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 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:
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, 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 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).
(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  sirtoct 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  sirtoct 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 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  sheat  x7o7/2o  gheat 
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) 
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  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 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 (nonlinear) types of Dynkin diagram structures (nor noncompact 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 fulldimensional 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 laminatruncates (cf. definition) are laminatrunc( x4x3o8o ) and laminatrunc( 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 pgonal 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 2021 a compact nonWythoffian though uniform hyperbolic was found by a guy calling himself "Grand Antiprism", the octsnich, having 2 octs and 8 snics per vertex.
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}^{++} 


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 D_{2}^{++} 
2loop ones more A_{2}^{++} 
simplicial ones more A_{2}^{++} 

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, (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 o8o4x *b3x x5o5o *b5/2o (nonconvex) 
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  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 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 ... x3x4o3o4o ... ... s3s4o3o4o ... 
... 
... 
... 
... 
... 
 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 
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  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 
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