, and the Pgons will have a circumradius of χ and an inradius of ψ.
The only regular startesselations have the symmetries oPoP/2o, here P being an odd integer greater than 5. All those startesselations would have density 3. (The case P = 5 already describes the spherical space tesselation or polyhedron sissid respectively gad.) In fact, xP/2oPo are derived as stellations of xPo3o. Dually, the edgeskeletons of xPoP/2o and of x3oPo are the same.
x7/2o7o  x7o7/2o 
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 o3x∞o o3o∞x x3x∞o x3o∞x o3x∞x x3x∞x x4o∞o o4x∞o o4o∞x x4x∞o x4o∞x o4x∞x ... 
x∞o∞o o∞x∞o x∞x∞o x∞o∞x x∞x∞x 
x3o3o∞*a o3x3o∞*a x3x3o∞*a x3o3x∞*a x3x3x∞*a ... x4o4x∞*a x4x4x∞*a ... 
x3o∞o∞*a o3o∞x∞*a x3x∞o∞*a x3o∞x∞*a x3x∞x∞*a ... 
x∞o∞o∞*a x∞x∞o∞*a x∞x∞x∞*a 
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) 
Here the restriction to finite tiles is much more effective, at least if being considered with respect to nonproduct 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  tridental 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 
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 

... 
s4o3o5o s4o3x5o s4o3o5x s4o3x5x ... 
... 
... 

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 nonconvex realms, i.e. asking for compact regular starhoneycombs, would come out to be hopeless either. In fact, the actual choice of any KeplerPoinsot polyhedron (as well for cell as for vertex figure) produces spherical curvatures only. – But this would not bother the other types of Dynkin diagram structures (nor noncompact linears)!
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 fulldimensional 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 belong here. The only known uniform laminatruncate (cf. definition) is laminatrunc( x4x3o8o ).
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 overextended 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 C_{2}^{++} & G_{2}^{++} 
tridental ones B_{2}^{++} 
loopntail ones some A_{2}^{++} 
loop ones D_{2}^{++} 
2loop ones more A_{2}^{++} 
simplicial ones more A_{2}^{++} 


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 
x3o6o3o o3x6o3o x3x6o3o x3o6x3o x3o6o3x o3x6x3o x3x6x3o x3x6o3x x3x6x3x x3o4o4o o3x4o4o o3o4x4o o3o4o4x x3x4o4o x3o4x4o x3o4o4x o3x4x4o o3x4o4x o3o4x4x x3x4x4o x3x4o4x x3o4x4x o3x4x4x x3x4x4x x4o4o4o o4x4o4o x4x4o4o x4o4x4o x4o4o4x o4x4x4o x4x4x4o x4x4o4x x4x4x4x 
o3x3o6o o3o3x6o o3o3o6x ... x4o3o6o o4o3o6x x4o3o6x ... o5o3o6x ... o6x3o6o o6x3x6o ... 
x3o3o *b6x ... x4o4o *b3o o4x4o *b3o o4o4o *b3x x4x4o *b3o x4o4x *b3o x4o4o *b3x o4x4o *b3x x4x4x *b3o x4x4o *b3x x4o4x *b3x x4x4x *b3x x4o4o *b4o o4x4o *b4o x4x4o *b4o x4o4x *b4o x4x4x *b4o x4o4x *b4x x4x4x *b4x 
x3o3o3o3*b o3x3o3o3*b o3o3x3o3*b x3x3o3o3*b x3o3x3o3*b o3x3x3o3*b o3o3x3x3*b x3x3x3o3*b x3o3x3x3*b o3x3x3x3*b x3x3x3x3*b x4o3o3o3*b o4x3o3o3*b o4o3x3o3*b x4x3o3o3*b x4o3x3o3*b o4x3x3o3*b o4o3x3x3*b x4x3x3o3*b x4o3x3x3*b o4x3x3x3*b x4x3x3x3*b x5o3o3o3*b o5x3o3o3*b o5o3x3o3*b x5x3o3o3*b x5o3x3o3*b o5x3x3o3*b o5o3x3x3*b x5x3x3o3*b x5o3x3x3*b o5x3x3x3*b x5x3x3x3*b x6o3o3o3*b o6x3o3o3*b o6o3x3o3*b x6x3o3o3*b x6o3x3o3*b o6x3x3o3*b o6o3x3x3*b x6x3x3o3*b x6o3x3x3*b o6x3x3x3*b x6x3x3x3*b 
... ... ... x3o6o3o6*a x3x6o3o6*a ... 
o3o3o4x4*a ... x3o4o4o4*a ... o3o4x4o4*a ... x4o4o4o4*a x4x4o4o4*a x4o4x4o4*a x4x4x4o4*a x4x4x4x4*a 
o3x3o3o3*a3*c x3x3o3o3*a3*c ... 
x3o3o3o3*a3*c *b3*d x3x3o3o3*a3*c *b3*d x3x3x3o3*a3*c *b3*d x3x3x3x3*a3*c *b3*d 
... o3o4o4s o3o4s4o s3s4o4o **) x3x4o4s s3s4o4x **) s3s4o4s' ... s4o4o4o o4s4o4o s4o4s4o s4o4o4s ... 
o3o3o6s ... s4o3o6o o4o3o6s s4o3o6x x4o3o6s s4o3o6s' ... o5o3o6s ... 
x3o3o *b6s ... s4o4s *b3o x4s4o *b3s ... s4o4o *b4o o4s4o *b4o s4o4s *b4o ... 
s4o3o3o3*b s4o3x3o3*b ... 
s3s6o3o6*a ... 
o3o3o4s4*a ... o3o4s4o4*a ... s4o4o4o4*a s4o4s4o4*a ... 
... 
... 
**) Even so 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, (P2)(Q2) ≤ 4 and (Q2)(R2) ≤ 4. Further, those numbers again can be used to derive the according geometry: Any xPoQoRo consists of xPoQocells 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/h_{Q,R}) cosh(ψ) = sin(π/P) cos(π/R) / sin(π/h_{P,Q}) cosh(χ) = cos(π/P) cos(π/Q) cos(π/R) / sin(π/h_{P,Q}) sin(π/h_{Q,R}) with: cos^{2}(π/h_{P,Q}) = cos^{2}(π/P) + cos^{2}(π/Q)
(The last equation clearly evaluates into h_{P,2} = P, h_{2,Q} = Q, h_{3,3} = 4, h_{3,4} = h_{4,3} = 6, h_{3,5} = h_{5,3} = 10, h_{3,6} = h_{4,4} = h_{6,3} = ∞. 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.)
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  tridental ones  loopntail ones  loop ones  2loop 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 
x8o4o *b3o 
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) 
(For honeycombs with more general fundamental domains cf. Coxeter domains.)
 4D Tetracombs (up) 
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 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  
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 startetracombs within class 1.
Beyond 4D, there will be no irreducible symmetry within class 1 anymore.
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  loopntail ones  loop ones  2loop 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 ... 
... 
... 
... 
... 
... 
 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  loopntail 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 
... o3o3o3o4s3s 
 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  bitridental ones  loopntail 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  bitridental ones  loopntail 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  bitridental ones  loopntail 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  bitridental ones 

o3o3o3o3o3o3o3o3o *c3o o3o3o3o3o3o3o3o4o *c3o 
o3o3o3o3o3o3o3o *b3o *f3o 
© 20042017  top of page 