Site Map  Polytopes  Dynkin Diagrams  Vertex Figures, etc.  Incidence Matrices  Index 
As a foregoing consideration one might ask whether euclidean space curvature also does allow for nonsimplicial reflection groups, such as the hyperbolic one does. In fact, one should state, that this is the case here indeed. But, on the other hand, that extraordinary regime of varities breaks down in here to one rather tame effect: For D=2 we just can have a tetragonal fundamental domain, with all angles being α=π/2, i.e. a rectangle. In Weyl notation that "extraordinary" symmetry is nothing but A_{1}×A_{1}.
For higher dimensions this same effect generalizes to cartesian honeycomb products of lower dimensional euclidean tesselations, where the fundamental domains derive likewise by orthogonal prism products of those of the factor tesselations. In terms of symmetries this generally is called a reducible (afine) symmetry. In Weyl group description those are marked by an ×.
Just as in hyperbolic geometry, nonsimplicial reflection groups in here also provide some additional degrees of freedom: a reducible symmetry with n factors, allows for n1 independent relative scalings of the orthogonal factor domains. (Sure, the absolut size of the first factor in euclidean space is not restricted either.)
Euclidean tesselations are closely related to cell complexes based on lattices. Best known is the Voronoi complex. This is an isohedral (1 single cell class) tesselation of space. Each cell is the closure of all points nearer to any specific lattice point than to any other.
The locally dual complex is the Delone complex. The vertices thus are the lattice points again. Two lattice points are joined by an edge iff the corresponding Voronoi cells are adjoined at an facet, etc. There might be several different types of Delone cells, corresponding to the symmetry inequivalent types of vertices of the Voronoi cell. Both these complexes by definition use convex cells only.
Beside to the (unmarked) Dynkin symbols of the symmetry groups the corresponding representations as Weyl groups and as Coxeter groups are given.
Finally, besides of (marked) Dynkin symbols and Bowers acronyms for individual tesselations the Olshevsky numbers ("O...") are provided as well (up to dimension 4). Those refer onto his paper on Uniform Panoploid Tetracombs, assembled since 2003, made available in 2006.)
 1D Sequence (up) 
o∞o = A_{1} = W(2)  o∞'o 
x∞o  aze x∞x  aze 
x∞'o  aze x∞'x  ∞covered edge 
A_{1}, taken as lattice, defines the intervals inbetween as Delone cells, while the Voronoi cells are those intervals shifted by half a unit length. Both complexes are equivalent (relatively shifted) to the apeirogon.
Generally {n/d} and {(n/d)'} = {n/(nd)} denote pairs of pro and retrograde polygons. That is, when considered without context, their geometric realization is the same. Even so, as abstract polytopes they have to be distinguished, esp. when used as link marks within Dynkin diagrams. – It then is only by transition to infinite values of this quotient of n/d, that {n/d} = xN/Do and {2n/d} = xN/Dx become geometrically identical. But this does not hold true for {n/(nd)} = xN/(ND)o, were the corresponding limit of the mark value becomes 1 (the retrograde aze), and {2n/(nd)} = x2N/(ND)o, were the limit of that mark value becomes 2 (the ∞covered edge). These rather behave like a folding rule, in its unfolded resp. its folded state. That is, x∞x is a wellbehaved polytope, whereas x∞'x rather would be some highly degenerate Grünbaumian.
 2D Tilings (up) 
o3o6o = G_{2} = V(3) 
o4o4o = C_{2} = R(3) 
o3o3o3*a = A_{2} = P(3) 
o∞o o∞o = A_{1}×A_{1} = W(2)^{2} 
x3o6o  trat  O2 o3x6o  that  O5 o3o6x  hexat  O3 x3x6o  hexat  O3 x3o6x  rothat  O8 o3x6x  toxat  O7 x3x6x  othat  O9 
x4o4o  squat  O1 o4x4o  squat  O1 x4x4o  tosquat  O6 x4o4x  squat  O1 x4x4x  tosquat  O6 
x3o3o3*a  trat  O2 x3x3o3*a  that  O5 x3x3x3*a  hexat  O3 
x∞o x∞o  squat  O1 x∞x x∞o  squat  O1 x∞x x∞x  squat  O1 
snubs of o3o6o 
snubs of o4o4o 
snubs of o3o3o3*a 
snubs of o∞o o∞o 
s6o3o  trat  O2 s6x3o  that  O5 s6o3x  that  O5 s6x3x  hexat  O3 s3s6o  trat  O2 s3s6x  rothat  O8 s3s6s  snathat  O11 β6β3o  2that+∞{3} (?) β6β3x  2rothat (?) β3x6o  2that (?) β3o6x  shothat β3x6x  2toxat (?) x3β6x  2rothat (?) 
s4o4o  squat  O1 s4x4o  squat  O1 s4o4x  tosquat  O6 s4x4x  tosquat  O6 o4s4o  squat  O1 x4s4o  squat  O1 x4s4x  squat  O1 s4s4o  snasquat  O10 s4s4x  squat  O1 s4o4s  squat  O1 s4x4s  squat  O1 s4s4s  snasquat  O10 s4o4s'  snasquat  O10 s4x4s'  squat  O1 ss'4o4x  squat  O1 
s3s3s3*a  trat  O2 
x∞s2s∞o  hexat  O3 
other (convex) uniforms  
elong( x3o6o )  etrat  O4 x∞o x  azip x∞x x  azip s∞o2s  azap s∞s2s  azap 
As lattice A_{2} and G_{2} are equivalent. The Voronoi cell is the hexagon, the Delone cell is the regular triangle. The Voronoi complex is hexat, the Delonai complex is trat.
The lattice C_{2} clearly has squares for Voronoi and Delone cells. Both complexes are relatively shifted representants of squat.
The lattica A_{2} (or the vertex set of trat) can be represented by the Eisenstein integers
E = Z[ω] = {e_{0} + e_{1}ω  e_{0},e_{1}∈Z}, ω = (1+sqrt(3))/2.
Similarily the lattice C_{2} (or the vertex set of squat) is represented by the
Gaussian integers G = Z[i] = {g_{0} + g_{1}i  g_{0},g_{1}∈Z}, i = sqrt(1).
(A nice applet for experimental tiling of the 2D euclidean plane (as well as hyperbolic ones) is tyler.)
Beyond the 3 regular tesselations and the 8 semiregular ones there are nonconvex euclidean tilings too. Those will add for additional possibilities the usage of
But a corresponding research, if desired to be meaningful, would ask for some special care in general. Two reasons are provided in what follows.
Even so euclidean tilings, in contrast to spherical space tesselations (polyhedra), allow for an infinitude of vertices, we still restrict to a global discreteness of the vertex set. Dense modules will be rejected. Thus the restriction derived from curvature, e.g. for linear diagrams (P2)(Q2) = 4, which for a rational P=n/d would resolve to Q=2n/(n2d), is not enough for that purpose: for instance the reflection groups o5o10/3o, o5/2o10o, o7o14/5o, o7/2o14/3o, o7/3o14o, etc. all would bow to this curvature condition. But those all would produce dense modules! These therefore are to be omitted from further consideration.
Not even a mere investigation of possible vertex configurations here is sufficient (even so the respective tilings will be enlisted by them). Because in general those local configurations either would not allow for an infinitely extending, uniform, nonGrünbaumian tiling at all, or they would result (globally) again in a dense vertex set.
a) Consideration by additionally allowed symmetry groups
o3/2o3/2o3*a  o4o4/3o  o4/3o4/3o 

x3/2o3/2o3*a  trat ° o3/2x3/2o3*a  trat ° x3/2x3/2o3*a  (∞covered {3}) x3/2o3/2x3*a  that ° x3/2x3/2x3*a  [Grünbaumian] 
x4o4/3o  squat ° o4x4/3o  squat ° o4o4/3x  squat ° x4x4/3o  tosquat ° x4o4/3x  (∞covered {4}) o4x4/3x  quitsquat x4x4/3x  qrasquit 
x4/3o4/3o  squat ° o4/3x4/3o  squat ° x4/3x4/3o  quitsquat x4/3o4/3x  squat ° x4/3x4/3x  quitsquat s4/3s4/3s  rasisquat 
o3/2o6o  o3o6/5o  o3/2o6/5o 
x3/2o6o  trat ° o3/2x6o  that ° o3/2o6x  hexat ° x3/2x6o  [Grünbaumian] x3/2o6x  qrothat o3/2x6x  toxat ° x3/2x6x  [Grünbaumian] 
x3o6/5o  trat ° o3x6/5o  that ° o3o6/5x  hexat ° x3x6/5o  hexat ° x3o6/5x  qrothat o3x6/5x  quothat x3x6/5x  quitothit 
x3/2o6/5o  trat ° o3/2x6/5o  that ° o3/2o6/5x  hexat ° x3/2x6/5o  [Grünbaumian] x3/2o6/5x  rothat ° o3/2x6/5x  quothat x3/2x6/5x  [Grünbaumian] 
o3/2o6o6*a  o3o6o6/5*a  o3/2o6/5o6/5*a 
x3/2o6o6*a  chatit o3/2o6x6*a  2hexat (?) x3/2x6o6*a  [Grünbaumian] x3/2o6x6*a  shothat x3/2x6x6*a  [Grünbaumian] 
x3o6o6/5*a  chatit o3x6o6/5*a  chatit o3o6x6/5*a  2hexat (?) x3x6o6/5*a  (∞covered {6}) x3o6x6/5*a  ghothat o3x6x6/5*a  shothat x3x6x6/5*a  thotithit 
x3/2o6/5o6/5*a  chatit o3/2o6/5x6/5*a  2hexat (?) x3/2x6/5o6/5*a  [Grünbaumian] x3/2o6/5x6/5*a  ghothat x3/2x6/5x6/5*a  [Grünbaumian] 
o3o3/2o∞*a  o3o3o∞'*a  o3/2o3/2o∞'*a 
x3o3/2o∞*a  ditatha o3o3/2x∞*a  ditatha x3x3/2o∞*a  chata x3o3/2x∞*a  tha o3x3/2x∞*a  [Grünbaumian] x3x3/2x∞*a  [Grünbaumian] 
x3o3o∞'*a  ditatha x3x3o∞'*a  chata x3o3x∞'*a  (contains ∞covered edge) x3x3x∞'*a  (contains ∞covered edge) 
x3/2o3/2o∞'*a  ditatha x3/2x3/2o∞'*a  [Grünbaumian] x3/2o3/2x∞'*a  (contains ∞covered edge) x3/2x3/2x∞'*a  [Grünbaumian] 
o4o4/3o∞*a  o4o4o∞'*a  o4/3o4/3o∞'*a 
x4o4/3o∞*a  cosa o4o4/3x∞*a  cosa x4x4/3o∞*a  gossa x4o4/3x∞*a  sha o4x4/3x∞*a  sossa x4x4/3x∞*a  satsa s4s4/3s∞*a  snassa 
x4o4o∞'*a  cosa x4x4o∞'*a  gossa x4o4x∞'*a  (contains ∞covered edge) x4x4x∞'*a  (contains ∞covered edge) 
x4/3o4/3o∞'*a  cosa x4/3x4/3o∞'*a  sossa x4/3o4/3x∞'*a  (contains ∞covered edge) x4/3x4/3x∞'*a  (contains ∞covered edge) 
o6o6/5o∞*a  o6o6o∞'*a  o6/5o6/5o∞'*a 
x6o6/5o∞*a  cha o6o6/5x∞*a  cha x6x6/5o∞*a  ghaha x6o6/5x∞*a  hoha o6x6/5x∞*a  shaha x6x6/5x∞*a  hatha 
x6o6o∞'*a  cha x6x6o∞'*a  ghaha x6o6x∞'*a  (contains ∞covered edge) x6x6x∞'*a  (contains ∞covered edge) 
x6/5o6/5o∞'*a  cha x6/5x6/5o∞'*a  shaha x6/5o6/5x∞'*a  (contains ∞covered edge) x6/5x6/5x∞'*a  (contains ∞covered edge) 
o o∞o  other  
x x∞o  azip ° x x∞x  azip ° s s∞o  azap ° s s∞s  azap ° 
sost (nonorientable member of sossa regiment) rassersa (own regiment, a sossa superregiment) rarsisresa (in rassersa regiment) rosassa (in rassersa subregiment) rorisassa (in rosassa regiment) sraht (nonorientable member of rothatregiment graht (nonorientable member of ghothatregiment huht (nonorientable member of shaharegiment) retrat (retroelongated trat) 
° : Those such marked tilings here come out again to be convex after all.
' (in the context of P') : Refering to the inversion between prograde and retrograde polygons {n/d} ↔ {n/(nd)}. (Note that as abstract polytopes {∞} and {∞'} are well to be distinguished, and thus esp. their usage as link marks within Dynkin diagrams, but that their geometric realizations within a euclidean space context are exactly the same aze.)
b) Consideration by additionally allowed vertex configurations
using {∞} *  using {n/d}  using both 

[4^{2},∞]  x x∞o, azip (∞prism, convex) [3^{3},∞]  s2s∞o, azap (∞antipr., convex) [(3,∞)^{3}]/2  x3o3/2o∞*a, ditatha (in tratreg.) 
[3/2,12/5,12/5]  o3x6/5x, quothat (in toxatarmy, reg. 11) [4,8/5,8/5]  o4x4/3x, quitsquat (in tosquatarmy, reg. 10) [4,6/5,12/5]  x3x6/5x, quitothit (in othatarmy, reg. 8) [4,8/3,8/7]  x4x4/3x, qrasquit (in tosquatarmy, reg. 7) [6,12/5,12/11]  x3x6x6/5*a, thotithit (in toxatsubarmy, reg. 9) [3/2,4,6/5,4]  x3/2o6x, qrothat (in rothatarmy, reg. 4) [3,12/5,6/5,12/5]  x3o6x6/5*a, ghothat (in rothatarmy, reg. 4) [3/2,12,6,12]  x3/2o6x6*a, shothat (in rothatregiment, 1) [4,12,4/3,12/11]/0  sraht (in rothatregiment, 1) [4,12/5,4/3,12/7]/0  graht (in rothatarmy, reg. 4) [8/3,8,8/5,8/7]/0  sost (in tosquatarmy, reg. 2) [12/5,12,12/7,12/11]/0  huht (in toxatarmy, reg. 3) [3,3,3,4/3,4/3]  retrat (in etratsuperreg., 12) [3,3,4/3,3,4/3]  s4/3s4/3s, rasisquat (in snasquatarmy, reg. 13) 
[8,8/3,∞]  x4x4/3x∞*a, satsa (in tosquatsubarmy, reg. 5) [12,12/5,∞]  x6x6/5x∞*a, hatha (in toxatsubarmy, reg. 6) [3,∞,3/2,∞]/0  x3o3/2x∞*a, tha (in thatregiment) [4,∞,4/3,∞]/0  x4o4/3x∞*a, sha (in squatregiment) [6,∞,6/5,∞]/0  x6o6/5x∞*a, hoha (in thatregiment) [8,4/3,8,∞]  x4x4/3o∞*a, gossa (in tosquatarmy, reg. 2) [8/3,4,8/3,∞]  o4x4/3x∞*a, sossa (in tosquatarmy, reg. 2) [12,6/5,12,∞]  x6x6/5o∞*a, ghaha (in toxatarmy, reg. 3) [12/5,6,12/5,∞]  o6x6/5x∞*a, shaha (in toxatarmy, reg. 3) [3,4,3,4/3,3,∞]  snassa (in snasquatarmy, reg. 14) [4,8,8/3,4/3,∞]  rorisassa (in tosquatarmy, in rassersasubreg. †) [4,8/3,8,4/3,∞]  rosassa (in tosquatarmy, in rassersasubreg. †) [4,8,4/3,8,4/3,∞]  rarsisresa (in sossasuperreg. †) [4,8/3,4,8/3,4/3,∞]  rassersa (in sossasuperreg. †) 
* Note, that some authors would count in the "tiling by 2 azes",
but there the 2 tiles would connect at more than a single edge, and therefore in the enlisting is not shown.
† Those were found by H. Sakamoto, cf. his website.
What he calls "complex" has nothing to do with the complex numbers, rather it means "complicated".
As such it refers to tilings of the euclidean plane, which are nonWythoffian.
The provided regiment numbers (for regiments, which occur exclusively within this section) correspond to the listing by J. McNeill. (The corresponding individual numberlinks would point to his respective page.)
 3D Honeycombs (up) 
o4o3o4o = C_{3} = R(4) 
o3o3o *b4o = B_{3} = S(4) 
o3o3o3o3*a = A_{3} = P(4) 
o∞o o3o6o = A_{1}×G_{2} = W(2)×V(3) 
x4o3o4o  chon  O1 o4x3o4o  rich  O15 x4x3o4o  tich  O14 x4o3x4o  srich  O17 x4o3o4x  chon  O1 o4x3x4o  batch  O16 x4x3x4o  grich  O18 x4x3o4x  prich  O19 x4x3x4x  otch  O20 s4o3o4o  octet  O21 s4x3o4o  rich  O15 s4o3x4o  tatoh  O25 s4o3o4x  ratoh  O26 s4o3x4x  gratoh  O28 s4x3x4o  batch  O16 s4x3o4x  srich  O17 s4x3x4x  grich  O18 s4o3o4s  octet  O21 s4o3o4s'  batatoh  O27 s4s3s4o  (?)  ** 
x3o3o *b4o  octet  O21 o3x3o *b4o  rich  O15 o3o3o *b4x  chon  O1 x3x3o *b4o  tatoh  O25 x3o3x *b4o  rich  O15 x3o3o *b4x  ratoh  O26 o3x3o *b4x  tich  O14 x3x3x *b4o  batch  O16 x3x3o *b4x  gratoh  O28 x3o3x *b4x  srich  O17 x3x3x *b4x  grich  O18 o3o3o *b4s  octet  O21 s3s3s *b4s  (?)  ** 
x3o3o3o3*a  octet  O21 x3x3o3o3*a  batatoh  O27 x3o3x3o3*a  rich  O15 x3x3x3o3*a  tatoh  O25 x3x3x3x3*a  batch  O16 
x∞o x3o6o  tiph  O2 x∞o o3x6o  thiph  O5 x∞o o3o6x  hiph  O3 x∞x x3o6o  tiph  O2 x∞x o3x6o  thiph  O5 x∞x o3o6x  hiph  O3 x∞o x3x6o  hiph  O3 x∞o x3o6x  rothaph  O8 x∞o o3x6x  thaph  O7 x∞x x3x6o  hiph  O3 x∞x x3o6x  rothaph  O8 x∞x o3x6x  thaph  O7 x∞o x3x6x  otathaph  O9 x∞x x3x6x  otathaph  O9 x∞o s3s6s  snathaph  O11 x∞x s3s6s  snathaph  O11 s∞o2o3o6s  ditoh  * s∞x2o3o6s  editoh  * s∞o2s3s6o  ditoh  * s∞o2x3o6s  gyrich  ° 
o∞o o4o4o = A_{1}×C_{2} = W(2)×R(3) 
o∞o o3o3o3*c = A_{1}×A_{2} = W(2)×P(3) 
o∞o o∞o o∞o = A_{1}×A_{1}×A_{1} = W(2)^{3}  other uniforms 
x∞o x4o4o  chon  O1 x∞o o4x4o  chon  O1 x∞x x4o4o  chon  O1 x∞x o4x4o  chon  O1 x∞o x4x4o  tassiph  O6 x∞o x4o4x  chon  O1 x∞x x4x4o  tassiph  O6 x∞x x4o4x  chon  O1 x∞o x4x4x  tassiph  O6 x∞x x4x4x  tassiph  O6 x∞o s4s4s  sassiph  O10 x∞x s4s4s  sassiph  O10 s∞o2s4o4o  octet  O21 s∞o2o4s4o  octet  O21 s∞x2s4o4o  pextoh  * s∞x2o4s4o  pextoh  * s∞o2s4o4x  pacratoh  * 
x∞o x3o3o3*c  tiph  O2 x∞x x3o3o3*c  tiph  O2 x∞o x3x3o3*c  thiph  O5 x∞x x3x3o3*c  thiph  O5 x∞o x3x3x3*c  hiph  O3 x∞x x3x3x3*c  hiph  O3 x∞o s3s3s3*c  tiph  O2 x∞x s3s3s3*c  tiph  O2 s∞o2s3s3s3*c  ditoh  * 
x∞o x∞o x∞o  chon  O1 x∞x x∞o x∞o  chon  O1 x∞x x∞x x∞o  chon  O1 x∞x x∞x x∞x  chon  O1 
gyro( x3o3o3o3*a )  gytoh  O22 elong( x3o3o3o3*a )  etoh  O23 gyroelong( x3o3o3o3*a )  gyetoh  O24 gyro( x3o3o *b4o )  gytoh  O22 elong( x3o3o *b4o )  etoh  O23 gyroelong( x3o3o *b4o )  gyetoh  O24 gyro( x∞o x3o6o )  gytoph  O12 elong( x∞o x3o6o )  etoph  O4 gyroelong( x∞o x3o6o )  gyetaph  O13 x∞o elong( x3o6o )  etoph  O4 x∞x elong( x3o6o )  etoph  O4 gyro( x∞o x3o3o3*c )  gytoph  O12 elong( x∞o x3o3o3*c )  etoph  O4 gyroelong( x∞o x3o3o3*c )  gyetaph  O13 x∞o elong( x3o3o3*c )  etoph  O4 x∞x elong( x3o3o3*c )  etoph  O4 x∞o xno  nazedip x∞x xno  nazedip x∞x sns  nazedip 
* These alternated facetings well can be varied to get all equal edge lengths, but still will not become uniform, because they incorporate Johnson solids. They not even will become scaliform then, because some of the used cells are not orbiform. Hence those count at most as CRF honeycombs. – This is why those do not have Olshevsky numbers either.
** Those not even are relaxable to unit edges only.
° Those on the other hand would become true scaliforms.
In 3D there are several different lattices, known as the Bravais lattices of crystallography. Only the higher symmetrical ones are related to uniform tesselations. As lattices A_{1}×G_{2} and A_{1}×A_{2} again are equivalent, the Voronoi complex of it is hiph, the Delone complex is tiph.
In the cubical area there are 3 different lattices. First there is the primitive cubical one, C_{3}. Its Voronoi complex and its Delone complex both are relatively shifted chon.
Next there is the bodycentered cubical (bcc) lattice. Thus it is the union of the primitive cubical lattice plus its Voronoi cell vertices ("holes"). The Voronoi complex here is batch. The Delone complex would be a squashed variant of octet.
Finally there is the facecentered cubical (fcc) lattice, as lattice equivalently derived from A_{3} or B_{3}, and alternatively described as mod 2 of the sum of the vertex coordinates of a (smaller) primitive cubical lattice. Its Voronoi complex is radh and the Voronoi cell the rhombic dodecahedron (rad). The Delone complex is octet, with 2 different Delone cells, oct corresponding to the 4fold vertices of rad ("deep holes"), while tet corresponds to the 3fold vertices of rad ("shallow holes").

In 2005 J. McNeill did a research on elementary honeycombs (cf. this crosslink to his website), that is (nonuniform) scaliform honeycombs using elementary solids only. Here elementary in turn means: regular faced solids, which are not subdivisable. He then listed 13 such honeycombs (plus some more 4 nonelementary ones). But in fact, 4 of the vertex surroundings, he is displaying, do further split into 3 different modes each (i.e. into a 2periodic alternating one, resp. into right or lefthanded, larger periodic helical ones). Thus the count increases rather to 21 scaliform elementary honeycombs. The therein being used solids are: Pn  ngonal prism Qn  ngonal cupola (Q3 is not elementary) Sn  ngonal antiprism (S3 is not elementary) T  tetrahedron Yn  ngonal pyramid (As he uses in their original namings the attributes ortho resp. gyro heavily, but most often in a different sense as those occur in the Johnson solids, I try to avoid those terms then completely.) The nonuniform partial Stott expansion examples have no elementary counterparts, as the coes are used in an axial surrounding, which selects their 4fold symmetry (not their 3fold one). 
Beyond the convex honeycombs there are nonconvex euclidean honeycombs too. Those will add for additional possibilities the usage of
(Several ones can be seen to be Grünbaumian a priori right from having both nodes being marked at links with even denominator. The same holds true for x∞'x, which by itself would be nothing but an infinitely many covered edge. All these then are represented below accordingly. Others might be recognised to be Grünbaumian a posteriory, e.g. when using such polytopes for cells or as vertex figure. These then are mentioned by referenciation to at least one of those elements.)
_o_ _P  3/2 o< 3 >o P_  _3 o 
_o_ _3  P_ o< 3/2 >o 3/2  _P o 

o3o3o3o3/2*a3*c  o3/2o3/2o3o3/2*a3*c  o3o3/2o3o3*a3/2*c  o3o3/2o3/2o3/2*a3/2*c 

x3o3o3o3/2*a3*c  (contains 2tet) o3x3o3o3/2*a3*c  datha ditetratihemiapeiratic HC o3o3x3o3/2*a3*c  (contains 2tet) o3o3o3x3/2*a3*c  (contains 2tet) x3x3o3o3/2*a3*c  (contains 2tet) x3o3x3o3/2*a3*c  ridatha rectified ditetratihemiapeiratic HC x3o3o3x3/2*a3*c  [Grünbaumian] o3x3x3o3/2*a3*c  (contains 2tet) o3x3o3x3/2*a3*c  (contains 2tet) o3o3x3x3/2*a3*c  tetatit tritrigonary apeiratitritetratic HC x3x3x3o3/2*a3*c  tedatha truncated ditetratihemiapeiratic HC x3x3o3x3/2*a3*c  [Grünbaumian] x3o3x3x3/2*a3*c  [Grünbaumian] o3x3x3x3/2*a3*c  (contains 2thah) x3x3x3x3/2*a3*c  [Grünbaumian] 
x3/2o3/2o3o3/2*a3*c  (contains 2tet) o3/2x3/2o3o3/2*a3*c  datha o3/2o3/2x3o3/2*a3*c  (contains 2tet) o3/2o3/2o3x3/2*a3*c  (contains 2tet) x3/2x3/2o3o3/2*a3*c  [Grünbaumian] x3/2o3/2x3o3/2*a3*c  ridatha x3/2o3/2o3x3/2*a3*c  [Grünbaumian] o3/2x3/2x3o3/2*a3*c  [Grünbaumian] o3/2x3/2o3x3/2*a3*c  (contains 2tet) o3/2o3/2x3x3/2*a3*c  tetatit x3/2x3/2x3o3/2*a3*c  [Grünbaumian] x3/2x3/2o3x3/2*a3*c  [Grünbaumian] x3/2o3/2x3x3/2*a3*c  (contains 2tut) o3/2x3/2x3x3/2*a3*c  [Grünbaumian] x3/2x3/2x3x3/2*a3*c  [Grünbaumian] 
x3o3/2o3o3*a3/2*c  (contains 2tet) o3x3/2o3o3*a3/2*c  (has 2octverf) o3o3/2x3o3*a3/2*c  (contains 2tet) o3o3/2o3x3*a3/2*c  (contains 2tet) x3x3/2o3o3*a3/2*c  (contains 2tet) x3o3/2x3o3*a3/2*c  [Grünbaumian] x3o3/2o3x3*a3/2*c  (contains ∞covered {3}) o3x3/2x3o3*a3/2*c  [Grünbaumian] o3x3/2o3x3*a3/2*c  (contains 2tet) o3o3/2x3x3*a3/2*c  tetatit x3x3/2x3o3*a3/2*c  [Grünbaumian] x3x3/2o3x3*a3/2*c  (contains 2thah) x3o3/2x3x3*a3/2*c  [Grünbaumian] o3x3/2x3x3*a3/2*c  [Grünbaumian] x3x3/2x3x3*a3/2*c  [Grünbaumian] 
x3o3/2o3/2o3/2*a3/2*c  (contains 2tet) o3x3/2o3/2o3/2*a3/2*c  (has 2octverf) o3o3/2x3/2o3/2*a3/2*c  (contains 2tet) o3o3/2o3/2x3/2*a3/2*c  (contains 2tet) x3x3/2o3/2o3/2*a3/2*c  (contains 2tet) x3o3/2x3/2o3/2*a3/2*c  [Grünbaumian] x3o3/2o3/2x3/2*a3/2*c  (contains 2oct) o3x3/2x3/2o3/2*a3/2*c  [Grünbaumian] o3x3/2o3/2x3/2*a3/2*c  (contains 2tet) o3o3/2x3/2x3/2*a3/2*c  [Grünbaumian] x3x3/2x3/2o3/2*a3/2*c  [Grünbaumian] x3x3/2o3/2x3/2*a3/2*c  [Grünbaumian] x3o3/2x3/2x3/2*a3/2*c  [Grünbaumian] o3x3/2x3/2x3/2*a3/2*c  [Grünbaumian] x3x3/2x3/2x3/2*a3/2*c  [Grünbaumian] 
o3o   3/2 P'   oPo 
o3/2o   3/2 3/2   o3/2o 
_o _3  o3o< ∞ 3/2  o 

o3o3o3/2o3/2*a  o3o3/2o3o3/2*a  o3/2o3/2o3/2o3/2*a  o3o3o∞o3/2*b 
x3o3o3/2o3/2*a  octet ° o3x3o3/2o3/2*a  octet ° o3o3o3/2x3/2*a  octet ° x3x3o3/2o3/2*a  batatoh ° x3o3x3/2o3/2*a  rich ° x3o3o3/2x3/2*a  [Grünbaumian] o3x3o3/2x3/2*a  (contains 2thah) x3x3x3/2o3/2*a  tatoh ° x3x3o3/2x3/2*a  [Grünbaumian] x3o3x3/2x3/2*a  [Grünbaumian] x3x3x3/2x3/2*a  [Grünbaumian] 
x3o3/2o3o3/2*a  x3x3/2o3o3/2*a  x3o3/2x3o3/2*a  x3o3/2o3x3/2*a  [Grünbaumian] x3x3/2x3o3/2*a  [Grünbaumian] x3x3/2x3x3/2*a  [Grünbaumian] 
x3/2o3/2o3/2o3/2*a  x3/2x3/2o3/2o3/2*a  [Grünbaumian] x3/2o3/2x3/2o3/2*a  x3/2x3/2x3/2o3/2*a  [Grünbaumian] x3/2x3/2x3/2x3/2*a  [Grünbaumian] 
o3o3x∞o3/2*b  o3o3o∞x3/2*b  x3o3x∞o3/2*b  x3o3o∞x3/2*b  (contains 2thah) o3x3x∞o3/2*b  o3x3o∞x3/2*b  [Grünbaumian] o3o3x∞x3/2*b  x3x3x∞o3/2*b  x3x3o∞x3/2*b  [Grünbaumian] x3o3x∞x3/2*b  (contains 2thah) o3x3x∞x3/2*b  [Grünbaumian] x3x3x∞x3/2*b  [Grünbaumian] 
_o _3  o3/2o< ∞ 3/2  o 
_o _P  o3o< ∞' P_  o 
_o _3  o3/2o< ∞' 3_  o 

o3/2o3o∞o3/2*b  o3o3o∞'o3*b  o3o3/2o∞'o3/2*b  o3/2o3o∞'o3*b 
o3/2o3x∞o3/2*b  o3/2o3o∞x3/2*b  x3/2o3x∞o3/2*b  (contains 2thah) x3/2o3o∞x3/2*b  o3/2x3x∞o3/2*b  o3/2x3o∞x3/2*b  [Grünbaumian] o3/2o3x∞x3/2*b  x3/2x3x∞o3/2*b  [Grünbaumian] x3/2x3o∞x3/2*b  [Grünbaumian] x3/2o3x∞x3/2*b  (contains 2thah) o3/2x3x∞x3/2*b  [Grünbaumian] x3/2x3x∞x3/2*b  [Grünbaumian] 
o3o3x∞'o3*b  x3o3x∞'o3*b  o3x3x∞'o3*b  o3o3x∞'x3*b  [Grünbaumian] x3x3x∞'o3*b  x3o3x∞'x3*b  [Grünbaumian] o3x3x∞'x3*b  [Grünbaumian] x3x3x∞'x3*b  [Grünbaumian] 
o3o3/2x∞'o3/2*b  x3o3/2x∞'o3/2*b  o3x3/2x∞'o3/2*b  [Grünbaumian] o3o3/2x∞'x3/2*b  [Grünbaumian] x3x3/2x∞'o3/2*b  [Grünbaumian] x3o3/2x∞'x3/2*b  [Grünbaumian] o3x3/2x∞'x3/2*b  [Grünbaumian] x3x3/2x∞'x3/2*b  [Grünbaumian] 
o3/2o3x∞'o3*b  x3/2o3x∞'o3*b  o3/2x3x∞'o3*b  o3/2o3x∞'x3*b  [Grünbaumian] x3/2x3x∞'o3*b  [Grünbaumian] x3/2o3x∞'x3*b  [Grünbaumian] o3/2x3x∞'x3*b  [Grünbaumian] x3/2x3x∞'x3*b  [Grünbaumian] 
_o 3/2  o3/2o< ∞' 3/2  o  
o3/2o3/2o∞'o3/2*b  
o3/2o3/2x∞'o3/2*b  x3/2o3/2x∞'o3/2*b  o3/2x3/2x∞'o3/2*b  [Grünbaumian] o3/2o3/2x∞'x3/2*b  [Grünbaumian] x3/2x3/2x∞'o3/2*b  [Grünbaumian] x3/2o3/2x∞'x3/2*b  [Grünbaumian] o3/2x3/2x∞'x3/2*b  [Grünbaumian] x3/2x3/2x∞'x3/2*b  [Grünbaumian] 

oPo3o4/3o 
o4o3/2oPo 

o4o3o4/3o  o4/3o3o4/3o  o4o3/2o4o  o4o3/2o4/3o 
x4o3o4/3o  chon ° o4x3o4/3o  rich ° o4o3x4/3o  rich ° o4o3o4/3x  chon ° x4x3o4/3o  tich ° x4o3x4/3o  srich ° x4o3o4/3x  chon ° o4x3x4/3o  batch ° o4x3o4/3x  querch quasirhombated cubic HC o4o3x4/3x  quitch x4x3x4/3o  grich ° x4x3o4/3x  paqrich x4o3x4/3x  quiprich o4x3x4/3x  gaqrich x4x3x4/3x  gaquapech 
x4/3o3o4/3o  chon ° o4/3x3o4/3o  rich ° x4/3x3o4/3o  quitch x4/3o3x4/3o  querch x4/3o3o4/3x  chon ° o4/3x3x4/3o  batch ° x4/3x3x4/3o  gaqrich x4/3x3o4/3x  quipqrich x4/3x3x4/3x  quequapech 
x4o3/2o4o  o4x3/2o4o  x4x3/2o4o  x4o3/2x4o  x4o3/2o4x  o4x3/2x4o  [Grünbaumian] x4x3/2x4o  [Grünbaumian] x4x3/2o4x  x4x3/2x4x  [Grünbaumian] 
x4o3/2o4/3o  o4x3/2o4/3o  o4o3/2x4/3o  o4o3/2o4/3x  x4x3/2o4/3o  x4o3/2x4/3o  x4o3/2o4/3x  o4x3/2x4/3o  [Grünbaumian] o4x3/2o4/3x  o4o3/2x4/3x  x4x3/2x4/3o  [Grünbaumian] x4x3/2o4/3x  x4o3/2x4/3x  o4x3/2x4/3x  [Grünbaumian] x4x3/2x4/3x  [Grünbaumian] 
o4/3o3/2o4/3o 
o_ 3_ >o4/3o _3 o 
o_ 3_ >oPo 3/2 o 

o4/3o3/2o4/3o  o3o3o *b4/3o  o3o3/2o *b4o  o3o3/2o *b4/3o 
x4/3o3/2o4/3o  o4/3x3/2o4/3o  x4/3x3/2o4/3o  x4/3o3/2x4/3o  x4/3o3/2o4/3x  o4/3x3/2x4/3o  [Grünbaumian] x4/3x3/2x4/3o  [Grünbaumian] x4/3x3/2o4/3x  x4/3x3/2x4/3x  [Grünbaumian] 
x3o3o *b4/3o  octet ° o3x3o *b4/3o  rich ° o3o3o *b4/3x  chon ° x3x3o *b4/3o  tatoh ° x3o3x *b4/3o  rich ° x3o3o *b4/3x  qrahch quasirhombic demicubic HC o3x3o *b4/3x  quitch x3x3x *b4/3o  batch ° x3x3o *b4/3x  gaqrahch great quasirhombic demicubic HC x3o3x *b4/3x  querch x3x3x *b4/3x  gaqrich 
x3o3/2o *b4o  o3x3/2o *b4o  o3o3/2x *b4o  o3o3/2o *b4x  x3x3/2o *b4o  x3o3/2x *b4o  x3o3/2o *b4x  o3x3/2x *b4o  [Grünbaumian] o3x3/2o *b4x  o3o3/2x *b4x  x3x3/2x *b4o  [Grünbaumian] x3x3/2o *b4x  x3o3/2x *b4x  o3x3/2x *b4x  [Grünbaumian] x3x3/2x *b4x  [Grünbaumian] 
x3o3/2o *b4/3o  o3x3/2o *b4/3o  o3o3/2x *b4/3o  o3o3/2o *b4/3x  x3x3/2o *b4/3o  x3o3/2x *b4/3o  x3o3/2o *b4/3x  o3x3/2x *b4/3o  [Grünbaumian] o3x3/2o *b4/3x  o3o3/2x *b4/3x  x3x3/2x *b4/3o  [Grünbaumian] x3x3/2o *b4/3x  x3o3/2x *b4/3x  o3x3/2x *b4/3x  [Grünbaumian] x3x3/2x *b4/3x  [Grünbaumian] 
o_ 3/2 >oPo 3/2 o 
oPo   P 4/3   o4o 

o3/2o3/2o *b4o  o3/2o3/2o *b4/3o  o3o3o4o4/3*a  o3/2o3/2o4o4/3*a 
x3/2o3/2o *b4o  o3/2x3/2o *b4o  o3/2o3/2o *b4x  x3/2x3/2o *b4o  [Grünbaumian] x3/2o3/2x *b4o  x3/2o3/2o *b4x  o3/2x3/2o *b4x  x3/2x3/2x *b4o  [Grünbaumian] x3/2x3/2o *b4x  [Grünbaumian] x3/2o3/2x *b4x  x3/2x3/2x *b4x  [Grünbaumian] 
x3/2o3/2o *b4/3o  o3/2x3/2o *b4/3o  o3/2o3/2o *b4/3x  x3/2x3/2o *b4/3o  [Grünbaumian] x3/2o3/2x *b4/3o  x3/2o3/2o *b4/3x  o3/2x3/2o *b4/3x  x3/2x3/2x *b4/3o  [Grünbaumian] x3/2x3/2o *b4/3x  [Grünbaumian] x3/2o3/2x *b4/3x  x3/2x3/2x *b4/3x  [Grünbaumian] 
x3o3o4o4/3*a  o3x3o4o4/3*a  (has ∞covered{4} verf) o3o3x4o4/3*a  o3o3o4x4/3*a  x3x3o4o4/3*a  x3o3x4o4/3*a  (contains ∞covered {4}) x3o3o4x4/3*a  o3x3x4o4/3*a  o3x3o4x4/3*a  o3o3x4x4/3*a  x3x3x4o4/3*a  (contains ∞covered {4}) x3x3o4x4/3*a  x3o3x4x4/3*a  o3x3x4x4/3*a  x3x3x4x4/3*a  
x3/2o3/2o4o4/3*a  o3/2x3/2o4o4/3*a  o3/2o3/2x4o4/3*a  o3/2o3/2o4x4/3*a  x3/2x3/2o4o4/3*a  [Grünbaumian] x3/2o3/2x4o4/3*a  x3/2o3/2o4x4/3*a  o3/2x3/2x4o4/3*a  [Grünbaumian] o3/2x3/2o4x4/3*a  o3/2o3/2x4x4/3*a  x3/2x3/2x4o4/3*a  [Grünbaumian] x3/2x3/2o4x4/3*a  [Grünbaumian] x3/2o3/2x4x4/3*a  o3/2x3/2x4x4/3*a  [Grünbaumian] x3/2x3/2x4x4/3*a  [Grünbaumian] 
o3o   3/2 P   oPo 
_o _3  o4o< P' P_  o 

o3o3/2o4o4*a  o3o3/2o4/3o4/3*a  o4o3o4o4/3*b  o4o3o4/3o4*b 
x3o3/2o4o4*a  o3x3/2o4o4*a  o3o3/2x4o4*a  o3o3/2o4x4*a  x3x3/2o4o4*a  x3o3/2x4o4*a  x3o3/2o4x4*a  o3x3/2x4o4*a  [Grünbaumian] o3x3/2o4x4*a  o3o3/2x4x4*a  x3x3/2x4o4*a  [Grünbaumian] x3x3/2o4x4*a  x3o3/2x4x4*a  o3x3/2x4x4*a  [Grünbaumian] x3x3/2x4x4*a  [Grünbaumian] 
x3o3/2o4/3o4/3*a  o3x3/2o4/3o4/3*a  o3o3/2x4/3o4/3*a  o3o3/2o4/3x4/3*a  x3x3/2o4/3o4/3*a  x3o3/2x4/3o4/3*a  x3o3/2o4/3x4/3*a  o3x3/2x4/3o4/3*a  [Grünbaumian] o3x3/2o4/3x4/3*a  o3o3/2x4/3x4/3*a  x3x3/2x4/3o4/3*a  [Grünbaumian] x3x3/2o4/3x4/3*a  x3o3/2x4/3x4/3*a  o3x3/2x4/3x4/3*a  [Grünbaumian] x3x3/2x4/3x4/3*a  [Grünbaumian] 
x4o3o4o4/3*b  (has oct+6{4}verf) o4x3o4o4/3*b  (contains oct+6{4}) o4o3x4o4/3*b  (contains oct+6{4}) o4o3o4x4/3*b  (contains 2cube) x4x3o4o4/3*b  (contains oct+6{4}) x4o3x4o4/3*b  (contains oct+6{4}) x4o3o4x4/3*b  (contains 2cube) o4x3x4o4/3*b  (contains 2cho) o4x3o4x4/3*b  wavicoca o4o3x4x4/3*b  scoca small cubaticubatiapeiratic HC x4x3x4o4/3*b  (contains 2cho) x4x3o4x4/3*b  gepdica great prismatodiscubatiapeiratic HC x4o3x4x4/3*b  (contains ∞covered {4}) o4x3x4x4/3*b  caquiteca cubatiquasitruncated cubatiapeiratic HC x4x3x4x4/3*b  caquitpica cubatiquasitrunciprismated cubatiapeiratic HC 
x4o3o4/3o4*b  (has oct+6{4}verf) o4x3o4/3o4*b  (contains oct+6{4}) o4o3x4/3o4*b  (contains oct+6{4}) o4o3o4/3x4*b  (contains 2cube) x4x3o4/3o4*b  (contains oct+6{4}) x4o3x4/3o4*b  (contains oct+6{4}) x4o3o4/3x4*b  (contains 2cube) o4x3x4/3o4*b  (contains 2cho) o4x3o4/3x4*b  rawvicoca retrosphenoverted cubaticubatiapeiratic HC o4o3x4/3x4*b  gacoco x4x3x4/3o4*b  (contains 2cho) x4x3o4/3x4*b  spadica small prismated discubatiapeiratic HC x4o3x4/3x4*b  skivpacoca small skewverted prismatocubaticubatiapeiratic HC o4x3x4/3x4*b  cuteca cubatitruncated cubatiapeiratic HC x4x3x4/3x4*b  cutpica cubatitrunciprismated cubatiapeiratic HC 
_o 3/2  o4o< P P_  o 
_o _3  o4/3o< P' P_  o 

o4o3/2o4o4*b  o4o3/2o4/3o4/3*b  o4/3o3o4o4/3*b  o4/3o3o4/3o4*b 
x4o3/2o4o4*b  o4x3/2o4o4*b  o4o3/2x4o4*b  o4o3/2o4x4*b  x4x3/2o4o4*b  x4o3/2x4o4*b  x4o3/2o4x4*b  o4x3/2x4o4*b  [Grünbaumian] o4x3/2o4x4*b  o4o3/2x4x4*b  x4x3/2x4o4*b  [Grünbaumian] x4x3/2o4x4*b  x4o3/2x4x4*b  o4x3/2x4x4*b  [Grünbaumian] x4x3/2x4x4*b  [Grünbaumian] 
x4o3/2o4/3o4/3*b  o4x3/2o4/3o4/3*b  o4o3/2x4/3o4/3*b  o4o3/2o4/3x4/3*b  x4x3/2o4/3o4/3*b  x4o3/2x4/3o4/3*b  x4o3/2o4/3x4/3*b  o4x3/2x4/3o4/3*b  [Grünbaumian] o4x3/2o4/3x4/3*b  o4o3/2x4/3x4/3*b  x4x3/2x4/3o4/3*b  [Grünbaumian] x4x3/2o4/3x4/3*b  x4o3/2x4/3x4/3*b  o4x3/2x4/3x4/3*b  [Grünbaumian] x4x3/2x4/3x4/3*b  [Grünbaumian] 
x4/3o3o4o4/3*b  (has oct+6{4}verf) o4/3x3o4o4/3*b  (contains oct+6{4}) o4/3o3x4o4/3*b  (contains oct+6{4}) o4/3o3o4x4/3*b  (contains 2cube) x4/3x3o4o4/3*b  (contains oct+6{4}) x4/3o3x4o4/3*b  (contains oct+6{4}) x4/3o3o4x4/3*b  (contains 2cube) o4/3x3x4o4/3*b  (contains 2cho) o4/3x3o4x4/3*b  wavicoca o4/3o3x4x4/3*b  scoca x4/3x3x4o4/3*b  (contains 2cho) x4/3x3o4x4/3*b  gapdica grand prismated discubatiapeiratic HC x4/3o3x4x4/3*b  gikkiv pacoca great skewverted prismatocubaticubatiapeiratic HC o4/3x3x4x4/3*b  caquiteca x4/3x3x4x4/3*b  gacquitpica great cubatiquasitrunciprismated cubatiapeiratic HC 
x4/3o3o4/3o4*b  (has oct+6{4}verf) o4/3x3o4/3o4*b  (contains oct+6{4}) o4/3o3x4/3o4*b  (contains oct+6{4}) o4/3o3o4/3x4*b  (contains 2cube) x4/3x3o4/3o4*b  (contains oct+6{4}) x4/3o3x4/3o4*b  (contains oct+6{4}) x4/3o3o4/3x4*b  (contains 2cube) o4/3x3x4/3o4*b  (contains 2cho) o4/3x3o4/3x4*b  rawvicoca o4/3o3x4/3x4*b  gacoco x4/3x3x4/3o4*b  (contains 2cho) x4/3x3o4/3x4*b  mapdica medial prismated discubatiapeiratic HC x4/3o3x4/3x4*b  (contains ∞covered {4}) o4/3x3x4/3x4*b  cuteca x4/3x3x4/3x4*b  gactipeca great cubatitrunciprismated cubatiapeiratic HC 
_o 3/2  o4/3o< P P_  o 
_o _4  oPo< ∞ 4/3  o 

o4/3o3/2o4o4*b  o4/3o3/2o4/3o4/3*b  o3o4o∞o4/3*b  o3/2o4o∞o4/3*b 
x4/3o3/2o4o4*b  o4/3x3/2o4o4*b  o4/3o3/2x4o4*b  o4/3o3/2o4x4*b  x4/3x3/2o4o4*b  x4/3o3/2x4o4*b  x4/3o3/2o4x4*b  o4/3x3/2x4o4*b  [Grünbaumian] o4/3x3/2o4x4*b  o4/3o3/2x4x4*b  x4/3x3/2x4o4*b  [Grünbaumian] x4/3x3/2o4x4*b  x4/3o3/2x4x4*b  o4/3x3/2x4x4*b  [Grünbaumian] x4/3x3/2x4x4*b  [Grünbaumian] 
x4/3o3/2o4/3o4/3*b  o4/3x3/2o4/3o4/3*b  o4/3o3/2x4/3o4/3*b  o4/3o3/2o4/3x4/3*b  x4/3x3/2o4/3o4/3*b  x4/3o3/2x4/3o4/3*b  x4/3o3/2o4/3x4/3*b  o4/3x3/2x4/3o4/3*b  [Grünbaumian] o4/3x3/2o4/3x4/3*b  o4/3o3/2x4/3x4/3*b  x4/3x3/2x4/3o4/3*b  [Grünbaumian] x4/3x3/2o4/3x4/3*b  x4/3o3/2x4/3x4/3*b  o4/3x3/2x4/3x4/3*b  [Grünbaumian] x4/3x3/2x4/3x4/3*b  [Grünbaumian] 
o3o4x∞o4/3*b  o3o4o∞x4/3*b  x3o4x∞o4/3*b  x3o4o∞x4/3*b  o3x4x∞o4/3*b  o3x4o∞x4/3*b  o3o4x∞x4/3*b  x3x4x∞o4/3*b  x3x4o∞x4/3*b  x3o4x∞x4/3*b  o3x4x∞x4/3*b  x3x4x∞x4/3*b  
o3/2o4x∞o4/3*b  o3/2o4o∞x4/3*b  x3/2o4x∞o4/3*b  x3/2o4o∞x4/3*b  o3/2x4x∞o4/3*b  o3/2x4o∞x4/3*b  o3/2o4x∞x4/3*b  x3/2x4x∞o4/3*b  [Grünbaumian] x3/2x4o∞x4/3*b  [Grünbaumian] x3/2o4x∞x4/3*b  o3/2x4x∞x4/3*b  x3/2x4x∞x4/3*b  [Grünbaumian] 
_o _P  o3o< ∞' P_  o 
_o _P  o3/2o< ∞' P_  o 

o3o4o∞'o4*b  o3o4/3o∞'o4/3*b  o3/2o4o∞'o4*b  o3/2o4/3o∞'o4/3*b 
o3o4x∞'o4*b  x3o4x∞'o4*b  o3x4x∞'o4*b  o3o4x∞'x4*b  [Grünbaumian] x3x4x∞'o4*b  x3o4x∞'x4*b  [Grünbaumian] o3x4x∞'x4*b  [Grünbaumian] x3x4x∞'x4*b  [Grünbaumian] 
o3o4/3x∞'o4/3*b  x3o4/3x∞'o4/3*b  o3x4/3x∞'o4/3*b  o3o4/3x∞'x4/3*b  [Grünbaumian] x3x4/3x∞'o4/3*b  x3o4/3x∞'x4/3*b  [Grünbaumian] o3x4/3x∞'x4/3*b  [Grünbaumian] x3x4/3x∞'x4/3*b  [Grünbaumian] 
o3/2o4x∞'o4*b  x3/2o4x∞'o4*b  o3/2x4x∞'o4*b  o3/2o4x∞'x4*b  [Grünbaumian] x3/2x4x∞'o4*b  [Grünbaumian] x3/2o4x∞'x4*b  [Grünbaumian] o3/2x4x∞'x4*b  [Grünbaumian] x3/2x4x∞'x4*b  [Grünbaumian] 
o3/2o4/3x∞'o4/3*b  x3/2o4/3x∞'o4/3*b  o3/2x4/3x∞'o4/3*b  o3/2o4/3x∞'x4/3*b  [Grünbaumian] x3/2x4/3x∞'o4/3*b  [Grünbaumian] x3/2o4/3x∞'x4/3*b  [Grünbaumian] o3/2x4/3x∞'x4/3*b  [Grünbaumian] x3/2x4/3x∞'x4/3*b  [Grünbaumian] 
° : Those such marked honeycombs come out again to be convex after all.
' (in the context of P') : Refering to the inversion between prograde and retrograde polygons {n/d} ↔ {n/(nd)}.
(Note that as abstract polytopes {∞} and {∞'} are well to be distinguished, and thus esp. their usage as link marks within Dynkin diagrams, but that their geometric realizations within a euclidean space context are exactly the same aze.)
 4D Tetracombs (up) 
o4o3o3o4o = C_{4} = R(5) 
o3o3o *b3o4o = B_{4} = S(5) 
o3o3o *b3o *b3o = D_{4} = Q(5) 
o3o3o3o3o3*a = A_{4} = P(5) 
x4o3o3o4o  test  O1 o4x3o3o4o  rittit  O87 o4o3x3o4o  icot  O88 x4x3o3o4o  tattit  O89 x4o3x3o4o  srittit  O90 x4o3o3x4o  sidpitit  O91 x4o3o3o4x  test  O1 o4x3x3o4o  batitit  O92 o4x3o3x4o  ricot  O93 x4x3x3o4o  grittit  O94 x4x3o3x4o  potatit  O95 x4x3o3o4x  capotat  O96 x4o3x3x4o  prittit  O97 x4o3x3o4x  scartit  O98 o4x3x3x4o  ticot  O99 x4x3x3x4o  gippittit  O100 x4x3x3o4x  gicartit  O101 x4x3o3x4x  captatit  O102 x4x3x3x4x  otatit  O103 s4o3o3o4o  hext  O104 s4o3x3o4o  thext  O105 s4o3o3x4o  bricot  O106 s4o3o3o4x  siphatit  O108 s4o3x3x4o  bithit  O107 s4o3x3o4x  pithatit  O109 s4o3o3x4x  pirhatit  O110 s4o3x3x4x  giphatit  O111 s4o3x3o4s  cesratit  **) s4o3o3o4s  hext  O104 o4s3s3s4o  sadit  O133 s4o3o3o4s'  rittit  O87 
x3o3o *b3o4o  hext  O104 o3x3o *b3o4o  icot  O88 o3o3o *b3x4o  rittit  O87 o3o3o *b3o4x  test  O1 x3x3o *b3o4o  thext  O105 x3o3x *b3o4o  rittit  O87 x3o3o *b3x4o  bricot  O106 x3o3o *b3o4x  siphatit  O108 o3x3o *b3x4o  batitit  O92 o3x3o *b3o4x  srittit  O90 o3o3o *b3x4x  tattit  O89 x3x3x *b3o4o  batitit  O92 x3x3o *b3x4o  bithit  O107 x3x3o *b3o4x  pithatit  O109 x3o3x *b3x4o  ricot  O93 x3o3x *b3o4x  sidpitit  O91 x3o3o *b3x4x  pirhatit  O110 o3x3o *b3x4x  grittit  O94 x3x3x *b3x4o  ticot  O99 x3x3x *b3o4x  prittit  O97 x3x3o *b3x4x  giphatit  O111 x3o3x *b3x4x  potatit  O95 x3x3x *b3x4x  gippittit  O100 x3o3o *b3o4s  rittit  O87 s3s3s *b3s4o  sadit  O133 
x3o3o *b3o *b3o  hext  O104 o3x3o *b3o *b3o  icot  O88 x3x3o *b3o *b3o  thext  O105 x3o3x *b3o *b3o  rittit  O87 x3x3x *b3o *b3o  batitit  O92 x3o3x *b3x *b3o  bricot  O106 x3x3x *b3x *b3o  bithit  O107 x3o3x *b3x *b3x  ricot  O93 x3x3x *b3x *b3x  ticot  O99 s3s3s *b3s *b3s  sadit  O133 
x3o3o3o3o3*a  cypit  O134 x3x3o3o3o3*a  cytopit  O135 x3o3x3o3o3*a  scyropot  O136 x3x3x3o3o3*a  gocyropit  O137 x3x3o3x3o3*a  cypropit  O138 x3x3x3x3o3*a  gocypapit  O139 x3x3x3x3x3*a  otcypit  O140 
o3o3o4o3o = F_{4} = U(5) 
o∞o o4o3o4o = A_{1}×C_{3} = W(2)×R(4) 
o∞o o3o3o *d4o = A_{1}×B_{3} = W(2)×S(4) 
o∞o o3o3o3o3*c = A_{1}×A_{3} = W(2)×P(4) 
x3o3o4o3o  hext  O104 o3x3o4o3o  icot  O88 o3o3x4o3o  bricot  O106 o3o3o4x3o  ricot  O93 o3o3o4o3x  icot  O88 x3x3o4o3o  thext  O105 x3o3x4o3o  ricot  O93 x3o3o4x3o  spaht  O122 x3o3o4o3x  scicot  O121 o3x3x4o3o  bithit  O107 o3x3o4x3o  sibricot  O116 o3x3o4o3x  spict  O115 o3o3x4x3o  baticot  O113 o3o3x4o3x  sricot  O112 o3o3o4x3x  ticot  O99 x3x3x4o3o  ticot  O99 x3x3o4x3o  pataht  O128 x3x3o4o3x  capicot  O127 x3o3x4x3o  praht  O125 prismatorhombated demitesseractic TC, great tetracontaoctachoric TC x3o3x4o3x  scaricot  O124 x3o3o4x3x  capoht  O123 o3x3x4x3o  gibricot  O119 o3x3x4o3x  pricot  O118 o3x3o4x3x  paticot  O117 o3o3x4x3x  gricot  O114 x3x3x4x3o  gipaht  O131 great prismated demitesseractic TC x3x3x4o3x  gicaricot  O130 x3x3o4x3x  capticot  O129 x3o3x4x3x  gicaroht  O126 o3x3x4x3x  gippict  O120 x3x3x4x3x  otit  O132 o3o3o4s3s  sadit  O133 s3s3s4o3o  sadit  O133 x3o3o4s3s  capshot  **) o3x3o4s3s  paltite  **) o3o3x4s3s  sricot  O112 s3s3s4o3x  capirsit  **) x3x3o4s3s  capsthat  **) x3o3x4s3s  scaricot  O124 o3x3x4s3s  pricot  O118 x3x3x4s3s  gicaricot  O130 
x∞o x4o3o4o  test  O1 x∞o o4x3o4o  ricpit  O15 rectifiedcubic prismatic TC x∞x x4o3o4o  test  O1 x∞x o4x3o4o  ricpit  O15 x∞o x4x3o4o  ticpit  O14 truncatedcubic prismatic TC x∞o x4o3x4o  cacpit  O17 cantellatedcubic prismatic TC x∞o x4o3o4x  test  O1 x∞o o4x3x4o  bitticpit  O16 bitruncatedcubic prismatic TC x∞x x4x3o4o  ticpit  O14 x∞x x4o3x4o  cacpit  O17 x∞x x4o3o4x  test  O1 x∞x o4x3x4o  bitticpit  O16 x∞o x4x3x4o  catcupit  O18 cantitruncatedcubic prismatic TC x∞o x4x3o4x  rutcupit  O19 runcitruncatedcubic prismatic TC x∞x x4x3x4o  catcupit  O18 x∞x x4x3o4x  rutcupit  O19 x∞o x4x3x4x  otacpit  O20 omnitruncatedcubic prismatic TC x∞x x4x3x4x  otacpit  O20 
x∞o x3o3o *d4o  acpit  O21 alternatedcubic prismatic TC x∞o o3x3o *d4o  ricpit  O15 x∞o o3o3o *d4x  test  O1 x∞x x3o3o *d4o  acpit  O21 x∞x o3x3o *d4o  ricpit  O15 x∞x o3o3o *d4x  test  O1 x∞o x3x3o *d4o  tacpit  O25 truncatedalternatedcubic prismatic TC x∞o x3o3x *d4o  ricpit  O15 x∞o x3o3o *d4x  racpit  O26 runcinatedalternatedcubic prismatic TC x∞o o3x3o *d4x  ticpit  O14 x∞x x3x3o *d4o  tacpit  O25 x∞x x3o3x *d4o  ricpit  O15 x∞x x3o3o *d4x  racpit  O26 x∞x o3x3o *d4x  ticpit  O14 x∞o x3x3x *d4o  bitticpit  O16 x∞o x3x3o *d4x  rucacpit  O28 runcicanticcubic prismatic TC x∞o x3o3x *d4x  cacpit  O17 x∞x x3x3x *d4o  bitticpit  O16 x∞x x3x3o *d4x  rucacpit  O28 x∞x x3o3x *d4x  cacpit  O17 x∞o x3x3x *d4x  catcupit  O18 x∞x x3x3x *d4x  catcupit  O18 
x∞o x3o3o3o3*c  acpit  O21 x∞x x3o3o3o3*c  acpit  O21 x∞o x3x3o3o3*c  quacpit  O27 quartercubic prismatic TC x∞o x3o3x3o3*c  ricpit  O15 x∞x x3x3o3o3*c  quacpit  O27 x∞x x3o3x3o3*c  ricpit  O15 x∞o x3x3x3o3*c  tacpit  O25 x∞x x3x3x3o3*c  tacpit  O25 x∞o x3x3x3x3*c  bitticpit  O16 x∞x x3x3x3x3*c  bitticpit  O16 
o3o6o o3o6o = G_{2}×G_{2} = V(3)^{2} 
o3o6o o4o4o = G_{2}×C_{2} = V(3)×R(3) 
o3o6o o3o3o3*d = G_{2}×A_{2} = V(3)×P(3) 
o4o4o o4o4o = C_{2}×C_{2} = R(3)^{2} 
x3o6o x3o6o  tribbit  O29 x3o6o o3x6o  tathibbit  O32 triangulartrihexagonal duoprismatic TC x3o6o o3o6x  thibbit  O30 triangularhexagonal duoprismatic TC o3x6o o3x6o  thabbit  O56 trihexagonal duoprismatic TC o3x6o o3o6x  hithibbit  O41 hexagonaltrihexagonal duoprismatic TC o3o6x o3o6x  hibbit  O39 hexagonal duoprismatic TC x3x6o x3o6o  thibbit  O30 x3x6o o3x6o  hithibbit  O41 x3x6o o3o6x  hibbit  O39 x3o6x x3o6o  trithit  O35 triangularrhombitrihexagonal TC x3o6x o3x6o  thrathibbit  O59 x3o6x o3o6x  harhibit  O44 hexagonalrhombihexagonal duoprismatic TC o3x6x x3o6o  tathobit  O34 triangulartomohexagonal duoprismatic TC o3x6x o3x6o  thathobit  O58 trihexagonaltomohexagonal duoprismatic TC o3x6x o3o6x  hithobit  O43 hexagonaltomohexagonal duoprismatic TC x3x6x x3o6o  totuthit  O36 triangularomnitruncatedtrihexagonal TC x3x6x o3x6o  thot thibbit  O60 x3x6x o3o6x  hot thibbit  O45 x3x6o x3x6o  hibbit  O39 x3x6o x3o6x  harhibit  O44 x3x6o o3x6x  hithobit  O43 x3o6x x3o6x  rithbit  O74 rhombitrihexagonal duoprismatic TC x3o6x o3x6x  thorahbit  O70 o3x6x o3x6x  thobit  O69 tomohexagonal duoprismatic TC x3x6x x3x6o  hot thibbit  O45 x3x6x x3o6x  rathotathibit  O75 x3x6x o3x6x  thoot thibbit  O71 x3x6x x3x6x  otathibbit  O78 omnitruncatedtrihexagonal duoprismatic TC x3o6o s3s6s  tisthit  O38 o3x6o s3s6s  thisthibbit  O62 trihexagonalsimotrihexagonal duoprismatic TC o3o6x s3s6s  hasithbit  O47 hexagonalsimotrihexagonal duoprismatic TC x3x6o s3s6s  hasithbit  O47 x3o6x s3s6s  rithsithbit  O77 o3x6x s3s6s  thosithbit  O73 tomohexagonalsimotrihexagonal duoprismatic TC x3x6x s3s6s  otsithbit  O80 omnitruncatedsimotrihexagonal duoprismatic TC s3s6s s'3s'6s'  sithbit  O83 simotrihexagonal duoprismatic TC 
x3o6o x4o4o  tisbat  O2 triangularsquare duoprismatic TC x3o6o o4x4o  tisbat  O2 o3x6o x4o4o  thisbit  O5 trihexagonalsquare duoprismatic TC o3x6o o4x4o  thisbit  O5 o3o6x x4o4o  shibbit  O3 squarehexagonal duoprismatic TC o3o6x o4x4o  shibbit  O3 x3x6o x4o4o  shibbit  O3 x3x6o o4x4o  shibbit  O3 x3o6x x4o4o  rithsibbit  O8 rhombitrihexagonalsquare duoprismatic TC x3o6x o4x4o  rithsibbit  O8 o3x6x x4o4o  thosbit  O7 tomohexagonalsquare duoprismatic TC o3x6x o4x4o  thosbit  O7 x3o6o x4x4o  tatosbit  O33 triangulartomosquare duoprismatic TC x3o6o x4o4x  tisbat  O2 o3x6o x4x4o  thatosbit  O57 trihexagonaltomosquare duoprismatic TC o3x6o x4o4x  thisbit  O5 o3o6x x4x4o  hitosbit  O42 hexagonaltomosquare duoprismatic TC o3o6x x4o4x  shibbit  O3 x3x6x x4o4o  otathisbit  O9 x3x6x o4x4o  otathisbit  O9 x3x6o x4x4o  hitosbit  O42 x3x6o x4o4x  shibbit  O3 x3o6x x4x4o  tosrithbit  O65 tomosquarerhombitrihexagonal duoprismatic TC x3o6x x4o4x  rithsibbit  O8 o3x6x x4x4o  tosthobit  O64 tomosquaretomohexagonal duoprismatic TC o3x6x x4o4x  thosbit  O7 x3o6o x4x4x  tatosbit  O33 o3x6o x4x4x  thatosbit  O57 o3o6x x4x4x  hitosbit  O42 x3x6x x4x4o  tosot thibbit  O66 x3x6x x4o4x  otathisbit  O9 x3x6o x4x4x  hitosbit  O42 x3o6x x4x4x  tosrithbit  O65 o3x6x x4x4x  tosthobit  O64 x3x6x x4x4x  tosot thibbit  O66 x3o6o s4s4s  tasist  O37 triangularsimosquare TC o3x6o s4s4s  thisosbit  O61 trihexagonalsimosquare duoprismatic TC o3o6x s4s4s  hisosbit  O46 hexagonalsimosquare duoprismatic TC x3x6o s4s4s  hisosbit  O46 x3o6x s4s4s  rithsisbit  O76 rhombitrihexagonalsimosquare duoprismatic TC o3x6x s4s4s  thosisbit  O72 tomohexagonalsimosquare duoprismatic TC x3x6x s4s4s  otsisbit  O79 omnitruncatedsimosquare duoprismatic TC s3s6s x4o4o  sithsobit  O11 simotrihexagonalsquare duoprismatic TC s3s6s o4x4o  sithsobit  O11 s3s6s x4x4o  tosasithbit  O68 tomosquaresimotrihexagonal duoprismatic TC s3s6s x4o4x  sithsobit  O11 s3s6s x4x4x  tosasithbit  O68 s3s6s s'4s'4s'  sissithbit  O82 simosquaresimotrihexagonal duoprismatic TC 
x3o6o x3o3o3*d  tribbit  O29 o3x6o x3o3o3*d  tathibbit  O32 o3o6x x3o3o3*d  thibbit  O30 x3x6o x3o3o3*d  thibbit  O30 x3o6x x3o3o3*d  trithit  O35 o3x6x x3o3o3*d  tathobit  O34 x3o6o x3x3o3*d  tathibbit  O32 o3x6o x3x3o3*d  thabbit  O56 o3o6x x3x3o3*d  hithibbit  O41 x3x6x x3o3o3*d  totuthit  O36 x3x6o x3x3o3*d  hithibbit  O41 x3o6x x3x3o3*d  thrathibbit  O59 o3x6x x3x3o3*d  thathobit  O58 x3o6o x3x3x3*d  thibbit  O30 o3x6o x3x3x3*d  hithibbit  O41 o3o6x x3x3x3*d  hibbit  O39 x3x6x x3x3o3*d  thot thibbit  O60 x3x6o x3x3x3*d  hibbit  O39 x3o6x x3x3x3*d  harhibit  O44 o3x6x x3x3x3*d  hithobit  O43 x3x6x x3x3x3*d  hot thibbit  O45 s3s6s x3o3o3*d  tisthit  O38 s3s6s x3x3o3*d  thisthibbit  O62 s3s6s x3x3x3*d  hasithbit  O47 
x4o4o x4o4o  test  O1 x4o4o o4x4o  test  O1 o4x4o o4x4o  test  O1 x4x4o x4o4o  tososbit  O6 tomosquaresquare duoprismatic TC x4x4o o4x4o  tososbit  O6 x4o4x x4o4o  test  O1 x4o4x o4x4o  test  O1 x4x4x x4o4o  tososbit  O6 x4x4x o4x4o  tososbit  O6 x4x4o x4x4o  tosbit  O63 tomosquare duoprismatic TC x4x4o x4o4x  tososbit  O6 x4o4x x4o4x  test  O1 x4x4x x4x4o  tosbit  O63 x4x4x x4o4x  tososbit  O6 x4x4x x4x4x  tosbit  O63 s4s4s x4o4o  sisosbit  O10 simosquaresquare duoprismatic TC s4s4s o4x4o  sisosbit  O10 s4s4s x4x4o  tosisasbit  O67 tomosquaresimosquare duoprismatic TC s4s4s x4o4x  sisosbit  O10 s4s4s x4x4x  tosisasbit  O67 s4s4s s4s4s  sisbit  O81 simosquare duoprismatic TC 
o4o4o o3o3o3*d = C_{2}×A_{2} = R(3)×P(3) 
o3o3o3*a o3o3o3*d = A_{2}×A_{2} = P(3)^{2} 
o∞o o∞o o3o6o = A_{1}×A_{1}×G_{2} = W(2)^{2}×V(3) 
o∞o o∞o o4o4o = A_{1}×A_{1}×C_{2} = W(2)^{2}×R(3) 
x4o4o x3o3o3*d  tisbat  O2 o4x4o x3o3o3*d  tisbat  O2 x4x4o x3o3o3*d  tatosbit  O33 x4o4x x3o3o3*d  tisbat  O2 x4o4o x3x3o3*d  thisbit  O5 o4x4o x3x3o3*d  thisbit  O5 x4x4x x3o3o3*d  tatosbit  O33 x4x4o x3x3o3*d  thatosbit  O57 x4o4x x3x3o3*d  thisbit  O5 x4o4o x3x3x3*d  shibbit  O3 o4x4o x3x3x3*d  shibbit  O3 x4x4x x3x3o3*d  thatosbit  O57 x4x4o x3x3x3*d  hitosbit  O42 x4o4x x3x3x3*d  shibbit  O3 x4x4x x3x3x3*d  hitosbit  O42 s4s4s x3o3o3*d  tasist  O37 s4s4s x3x3o3*d  thisosbit  O61 s4s4s x3x3x3*d  hisosbit  O46 
x3o3o3*a x3o3o3*d  tribbit  O29 x3o3o3*a x3x3o3*d  tathibbit  O32 x3x3o3*a x3x3o3*d  thabbit  O56 x3o3o3*a x3x3x3*d  thibbit  O30 x3x3o3*a x3x3x3*d  hithibbit  O41 x3x3x3*a x3x3x3*d  hibbit  O39 
x∞o x∞o x3o6o  tisbat  O2 x∞o x∞o o3x6o  thisbit  O5 x∞o x∞o o3o6x  shibbit  O3 x∞x x∞o x3o6o  tisbat  O2 x∞x x∞o o3x6o  thisbit  O5 x∞x x∞o o3o6x  shibbit  O3 x∞o x∞o x3x6o  shibbit  O3 x∞o x∞o x3o6x  rithsibbit  O8 x∞o x∞o o3x6x  thosbit  O7 x∞x x∞x x3o6o  tisbat  O2 x∞x x∞x o3x6o  thisbit  O5 x∞x x∞x o3o6x  shibbit  O3 x∞x x∞o x3x6o  shibbit  O3 x∞x x∞o x3o6x  rithsibbit  O8 x∞x x∞o o3x6x  thosbit  O7 x∞o x∞o x3x6x  otathisbit  O9 x∞x x∞x x3x6o  shibbit  O3 x∞x x∞x x3o6x  rithsibbit  O8 x∞x x∞x o3x6x  thosbit  O7 x∞x x∞o x3x6x  otathisbit  O9 x∞x x∞x x3x6x  otathisbit  O9 x∞o x∞o s3s6s  sithsobit  O11 
x∞o x∞o x4o4o  test  O1 x∞o x∞o o4x4o  test  O1 x∞x x∞o x4o4o  test  O1 x∞x x∞o o4x4o  test  O1 x∞o x∞o x4x4o  tososbit  O6 x∞o x∞o x4o4x  test  O1 x∞x x∞x x4o4o  test  O1 x∞x x∞x o4x4o  test  O1 x∞x x∞o x4x4o  tososbit  O6 x∞x x∞o x4o4x  test  O1 x∞o x∞o x4x4x  tososbit  O6 x∞x x∞x x4x4o  tososbit  O6 x∞x x∞x x4o4x  test  O1 x∞x x∞o x4x4x  tososbit  O6 x∞x x∞x x4x4x  tososbit  O6 x∞o x∞o s4s4s  sisosbit  O10 
o∞o o∞o o3o3o3*e = A_{1}×A_{1}×A_{2} = W(2)^{2}×P(3) 
o∞o o∞o o∞o o∞o = A_{1}×A_{1}×A_{1}×A_{1} = W(2)^{4}  other uniforms  
x∞o x∞o x3o3o3*e  tisbat  O2 x∞x x∞o x3o3o3*e  tisbat  O2 x∞o x∞o x3x3o3*e  thisbit  O5 x∞x x∞x x3o3o3*e  tisbat  O2 x∞x x∞o x3x3o3*e  thisbit  O5 x∞o x∞o x3x3x3*e  shibbit  O3 x∞x x∞x x3x3o3*e  thisbit  O5 x∞x x∞o x3x3x3*e  shibbit  O3 x∞x x∞x x3x3x3*e  shibbit  O3 
x∞o x∞o x∞o x∞o  test  O1 x∞x x∞o x∞o x∞o  test  O1 x∞x x∞x x∞o x∞o  test  O1 x∞x x∞x x∞x x∞o  test  O1 x∞x x∞x x∞x x∞x  test  O1 
elong( x3o3o3o3o3*a )  ecypit  O141 elongated cyclopentachoric TC, elongated pentachoricdispentachoric TC schmo( x3o3o3o3o3*a )  zucypit  O142 schmoozed cyclopentachoric TC, schmoozed pentachoricdispentachoric TC elongschmo( x3o3o3o3o3*a )  ezucypit  O143 elongated schmoozed cyclopentachoric TC, elongated schmoozed pentachoricdispentachoric TC elong( x3o6o x3o6o )  etbit  O31 elongated triangular duoprismatic TC elong( x3o6o o3x6o )  etothbit  O49 elongated triangulartrihexagonal duoprismatic TC elong( x3o6o o3o6x )  ethibit  O40 elongated triangularhexagonal duoprismatic TC elong( x3o6o x3x6o )  ethibit  O40 elong( x3o6o x3o6x )  etrithit  O52 elongated triangularrhombitrihexagonal TC elong( x3o6o o3x6x )  etathobit  O51 elongated triangulartomohexagonal duoprismatic TC elong( x3o6o x3x6x )  etotithat  O53 elongated triangularomnitruncatedtrihexagonal TC elong( x3o6o s3s6s )  etasithit  O55 elongated triangularsimotrihexagonal TC elong( elong( x3o6o x3o6o ))  betobit  O48 bielongated triangular duoprismatic TC elong( x3o6o elong( x3o6o ))  betobit  O48 x3o6o elong( x3o6o )  etbit  O31 o3x6o elong( x3o6o )  etothbit  O49 o3o6x elong( x3o6o )  ethibit  O40 x3x6o elong( x3o6o )  ethibit  O40 x3o6x elong( x3o6o )  etrithit  O52 o3x6x elong( x3o6o )  etathobit  O51 x3x6x elong( x3o6o )  etotithat  O53 s3s6s elong( x3o6o )  etasithit  O55 elong( x3o6o ) elong( x3o6o )  betobit  O48 elong( x3o6o x4o4o )  etsobit  O4 elongated triangularsquare duoprismatic TC elong( x3o6o x4x4o )  etatosbit  O50 elongated triangulartomosquare duoprismatic TC elong( x3o6o x4x4x )  etatosbit  O50 elong( x3o6o s4s4s )  etasist  O54 elongated triangularsimosquare TC gyro( x3o6o x4o4o )  gytosbit  O12 gyrated triangularsquare duoprismatic TC gyroelong( x3o6o x4o4o )  egytsobit  O13 elongated gyrated triangularsquare duoprismatic TC bigyro( x3o6o x4o4o )  bigytsbit  O84 bigyrated triangularsquare duoprismatic TC bigyroelong( x3o6o x4o4o )  ebiytsbit  O85 elongated bigyrated triangularsquare duoprismatic TC prismatogyro( x3o6o x4o4o )  pegytsbit  O86 prismatoelongated gyrated triangularsquare duoprismatic TC gyro( elong( x3o6o ) x4o4o )  egytsobit  O13 bigyro( elong( x3o6o ) x4o4o )  ebiytsbit  O85 elong( x3o6o ) x4o4o  etsobit  O4 elong( x3o6o ) x4x4o  etatosbit  O50 elong( x3o6o ) x4x4x  etatosbit  O50 elong( x3o6o ) s4s4s  etasist  O54 elong( x3o6o x3o3o3*d )  etbit  O31 elong( x3o6o x3x3o3*d )  etothbit  O49 elong( x3o6o x3x3x3*d )  ethibit  O40 elong( o3x6o x3o3o3*d )  etothbit  O49 elong( o3o6x x3o3o3*d )  ethibit  O40 elong( x3x6o x3o3o3*d )  ethibit  O40 elong( x3o6x x3o3o3*d )  etrithit  O52 elong( o3x6x x3o3o3*d )  etathobit  O51 elong( x3x6x x3o3o3*d )  etotithat  O53 elong (s3s6s x3o3o3*d )  etasithit  O55 elong( elong( x3o6o x3o3o3*d ))  betobit  O48 elong( x3o6o elong( x3o3o3*d ))  betobit  O48 elong( elong( x3o6o ) x3o3o3*d )  betobit  O48 x3o6o elong( x3o3o3*d )  etbit  O31 o3x6o elong( x3o3o3*d )  etothbit  O49 o3o6x elong( x3o3o3*d )  ethibit  O40 x3x6o elong( x3o3o3*d )  ethibit  O40 x3o6x elong( x3o3o3*d )  etrithit  O52 o3x6x elong( x3o3o3*d )  etathobit  O51 x3x6x elong( x3o3o3*d )  etotithat  O53 s3s6s elong( x3o3o3*d )  etasithit  O55 elong( x3o6o ) x3o3o3*d  etbit  O31 elong( x3o6o ) x3x3o3*d  etothbit  O49 elong( x3o6o ) x3x3x3*d  ethibit  O40 elong( x3o6o ) elong( x3o3o3*d )  betobit  O48 elong( x3o3o3*a x4x4o )  etatosbit  O50 elong( x3o3o3*a x4x4x )  etatosbit  O50 elong( x3o3o3*a s4s4s )  etasist  O54 gyro( x3o3o3*a x4o4o )  gytosbit  O12 gyroelong( x3o3o3*a x4o4o )  egytsobit  O13 bigyro( x3o3o3*a x4o4o )  bigytsbit  O84 bigyroelong( x3o3o3*a x4o4o )  ebiytsbit  O85 prismatogyro( x3o3o3*a x4o4o )  pegytsbit  O86 gyro( elong( x3o3o3*a ) x4o4o )  egytsobit  O13 bigyro( elong( x3o3o3*a ) x4o4o )  ebiytsbit  O85 elong( x3o3o3*a ) x4x4o  etatosbit  O50 elong( x3o3o3*a ) x4x4x  etatosbit  O50 elong( x3o3o3*a ) s4s4s  etasist  O54 elong( x3o3o3*a x3o3o3*d )  etbit  O31 elong( x3o3o3*a x3x3o3*d )  etothbit  O49 elong( x3o3o3*a x3x3x3*d )  ethibit  O40 elong( elong( x3o3o3*a x3o3o3*d ))  betobit  O48 elong( x3o3o3*a elong( x3o3o3*d ))  betobit  O48 x3o3o3*a elong( x3o3o3*d )  etbit  O31 x3x3o3*a elong( x3o3o3*d )  etothbit  O49 x3x3x3*a elong( x3o3o3*d )  ethibit  O40 elong( x3o3o3*a ) elong( x3o3o3*d )  betobit  O48 elong( x∞o x3o3o *d4o )  eacpit  O23 elongatedalternatedcubic prismatic TC gyro( x∞o x3o3o *d4o )  gyacpit  O22 gyratedalternatedcubic prismatic TC gyroelong( x∞o x3o3o *d4o )  gyeacpit  O24 gyratedelongatedalternatedcubic prismatic TC x∞o elong( x3o3o *d4o )  eacpit  O23 x∞o gyro( x3o3o *d4o )  gyacpit  O22 x∞o gyroelong( x3o3o *d4o )  gyeacpit  O24 elong( x∞o x3o3o3o3*c )  eacpit  O23 gyro( x∞o x3o3o3o3*c )  gyacpit  O22 gyroelong( x∞o x3o3o3o3*c )  gyeacpit  O24 x∞o elong( x3o3o3o3*c )  eacpit  O23 x∞o gyro( x3o3o3o3*c )  gyacpit  O22 x∞o gyroelong( x3o3o3o3*c )  gyeacpit  O24 gyro( x∞o x∞o x3o6o )  gytosbit  O12 gyroelong( x∞o x∞o x3o6o )  egytsobit  O13 bigyro( x∞o x∞o x3o6o )  bigytsbit  O84 bigyroelong( x∞o x∞o x3o6o )  ebiytsbit  O85 prismatogyro( x∞o x∞o x3o6o )  pegytsbit  O86 elong( x∞o gyro( x∞o x3o6o ))  egytsobit  O13 gyro( x∞o gyro( x∞o o3x6o ))  bigytsbit  O84 gyroelong( x∞o gyro( x∞o o3x6o ))  ebiytsbit  O85 gyro( x∞o x∞o elong( x3o6o ))  egytsobit  O13 bigyro( x∞o x∞o elong( x3o6o ))  ebiytsbit  O85 x∞o elong( x∞o x3o6o )  etsobit  O4 x∞o gyro( x∞o x3o6o )  gytosbit  O12 x∞o gyroelong( x∞o x3o6o )  egytsobit  O13 gyro( x∞o x∞o x3o3o3*e )  gytosbit  O12 gyroelong( x∞o x∞o x3o3o3*e )  egytsobit  O13 bigyro( x∞o x∞o x3o3o3*e )  bigytsbit  O84 bigyroelong( x∞o x∞o x3o3o3*e )  ebiytsbit  O85 prismatogyro( x∞o x∞o x3o3o3*e )  pegytsbit  O86 elong( x∞o gyro( x∞o x3o3o3*e ))  egytsobit  O13 gyro( x∞o gyro( x∞o x3x3o3*e ))  bigytsbit  O84 gyroelong( x∞o gyro( x∞o x3x3o3*e ))  ebiytsbit  O85 gyro( x∞o x∞o elong( x3o3o3*e ))  egytsobit  O13 bigyro( x∞o x∞o elong( x3o3o3*e ))  ebiytsbit  O85 x∞o gyro( x∞o x3o3o3*e )  gytosbit  O12 x∞o gyroelong( x∞o x3o3o3*e )  egytsobit  O13 ... schmo = ...00... gyro = ...0101... bigyro = ...012012... prismatogyro = ...01020102... 
** These alternated facetings well can be varied to get all equal edge lengths, but still will not become uniform. In fact those relaxations would qualify as scaliform tetracombs only. – This is why they do not have an Olshevsky number either.
In 4D too there are several different lattices.
In the pentachoric area there are 2 lattices, A_{4} and A_{4}*. The latter of which can be considered as a superposition of 5 of the former, each having a 1/5 rotated Dynkin diagram. The Delone complex of A_{4} is cypit. The Voronoi complex of A_{4}* is otcypit.
In the tesseractic area there is the primitive cubical one, C_{4}. Its Voronoi complex and its Delone complex both are relatively shifted test.
Next there is the bodycentered tesseractic (bct) lattice. Thus it is the union of the primitive cubical tesseractic plus its Voronoi cell vertices ("holes"). Alternatively this one can be described either as lattice B_{4}, as lattice D_{4}, or as lattice F_{4}. The Voronoi complex here is icot. The Delone complex is hext.
The lattice C_{4} = C_{2}×C_{2} (or the vertex set of test) can be
represented by the Hamiltonian integers Ham = Z[i,j] = {a_{0} + a_{1}i + a_{2}j + a_{3}k 
a_{0},a_{1},a_{2},a_{3}∈Z}, i^{2} = j^{2} = k^{2} = ijk = 1.
The lattice A_{2}×A_{2} (or the vertex set of tribbit) can be represented by
the hybrid integers Hyb = Z[ω,j] = {b_{0} + b_{1}ω + b_{2}j + b_{3}ωj 
b_{0},b_{1},b_{2},b_{3}∈Z}, ω = (1+sqrt(3))/2, j as above.
The lattice F_{4} (or the vertex set of hext) can be represented by Hurwitz integers
Hur = Z[u,v] = {c_{0} + c_{1}u + c_{2}v + c_{3}w 
c_{0},c_{1},c_{2},c_{3}∈Z}, u = (1ij+k)/2, v = (1+ijk)/2, w = (1i+jk)/2, uvw = 1.
 5D Pentacombs (up) 
So far Wythoffian elementary ones only.
o4o3o3o3o4o = C_{5} = R(6) 
o3o3o *b3o3o4o = B_{5} = S(6) 
o3o3o o3o3o *b3*e = D_{5} = Q(6) 
o3o3o3o3o3o3*a = A_{5} = P(6) 
x4o3o3o3o4o  penth o4x3o3o3o4o  rinoh o4o3x3o3o4o  brinoh x4x3o3o3o4o  tanoh x4o3x3o3o4o  sirnoh x4o3o3x3o4o  spanoh x4o3o3o3x4o  scanoh x4o3o3o3o4x  penth (stenoh) o4x3x3o3o4o  bittinoh o4x3o3x3o4o  sibranoh o4x3o3o3x4o  sibpanoh o4o3x3x3o4o  titanoh x4x3x3o3o4o  girnoh x4x3o3x3o4o  pattinoh x4x3o3o3x4o  catanoh x4x3o3o3o4x  tetanoh x4o3x3x3o4o  prinoh x4o3x3o3x4o  carnoh x4o3x3o3o4x  tepanoh x4o3o3x3x4o  cappinoh o4x3x3x3o4o  gibranoh o4x3x3o3x4o  biprinoh x4x3x3x3o4o  gippinoh x4x3x3o3x4o  cogrinoh x4x3x3o3o4x  tegranoh x4x3o3x3x4o  captinoh x4x3o3x3o4x  teptanoh x4x3o3o3x4x  tectanoh x4o3x3x3x4o  capranoh x4o3x3x3o4x  tepranoh o4x3x3x3x4o  gibpanoh x4x3x3x3x4o  gacnoh x4x3x3x3o4x  tegpenoh x4x3x3o3x4x  tecgranoh x4x3x3x3x4x  gatenoh 
x3o3o *b3o3o4o  hinoh o3x3o *b3o3o4o  brinoh o3o3o *b3x3o4o  brinoh o3o3o *b3o3x4o  rinoh o3o3o *b3o3o4x  penth x3x3o *b3o3o4o  thinoh x3o3x *b3o3o4o  rinoh x3o3o *b3x3o4o  sirhinoh x3o3o *b3o3x4o  siphinoh x3o3o *b3o3o4x  sachinoh o3x3o *b3x3o4o  titanoh o3x3o *b3o3x4o  sibranoh o3x3o *b3o3o4x  spanoh o3o3o *b3x3x4o  bittinoh o3o3o *b3x3o4x  sirnoh o3o3o *b3o3x4x  tanoh x3x3x *b3o3o4o  girnoh x3x3o *b3x3o4o  girhinoh x3x3o *b3o3x4o  pithinoh x3x3o *b3o3o4x  cathinoh x3o3x *b3x3o4o  sibranoh x3o3x *b3o3x4o  sibpanoh x3o3x *b3o3o4x  scanoh x3o3o *b3x3x4o  pirhinoh x3o3o *b3x3o4x  crahinoh x3o3o *b3o3x4x  caphinoh o3x3o *b3x3x4o  gibranoh o3x3o *b3x3o4x  prinoh o3x3o *b3o3x4x  pattinoh o3o3o *b3x3x4x  girnoh x3x3x *b3x3o4o  gibranoh x3x3x *b3o3x4o  biprinoh x3x3x *b3o3o4x  cappinoh x3x3o *b3x3x4o  giphinoh x3x3o *b3x3o4x  cograhnoh x3x3o *b3o3x4x  copthinoh x3o3x *b3x3x4o  biprinoh x3o3x *b3x3o4x  carnoh x3o3x *b3o3x4x  catanoh o3x3o *b3x3x4x  gippinoh x3x3x *b3x3x4o  gibpanoh x3x3x *b3x3o4x  capranoh x3x3x *b3o3x4x  captinoh x3x3o *b3x3x4x  gachinoh x3o3x *b3x3x4x  cogrinoh x3x3x *b3x3x4x  gacnoh 
x3o3o o3o3o *b3*e  hinoh o3x3o o3o3o *b3*e  brinoh x3x3o o3o3o *b3*e  thinoh x3o3x o3o3o *b3*e  rinoh x3o3o x3o3o *b3*e  spaquinoh x3o3o o3x3o *b3*e  sirhinoh o3x3o o3x3o *b3*e  titanoh x3x3x o3o3o *b3*e  bittinoh x3x3o x3o3o *b3*e  praquinoh x3x3o o3x3o *b3*e  girhinoh x3o3x x3o3o *b3*e  siphinoh x3o3x o3x3o *b3*e  sibranoh x3x3x x3o3o *b3*e  pirhinoh x3x3x o3x3o *b3*e  gibranoh x3x3o x3x3o *b3*e  gapquinoh x3x3o x3o3x *b3*e  pithinoh x3o3x x3o3x *b3*e  sibpanoh x3x3x x3x3o *b3*e  giphinoh x3x3x x3o3x *b3*e  biprinoh x3x3x x3x3x *b3*e  gibpanoh 
x3o3o3o3o3o3*a  cyxh x3x3o3o3o3o3*a  cytaxh x3o3x3o3o3o3*a  racyxh x3o3o3x3o3o3*a  spacyxh x3x3x3o3o3o3*a  tacyxh x3x3o3x3o3o3*a  cyprexh x3o3x3o3x3o3*a  sarcyxh x3x3x3x3o3o3*a  cygpoxh x3x3x3o3x3o3*a  parcyxh x3x3o3x3x3o3*a  bitcyxh x3x3x3x3x3o3*a  garcyxh x3x3x3x3x3x3*a  gapcyxh 
The lattice D_{5} is the vertex set of hinoh, or taken the other way round, that one is the Delone complex of this lattice. The vertex count of its vertex figure (rat) displays the highest kissing number of this dimension (40).
The lattice D_{5}* can be constructed either as union of the vertex sets of 4 hinoh (x3o3o o3o3o *b3*e + o3o3x o3o3o *b3*e + o3o3o x3o3o *b3*e + o3o3o o3o3x *b3*e), or as the union of the vertex sets of 2 penth (x4o3o3o3o4o + o4o3o3o3o4x). The latter description shows that this is the bodycentered penteractic lattice. Its kissing number is 10. Its Voronoi complex is titanoh.
The lattice A_{5} is the vertex set of cyxh, or taken the other way round, that one is the Delone complex of this lattice.
There are different overlays of that vertex set possible, which correspond to overlays by rotations of the respective diagrams: 2 such intervoven lattices would result from 2 diagrams with opposite nodes ringed (x3o3o3o3o3o3*a + o3o3o3x3o3o3*a), generating in the lattice A_{5}^{2}; 3 such intervoven lattices result from 3 diagrams with the ringed node in different triangular positions (x3o3o3o3o3o3*a + o3o3x3o3o3o3*a + o3o3o3o3x3o3*a), generating A_{5}^{3}; and finally, using 6 intervoven lattices, corresponding to either orientation of the diagram (x3o3o3o3o3o3*a + o3x3o3o3o3o3*a + o3o3x3o3o3o3*a + o3o3o3x3o3o3*a + o3o3o3o3x3o3*a + o3o3o3o3o3x3*a), generates A_{5}^{6} = A_{5}*. And the Voronoi complex of that last one would be gapcyxh.
 6D Hexacombs (up) 
So far just the quasiregular ones of the irreducible groups.
o4o3o3o3o3o4o = C_{6} = R(7) 
o3o3o *b3o3o3o4o = B_{6} = S(7) 
o3o3o o3o3o *b3o3*e = D_{6} = Q(7) 
o3o3o3o3o3o3o3*a = A_{6} = P(7) 
o3o3o3o3o *c3o3o = E_{6} = T(7) 
x4o3o3o3o3o4o  o4x3o3o3o3o4o  o4o3x3o3o3o4o  o4o3o3x3o3o4o  
x3o3o *b3o3o3o4o  o3x3o *b3o3o3o4o  o3o3o *b3x3o3o4o  o3o3o *b3o3x3o4o  o3o3o *b3o3o3x4o  o3o3o *b3o3o3o4x  
x3o3o o3o3o *b3o3*e  o3x3o o3o3o *b3o3*e  o3o3o o3o3o *b3x3*e  
x3o3o3o3o3o3o3*a  
x3o3o3o3o *c3o3o  o3x3o3o3o *c3o3o  o3o3x3o3o *c3o3o  
 7D Heptacombs (up) 
Just the quasiregular ones of the irreducible groups.
o4o3o3o3o3o3o4o = C_{7} = R(8) 
o3o3o *b3o3o3o3o4o = B_{7} = S(8) 
o3o3o o3o3o *b3o3o3*e = D_{7} = Q(8) 
o3o3o3o3o3o3o3o3*a = A_{7} = P(8) 
o3o3o3o3o3o3o *d3o = E_{7} = T(8) 
x4o3o3o3o3o3o4o  o4x3o3o3o3o3o4o  o4o3x3o3o3o3o4o  o4o3o3x3o3o3o4o  
x3o3o *b3o3o3o3o4o  o3x3o *b3o3o3o3o4o  o3o3o *b3x3o3o3o4o  o3o3o *b3o3x3o3o4o  o3o3o *b3o3o3x3o4o  o3o3o *b3o3o3o3x4o  o3o3o *b3o3o3o3o4x  
x3o3o o3o3o *b3o3o3*e  o3x3o o3o3o *b3o3o3*e  o3o3o o3o3o *b3x3o3*e  
x3o3o3o3o3o3o3o3*a  
x3o3o3o3o3o3o *d3o  o3x3o3o3o3o3o *d3o  o3o3x3o3o3o3o *d3o  o3o3o3x3o3o3o *d3o  o3o3o3o3o3o3o *d3x  
 8D Octacombs (up) 
Just the quasiregular ones of the irreducible groups.
o4o3o3o3o3o3o3o4o = C_{8} = R(9) 
o3o3o *b3o3o3o3o3o4o = B_{8} = S(9) 
o3o3o o3o3o *b3o3o3o3*e = D_{8} = Q(9) 
o3o3o3o3o3o3o3o3o3*a = A_{8} = P(9) 
o3o3o3o3o3o3o3o *c3o = E_{8} = T(9) 
o4o3o3o3o3o3o3o4o  o4o3o3o3o3o3o3o4o  o4o3o3o3o3o3o3o4o  o4o3o3o3o3o3o3o4o  o4o3o3o3o3o3o3o4o  
x3o3o *b3o3o3o3o3o4o  o3x3o *b3o3o3o3o3o4o  o3o3o *b3x3o3o3o3o4o  o3o3o *b3o3x3o3o3o4o  o3o3o *b3o3o3x3o3o4o  o3o3o *b3o3o3o3x3o4o  o3o3o *b3o3o3o3o3x4o  o3o3o *b3o3o3o3o3o4x  
x3o3o o3o3o *b3o3o3o3*e  o3x3o o3o3o *b3o3o3o3*e  o3o3o o3o3o *b3x3o3o3*e  o3o3o o3o3o *b3o3x3o3*e  
x3o3o3o3o3o3o3o3o3*a  
x3o3o3o3o3o3o3o *c3o  o3x3o3o3o3o3o3o *c3o  o3o3x3o3o3o3o3o *c3o  o3o3o3x3o3o3o3o *c3o  o3o3o3o3x3o3o3o *c3o  o3o3o3o3o3x3o3o *c3o  o3o3o3o3o3o3x3o *c3o  o3o3o3o3o3o3o3x *c3o  o3o3o3o3o3o3o3o *c3x  
The lattice C_{8} (or the vertex set of of {4,3^{6},4}) can be represented by the CaylayGraves integers Ocg = Z[i,j,e] = {a_{0} + a_{1}i + a_{2}j
+ a_{3}k + a_{4}e + a_{5}ie + a_{6}je + a_{7}ke  a_{0},...,a_{7}∈Z}.
The lattice E_{8} (or the vertex set of of {3^{5},3^{2,1}}) can be represented by (each of the 7 different types of) CoxeterDickson integers Ocd = Z[i,j,h] =
{b_{0} + b_{1}i + b_{2}j + b_{3}k + b_{4}h + b_{5}ih + b_{6}jh + b_{7}kh 
b_{0},...,b_{7}∈Z}, h = (i+j+k+e)/2.
The lattice F_{4}×F_{4} (or the vertex set of of {3,3,4,3}^{2}) can be represented by (each of the 7 different types of) coupled Hurwitz integers Och
= Z[u,v,e] = {c_{0} + c_{1}u + c_{2}v + c_{3}w + c_{4}e + c_{5}ue +
c_{6}ve + c_{7}we  c_{0},...,c_{7}∈Z}, u = (1ij+k)/2, v = (1+ijk)/2, w = (1i+jk)/2,
uvw = 1.
The lattice A_{2}×A_{2}×A_{2}×A_{2} (or the vertex set of of {3,6}^{4}) can be represented by the compound Eisenstein integers
Oce = Z[ω,j,e] = {d_{0} + d_{1}ω + d_{2}j + d_{3}ωj + d_{4}e +
d_{5}ωe + d_{6}je + d_{7}ωje  d_{0},...,d_{7}∈Z}, ω = (1+sqrt(3))/2.
© 20042015  top of page 