### Euclidean Tesselations

• 1D Tesselations (Sequence)
• 2D Tesselations (Tilings)
• 3D Tesselations (Honeycombs)
• 4D Tesselations (Tetracombs)
• 5D Tesselations (Pentacombs)
• 6D Tesselations (Hexacombs)
• 7D Tesselations (Heptacombs)
• 8D Tesselations (Octacombs)
• 9D Tesselations (Enneacombs)

As a foregoing consideration one might ask whether euclidean space curvature also does allow for non-simplicial 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 A1×A1.

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, non-simplicial reflection groups in here also provide some additional degrees of freedom: a reducible symmetry with n factors, allows for n-1 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 = A1 = W(2) o∞'o ```x∞o - aze x∞x - aze ``` ```x∞'o - aze x∞'x - ∞-covered edge ```

A1, 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/(n-d)} 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} = x-N/D-o and {2n/d} = x-N/D-x become geometrically identical. But this does not hold true for {n/(n-d)} = x-N/(N-D)-o, were the corresponding limit of the mark value becomes 1 (the retrograde aze), and {2n/(n-d)} = x-2N/(N-D)-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 well-behaved polytope, whereas x-∞'-x rather would be some highly degenerate Grünbaumian.

```----
2D  Tilings (up)
----
```
 o3o6o = G2 = V(3) o4o4o = C2 = R(3) o3o3o3*a = A2 = P(3) o∞o o∞o = A1×A1 = 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 A2 and G2 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 C2 clearly has squares for Voronoi and Delone cells. Both complexes are relatively shifted representants of squat.

The lattice A2 (or the vertex set of trat) can be represented by the Eisenstein integers E = Z[ω] = {e0 + e1ω | e0,e1Z}, ω = (-1+sqrt(-3))/2.
Similarily the lattice C2 (or the vertex set of squat) is represented by the Gaussian integers G = Z[i] = {g0 + g1i | g0,g1Z}, i = sqrt(-1).

(A nice applet for experimental tiling of the 2D euclidean plane (as well as hyperbolic ones) is tyler.)

#### Star Tilings

Beyond the 3 regular tesselations and the 8 semi-regular ones there are non-convex euclidean tilings too. Those will add for additional possibilities the usage of

• star polygons (polygrams), and/or the
• infinite tile (aze).

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 (P-2)(Q-2) = 4, which for a rational P=n/d would resolve to Q=2n/(n-2d), 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, non-Grünbaumian 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 - 2hoha (?)
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 (non-orientable member of sossa regiment)

rassersa (own regiment, a sossa superregiment)
rarsisresa (in rassersa regiment)

rosassa (in rassersa subregiment)
rorisassa (in rosassa regiment)

sraht (non-orientable member of rothat-regiment
graht (non-orientable member of ghothat-regiment

huht (non-orientable member of shaha-regiment)

retrat (retro-elongated 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/(n-d)}. (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
```[42,∞]     - x x∞o, azip
(∞-prism, convex)

[33,∞]     - s2s∞o, azap
(∞-antipr., convex)

[(3,∞)3]/2 - x3o3/2o∞*a, ditatha
(in trat-reg.)
```
```[3/2,12/5,12/5]        - o3x6/5x, quothat
(in toxat-army, reg. 11)
[4,8/5,8/5]            - o4x4/3x, quitsquat
(in tosquat-army, reg. 10)

[4,6/5,12/5]           - x3x6/5x, quitothit
(in othat-army, reg. 8)
[4,8/3,8/7]            - x4x4/3x, qrasquit
(in tosquat-army, reg. 7)
[6,12/5,12/11]         - x3x6x6/5*a, thotithit
(in toxat-subarmy, reg. 9)

[3/2,4,6/5,4]          - x3/2o6x, qrothat
(in rothat-army, reg. 4)
[3,12/5,6/5,12/5]      - x3o6x6/5*a, ghothat
(in rothat-army, reg. 4)
[3/2,12,6,12]          - x3/2o6x6*a, shothat
(in rothat-regiment, 1)

[4,12,4/3,12/11]/0     - sraht
(in rothat-regiment, 1)
[4,12/5,4/3,12/7]/0    - graht
(in rothat-army, reg. 4)
[8/3,8,8/5,8/7]/0      - sost
(in tosquat-army, reg. 2)
[12/5,12,12/7,12/11]/0 - huht
(in toxat-army, reg. 3)

[3,3,3,4/3,4/3]        - retrat
(in etrat-superreg., 12)

[3,3,4/3,3,4/3]        - s4/3s4/3s, rasisquat
(in snasquat-army, reg. 13)
```
```[8,8/3,∞]           - x4x4/3x∞*a, satsa
(in tosquat-subarmy, reg. 5)
[12,12/5,∞]         - x6x6/5x∞*a, hatha
(in toxat-subarmy, reg. 6)

[3,∞,3/2,∞]/0       - x3o3/2x∞*a, tha
(in that-regiment)
[4,∞,4/3,∞]/0       - x4o4/3x∞*a, sha
(in squat-regiment)
[6,∞,6/5,∞]/0       - hoha (or x6o6/5x∞*a, 2hoha)
(in that-regiment)

[8,4/3,8,∞]         - x4x4/3o∞*a, gossa
(in tosquat-army, reg. 2)
[8/3,4,8/3,∞]       - o4x4/3x∞*a, sossa
(in tosquat-army, reg. 2)
[12,6/5,12,∞]       - x6x6/5o∞*a, ghaha
(in toxat-army, reg. 3)
[12/5,6,12/5,∞]     - o6x6/5x∞*a, shaha
(in toxat-army, reg. 3)

[3,4,3,4/3,3,∞]     - snassa
(in snasquat-army, reg. 14)
[4,8,8/3,4/3,∞]     - rorisassa
(in tosquat-army,
in rassersa-subreg. †)
[4,8/3,8,4/3,∞]     - rosassa
(in tosquat-army,
in rassersa-subreg. †)

[4,8,4/3,8,4/3,∞]   - rarsisresa
(in sossa-superreg. †)
[4,8/3,4,8/3,4/3,∞] - rassersa
(in sossa-superreg. †)
```

* 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 non-Wythoffian.

The provided regiment numbers (for regiments, which occur exclusively within this section) correspond to the listing by J. McNeill. (The corresponding individual number-links would point to his respective page.)

```----
3D  Honeycombs (up)
----
```
 o4o3o4o = C3 = R(4) o3o3o *b4o = B3 = S(4) o3o3o3o3*a = A3 = P(4) o∞o o3o6o = A1×G2 = 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 o4s3s4o - (?) - ** s4s3s4o - (?) - ** s4s3s4s - snich - ** ``` ```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 *b4o - (?) - ** s3s3s *b4s - (?) - ** ``` ```x3o3o3o3*a - octet - O21 x3x3o3o3*a - batatoh - O27 x3o3x3o3*a - rich - O15 x3x3x3o3*a - tatoh - O25 x3x3x3x3*a - batch - O16 s3s3s3s3*a - (?) - ** ``` ```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 = A1×C2 = W(2)×R(3) o∞o o3o3o3*c = A1×A2 = W(2)×P(3) o∞o o∞o o∞o = A1×A1×A1 = 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 x-n-o - n-azedip x∞x x-n-o - n-azedip x∞x s-n-s - n-azedip ```

* 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 A1×G2 and A1×A2 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, C3. Its Voronoi complex and its Delone complex both are relatively shifted chon.

Next there is the body-centered cubical (bcc) lattice. Thus it is the union of the primitive cubical lattice plus its Voronoi cell vertices ("holes"). Alternatively it also is described by A3* or equivalently by D3*. The Voronoi complex here is batch. The Delone complex would be a squashed variant of octet.

Finally there is the face-centered cubical (fcc) lattice, as lattice equivalently described as A3 or D3, and likewise 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 4-fold vertices of rad ("deep holes"), while tet corresponds to the 3-fold vertices of rad ("shallow holes").

 octet derived honeycombs ```5Y4-4T-4P4 (cf. pextoh) 5Y4-4T-6P3-sq-para 5Y4-4T-6P3-sq-skew 10Y4-8T-0 (cf. octet) 10Y4-8T-1-alt 10Y4-8T-1-hel (r/l) 10Y4-8T-2-alt 10Y4-8T-2-hel (r/l) 10Y4-8T-3 (cf. gytoh, ditoh) 5Y4-4T-6P3-tri-0 (cf. etoh, gyeditoh) 5Y4-4T-6P3-tri-1-alt 5Y4-4T-6P3-tri-1-hel (r/l) 5Y4-4T-6P3-tri-2-alt 5Y4-4T-6P3-tri-2-hel (r/l) 5Y4-4T-6P3-tri-3 (cf. gyetoh, editoh) ``` ratoh derived honeycombs ```3Q4-T-2P8-P4 (cf. ratoh) 6Q4-2T (cf. pacratoh) ``` rich derived honeycombs (none is elementary) ```6Q3-2S3-gyro (cf. rich) 6Q3-2S3-ortho (cf. gyrich) 3Q3-S3-2P6-2P3-gyro (cf. erich) 3Q3-S3-2P6-2P3-ortho (cf. gyerich) ``` partial Stott expansion derived honeycombs (between rich and srich – none is elementary) ```pexrich pacsrich ``` pyrithohedral honeycombs (not elementary) ```cube-doe-bilbiro ``` In 2005 J. McNeill did a research on elementary honeycombs (cf. this cross-link to his website), that is (non-uniform) 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 non-elementary ones). But in fact, 4 of the vertex surroundings, he is displaying, do further split into 3 different modes each (i.e. into a 2-periodic alternating one, resp. into right- or left-handed, larger periodic helical ones). Thus the count increases rather to 21 scaliform elementary honeycombs. The therein being used solids are: ``` Pn - n-gonal prism Qn - n-gonal cupola (Q3 is not elementary) Sn - n-gonal antiprism (S3 is not elementary) T - tetrahedron Yn - n-gonal 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 non-uniform partial Stott expansion examples have no elementary counterparts, as the coes are used in an axial surrounding, which selects their 4-fold symmetry (not their 3-fold one).

#### Non-Convex Honeycombs

Beyond the convex honeycombs there are non-convex euclidean honeycombs too. Those will add for additional possibilities the usage of

• star polygons (polygrams), and/or the
• euclidean tilings, featuring like cells.

(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.)

o3o3o3o3/2*a3*c o3/2o3/2o3o3/2*a3*c o3o3/2o3o3*a3/2*c o3o3/2o3/2o3/2*a3/2*c ``` _o_ _P- | 3/2 o< 3 >o -P_ | _3- -o- ``` ``` _o_ _3- | -P_ o< 3/2 >o 3/2 | _P- -o- ``` ```x3o3o3o3/2*a3*c - (contains 2tet) o3x3o3o3/2*a3*c - datha o3o3x3o3/2*a3*c - (contains 2tet) o3o3o3x3/2*a3*c - (contains 2tet) x3x3o3o3/2*a3*c - (contains 2tet) x3o3x3o3/2*a3*c - ridatha 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 2oct-verf) 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 2oct-verf) 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] ``` ``` o---3---o | | 3/2 P' | | o---P---o ``` ``` o--3/2--o | | 3/2 3/2 | | o--3/2--o ``` ``` _o _3- | o-3-o< ∞ 3/2 | -o ``` ```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- | o-3/2-o< ∞ 3/2 | -o ``` ``` _o _P- | o-3-o< ∞' -P_ | -o ``` ``` _o _3- | o-3/2-o< ∞' -3_ | -o ``` ```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 | o-3/2-o< ∞' 3/2 | -o ``` ```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] ``` ``` o-P-o-3-o-4/3-o ``` ``` o-4-o-3/2-o-P-o ``` ```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] ``` ``` o-4/3-o-3/2-o-4/3-o ``` ``` o_ -3_ >o-4/3-o _3- o- ``` ``` o_ -3_ >o-P-o 3/2 o- ``` ```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 >o-P-o 3/2 o- ``` ``` o---P---o | | P 4/3 | | o---4---o ``` ```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 - getdoca 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] ``` ``` o---3---o | | 3/2 P | | o---P---o ``` ``` _o _3- | o-4-o< P' -P_ | -o ``` ```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 ``` ```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 - gacoca x4x3x4/3o4*b - (contains 2cho) x4x3o4/3x4*b - spadica small prismated discubatiapeiratic HC x4o3x4/3x4*b - skivpacoca o4x3x4/3x4*b - cuteca cubatitruncated cubatiapeiratic HC x4x3x4/3x4*b - cutpica ``` ``` _o 3/2 | o-4-o< P -P_ | -o ``` ``` _o _3- | o-4/3-o< P' -P_ | -o ``` ```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 o4/3x3x4x4/3*b - caquiteca x4/3x3x4x4/3*b - gacquitpica ``` ```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 - gacoca 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 ``` ``` _o 3/2 | o-4/3-o< P -P_ | -o ``` ``` _o _4- | o-P-o< ∞ 4/3 | -o ``` ```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 - wavicac sphenoverted cubitiapeiraticubatic HC o3o4x∞x4/3*b - x3x4x∞o4/3*b - x3x4o∞x4/3*b - sacpaca small cubatiprismatocubatiapeiratic HC x3o4x∞x4/3*b - o3x4x∞x4/3*b - dacta dicubatitruncated apeiratic HC x3x4x∞x4/3*b - dactapa dicubiatitruncated prismato-apeiratic HC ``` ```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- | o-3-o< ∞' -P_ | -o ``` ``` _o _P- | o-3/2-o< ∞' -P_ | -o ``` ```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] ``` ``` _o_ _P- | -4_ o< 3 >o -P_ | 4/3 -o- ``` ```x3o3o4/3o4*a3*c - [Grünbaumian] o3x3o4/3o4*a3*c - [Grünbaumian] o3o3x4/3o4*a3*c - [Grünbaumian] o3o3o4/3x4*a3*c - [Grünbaumian] x3x3o4/3o4*a3*c - [Grünbaumian] x3o3x4/3o4*a3*c - [Grünbaumian] x3o3o4/3x4*a3*c - getit cadoca o3x3x4/3o4*a3*c - [Grünbaumian] o3x3o4/3x4*a3*c - [Grünbaumian] o3o3x4/3x4*a3*c - stut cadoca x3x3x4/3o4*a3*c - [Grünbaumian] x3x3o4/3x4*a3*c - gikkivcadach x3o3x4/3x4*a3*c - dichac dicubatihexacubatic HC o3x3x4/3x4*a3*c - skivcadach x3x3x4/3x4*a3*c - dicroch dicubatirhombated cubihexagonal HC ``` ```x3/2o3/2o4/3o4*a3*c - [Grünbaumian] o3/2x3/2o4/3o4*a3*c - [Grünbaumian] o3/2o3/2x4/3o4*a3*c - [Grünbaumian] o3/2o3/2o4/3x4*a3*c - [Grünbaumian] x3/2x3/2o4/3o4*a3*c - [Grünbaumian] x3/2o3/2x4/3o4*a3*c - [Grünbaumian] x3/2o3/2o4/3x4*a3*c - getit cadoca o3/2x3/2x4/3o4*a3*c - [Grünbaumian] o3/2x3/2o4/3x4*a3*c - [Grünbaumian] o3/2o3/2x4/3x4*a3*c - stut cadoca x3/2x3/2x4/3o4*a3*c - [Grünbaumian] x3/2x3/2o4/3x4*a3*c - [Grünbaumian] x3/2o3/2x4/3x4*a3*c - dichac dicubatihexacubatic HC o3/2x3/2x4/3x4*a3*c - [Grünbaumian] x3/2x3/2x4/3x4*a3*c - [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/(n-d)}. (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 = C4 = R(5) o3o3o *b3o4o = B4 = S(5) o3o3o *b3o *b3o = D4 = Q(5) o3o3o3o3o3*a = A4 = 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 = F4 = U(5) o∞o o4o3o4o = A1×C3 = W(2)×R(4) o∞o o3o3o *d4o = A1×B3 = W(2)×S(4) o∞o o3o3o3o3*c = A1×A3 = 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 rectified-cubic prismatic TC x∞x x4o3o4o - test - O1 x∞x o4x3o4o - ricpit - O15 x∞o x4x3o4o - ticpit - O14 truncated-cubic prismatic TC x∞o x4o3x4o - cacpit - O17 cantellated-cubic prismatic TC x∞o x4o3o4x - test - O1 x∞o o4x3x4o - bitticpit - O16 bitruncated-cubic 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 cantitruncated-cubic prismatic TC x∞o x4x3o4x - rutcupit - O19 runcitruncated-cubic prismatic TC x∞x x4x3x4o - catcupit - O18 x∞x x4x3o4x - rutcupit - O19 x∞o x4x3x4x - otacpit - O20 omnitruncated-cubic prismatic TC x∞x x4x3x4x - otacpit - O20 ``` ```x∞o x3o3o *d4o - acpit - O21 alternated-cubic 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 truncated-alternated-cubic prismatic TC x∞o x3o3x *d4o - ricpit - O15 x∞o x3o3o *d4x - racpit - O26 runcinated-alternated-cubic 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 runcicantic-cubic 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 quarter-cubic 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 = G2×G2 = V(3)2 o3o6o o4o4o = G2×C2 = V(3)×R(3) o3o6o o3o3o3*d = G2×A2 = V(3)×P(3) o4o4o o4o4o = C2×C2 = R(3)2 ```x3o6o x3o6o - tribbit - O29 x3o6o o3x6o - tathibbit - O32 triangular-trihexagonal duoprismatic TC x3o6o o3o6x - thibbit - O30 triangular-hexagonal duoprismatic TC o3x6o o3x6o - thabbit - O56 trihexagonal duoprismatic TC o3x6o o3o6x - hithibbit - O41 hexagonal-trihexagonal 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 triangular-rhombitrihexagonal TC x3o6x o3x6o - thrathibbit - O59 x3o6x o3o6x - harhibit - O44 hexagonal-rhombihexagonal duoprismatic TC o3x6x x3o6o - tathobit - O34 triangular-tomohexagonal duoprismatic TC o3x6x o3x6o - thathobit - O58 trihexagonal-tomohexagonal duoprismatic TC o3x6x o3o6x - hithobit - O43 hexagonal-tomohexagonal duoprismatic TC x3x6x x3o6o - totuthit - O36 triangular-omnitruncated-trihexagonal 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 omnitruncated-trihexagonal duoprismatic TC x3o6o s3s6s - tisthit - O38 o3x6o s3s6s - thisthibbit - O62 trihexagonal-simotrihexagonal duoprismatic TC o3o6x s3s6s - hasithbit - O47 hexagonal-simotrihexagonal duoprismatic TC x3x6o s3s6s - hasithbit - O47 x3o6x s3s6s - rithsithbit - O77 o3x6x s3s6s - thosithbit - O73 tomohexagonal-simotrihexagonal duoprismatic TC x3x6x s3s6s - otsithbit - O80 omnitruncated-simotrihexagonal duoprismatic TC s3s6s s'3s'6s' - sithbit - O83 simotrihexagonal duoprismatic TC ``` ```x3o6o x4o4o - tisbat - O2 triangular-square duoprismatic TC x3o6o o4x4o - tisbat - O2 o3x6o x4o4o - thisbit - O5 trihexagonal-square duoprismatic TC o3x6o o4x4o - thisbit - O5 o3o6x x4o4o - shibbit - O3 square-hexagonal duoprismatic TC o3o6x o4x4o - shibbit - O3 x3x6o x4o4o - shibbit - O3 x3x6o o4x4o - shibbit - O3 x3o6x x4o4o - rithsibbit - O8 rhombitrihexagonal-square duoprismatic TC x3o6x o4x4o - rithsibbit - O8 o3x6x x4o4o - thosbit - O7 tomohexagonal-square duoprismatic TC o3x6x o4x4o - thosbit - O7 x3o6o x4x4o - tatosbit - O33 triangular-tomosquare duoprismatic TC x3o6o x4o4x - tisbat - O2 o3x6o x4x4o - thatosbit - O57 trihexagonal-tomosquare duoprismatic TC o3x6o x4o4x - thisbit - O5 o3o6x x4x4o - hitosbit - O42 hexagonal-tomosquare 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 tomosquare-rhombitrihexagonal duoprismatic TC x3o6x x4o4x - rithsibbit - O8 o3x6x x4x4o - tosthobit - O64 tomosquare-tomohexagonal 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 triangular-simosquare TC o3x6o s4s4s - thisosbit - O61 trihexagonal-simosquare duoprismatic TC o3o6x s4s4s - hisosbit - O46 hexagonal-simosquare duoprismatic TC x3x6o s4s4s - hisosbit - O46 x3o6x s4s4s - rithsisbit - O76 rhombitrihexagonal-simosquare duoprismatic TC o3x6x s4s4s - thosisbit - O72 tomohexagonal-simosquare duoprismatic TC x3x6x s4s4s - otsisbit - O79 omnitruncated-simosquare duoprismatic TC s3s6s x4o4o - sithsobit - O11 simotrihexagonal-square duoprismatic TC s3s6s o4x4o - sithsobit - O11 s3s6s x4x4o - tosasithbit - O68 tomosquare-simotrihexagonal duoprismatic TC s3s6s x4o4x - sithsobit - O11 s3s6s x4x4x - tosasithbit - O68 s3s6s s'4s'4s' - sissithbit - O82 simosquare-simotrihexagonal 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 tomosquare-square 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 simosquare-square duoprismatic TC s4s4s o4x4o - sisosbit - O10 s4s4s x4x4o - tosisasbit - O67 tomosquare-simosquare duoprismatic TC s4s4s x4o4x - sisosbit - O10 s4s4s x4x4x - tosisasbit - O67 s4s4s s4s4s - sisbit - O81 simosquare duoprismatic TC ``` o4o4o o3o3o3*d = C2×A2 = R(3)×P(3) o3o3o3*a o3o3o3*d = A2×A2 = P(3)2 o∞o o∞o o3o6o = A1×A1×G2 = W(2)2×V(3) o∞o o∞o o4o4o = A1×A1×C2 = 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 = A1×A1×A2 = W(2)2×P(3) o∞o o∞o o∞o o∞o = A1×A1×A1×A1 = 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 pentachoric-dispentachoric TC schmo( x3o3o3o3o3*a ) - zucypit - O142 schmoozed cyclopentachoric TC, schmoozed pentachoric-dispentachoric TC elongschmo( x3o3o3o3o3*a ) - ezucypit - O143 elongated schmoozed cyclopentachoric TC, elongated schmoozed pentachoric-dispentachoric TC elong( x3o6o x3o6o ) - etbit - O31 elongated triangular duoprismatic TC elong( x3o6o o3x6o ) - etothbit - O49 elongated triangular-trihexagonal duoprismatic TC elong( x3o6o o3o6x ) - ethibit - O40 elongated triangular-hexagonal duoprismatic TC elong( x3o6o x3x6o ) - ethibit - O40 elong( x3o6o x3o6x ) - etrithit - O52 elongated triangular-rhombitrihexagonal TC elong( x3o6o o3x6x ) - etathobit - O51 elongated triangular-tomohexagonal duoprismatic TC elong( x3o6o x3x6x ) - etotithat - O53 elongated triangular-omnitruncated-trihexagonal TC elong( x3o6o s3s6s ) - etasithit - O55 elongated triangular-simotrihexagonal 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 triangular-square duoprismatic TC elong( x3o6o x4x4o ) - etatosbit - O50 elongated triangular-tomosquare duoprismatic TC elong( x3o6o x4x4x ) - etatosbit - O50 elong( x3o6o s4s4s ) - etasist - O54 elongated triangular-simosquare TC gyro( x3o6o x4o4o ) - gytosbit - O12 gyrated triangular-square duoprismatic TC gyroelong( x3o6o x4o4o ) - egytsobit - O13 elongated gyrated triangular-square duoprismatic TC bigyro( x3o6o x4o4o ) - bigytsbit - O84 bigyrated triangular-square duoprismatic TC bigyroelong( x3o6o x4o4o ) - ebiytsbit - O85 elongated bigyrated triangular-square duoprismatic TC prismatogyro( x3o6o x4o4o ) - pegytsbit - O86 prismatoelongated gyrated triangular-square 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 elongated-alternated-cubic prismatic TC gyro( x∞o x3o3o *d4o ) - gyacpit - O22 gyrated-alternated-cubic prismatic TC gyroelong( x∞o x3o3o *d4o ) - gyeacpit - O24 gyrated-elongated-alternated-cubic 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- = ...-0-0-... -gyro- = ...-0-1-0-1-... -bigyro- = ...-0-1-2-0-1-2-... -prismatogyro- = ...-0-1-0-2-0-1-0-2-... ```

** 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, A4 and A4*. 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 A4 is cypit. The Voronoi complex of A4* is otcypit.

In the tesseractic area there is the primitive cubical one, C4. Its Voronoi complex and its Delone complex both are relatively shifted test.

Next there is the body-centered 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 B4, as lattice D4, or as lattice F4. The Voronoi complex here is icot. The Delone complex is hext.

The lattice C4 = C2×C2 (or the vertex set of test) can be represented by the Hamiltonian integers Ham = Z[i,j] = {a0 + a1i + a2j + a3k | a0,a1,a2,a3Z}, i2 = j2 = k2 = ijk = -1.
The lattice A2×A2 (or the vertex set of tribbit) can be represented by the hybrid integers Hyb = Z[ω,j] = {b0 + b1ω + b2j + b3ωj | b0,b1,b2,b3Z}, ω = (-1+sqrt(-3))/2, j as above.
The lattice F4 (or the vertex set of hext) can be represented by Hurwitz integers Hur = Z[u,v] = {c0 + c1u + c2v + c3w | c0,c1,c2,c3Z}, u = (1-i-j+k)/2, v = (1+i-j-k)/2, w = (1-i+j-k)/2, uvw = 1.

 ```:xoqo:4:xxox:3:oooo:4:oxxx:&##x :xoqo:4:xxox:3:xxxx:4:oxxx:&##x ... ``` Non-uniform tetracombs using axial CRF polychora for building blocks easily can be derived from infinite periodic stacks of 3D honeycombs, provided those use the same axial symmetry and have the same periodicity scaling in each (independent) around-symmetrical direction. The (infinite) segments then are filled from according lace prisms. As these are piled up to linear columns each, those can be considered to become multistratic blends, as long as those still remain convex.

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

So far Wythoffian elementary ones only.

 o4o3o3o3o4o = C5 = R(6) o3o3o *b3o3o4o = B5 = S(6) o3o3o o3o3o *b3*e = D5 = Q(6) o3o3o3o3o3o3*a = A5 = 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 D5 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 D5* 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 body-centered penteractic lattice. Its kissing number is 10. Its Voronoi complex is titanoh.

The lattice A5 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 A52; 3 such intervoven lattices result from 3 diagrams with the ringed node in different triangular positions (x3o3o3o3o3o3*a  +  o3o3x3o3o3o3*a  +  o3o3o3o3x3o3*a), generating A53; 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 A56 = A5*. 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 = C6 = R(7) o3o3o *b3o3o3o4o = B6 = S(7) o3o3o o3o3o *b3o3*e = D6 = Q(7) o3o3o3o3o3o3o3*a = A6 = P(7) o3o3o3o3o *c3o3o = E6 = T(7) ```x4o3o3o3o3o4o - axh o4x3o3o3o3o4o - raxh o4o3x3o3o3o4o - braxh o4o3o3x3o3o4o - traxh ``` ```x3o3o *b3o3o3o4o - haxh o3x3o *b3o3o3o4o - braxh o3o3o *b3x3o3o4o - traxh o3o3o *b3o3x3o4o - braxh o3o3o *b3o3o3x4o - raxh o3o3o *b3o3o3o4x - axh ``` ```x3o3o o3o3o *b3o3*e - haxh o3x3o o3o3o *b3o3*e - braxh o3o3o o3o3o *b3x3*e - traxh ``` ```x3o3o3o3o3o3o3*a - cyloh ... x3x3x3x3x3x3x3*a - gapcyloh ``` ```x3o3o3o3o *c3o3o - jakoh o3x3o3o3o *c3o3o - rojkoh o3o3x3o3o *c3o3o - ramoh ```

The lattice D6 is the vertex set of haxh, or taken the other way round, that one is the Delone complex of this lattice.

The lattice A6 is the vertex set of cyloh, or taken the other way round, that one is the Delone complex of this lattice.

The lattice E6 is the vertex set of jakoh, or taken the other way round, that one is the Delone complex of this lattice. The vertex count of its vertex figure (mo) displays the highest kissing number of this dimension (72).

The lattice D6* can be constructed either as union of the vertex sets of 4 haxh (x3o3o o3o3o *b3o3*e  +  o3o3x o3o3o *b3o3*e  +  o3o3o x3o3o *b3o3*e  +  o3o3o o3o3x *b3o3*e), or as the union of the vertex sets of 2 axh (x4o3o3o3o3o4o  +  o4o3o3o3o3o4x). The latter description shows that this is the body-centered hexeractic lattice. Its kissing number is 12. Its Voronoi complex is traxh.

The lattice A6* can be constructed as union of the vertex sets of 7 cyloh (x3o3o3o3o3o3o3*a  +  o3x3o3o3o3o3o3*a  +  o3o3x3o3o3o3o3*a  +  o3o3o3x3o3o3o3*a  +  o3o3o3o3x3o3o3*a  +  o3o3o3o3o3x3o3*a  +  o3o3o3o3o3o3x3*a). Its Voronoi complex is gapcyloh (the omnitruncate).

The lattice E6* can be constructed as union of the vertex sets of 3 jakoh (x3o3o3o3o *c3o3o  +  o3o3o3o3x *c3o3o  +  o3o3o3o3o *c3o3x). Its Voronoi complex is ramoh.

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

Just the quasiregular ones of the irreducible groups.

 o4o3o3o3o3o3o4o = C7 = R(8) o3o3o *b3o3o3o3o4o = B7 = S(8) o3o3o o3o3o *b3o3o3*e = D7 = Q(8) o3o3o3o3o3o3o3o3*a = A7 = P(8) o3o3o3o3o3o3o *d3o = E7 = T(8) ```x4o3o3o3o3o3o4o - hepth o4x3o3o3o3o3o4o - rash o4o3x3o3o3o3o4o - brash o4o3o3x3o3o3o4o - trash ``` ```x3o3o *b3o3o3o3o4o - hash o3x3o *b3o3o3o3o4o - brash o3o3o *b3x3o3o3o4o - trash o3o3o *b3o3x3o3o4o - trash o3o3o *b3o3o3x3o4o - brash o3o3o *b3o3o3o3x4o - rash o3o3o *b3o3o3o3o4x - hepth ``` ```x3o3o o3o3o *b3o3o3*e - hash o3x3o o3o3o *b3o3o3*e - brash o3o3o o3o3o *b3x3o3*e - trash ``` ```x3o3o3o3o3o3o3o3*a - cyooh ``` ```x3o3o3o3o3o3o *d3o - naquoh o3x3o3o3o3o3o *d3o - rinquoh o3o3x3o3o3o3o *d3o - brinquoh o3o3o3x3o3o3o *d3o - rolinoh o3o3o3o3o3o3o *d3x - linoh ```

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

Just the quasiregular ones of the irreducible groups.

 o4o3o3o3o3o3o3o4o = C8 = R(9) o3o3o *b3o3o3o3o3o4o = B8 = S(9) o3o3o o3o3o *b3o3o3o3*e = D8 = Q(9) o3o3o3o3o3o3o3o3o3*a = A8 = P(9) o3o3o3o3o3o3o3o *c3o = E8 = T(9) ```x4o3o3o3o3o3o3o4o - octh o4x3o3o3o3o3o3o4o - rocth o4o3x3o3o3o3o3o4o - broch o4o3o3x3o3o3o3o4o - troch o4o3o3o3x3o3o3o4o - teroch ``` ```x3o3o *b3o3o3o3o3o4o - hocth o3x3o *b3o3o3o3o3o4o - broch o3o3o *b3x3o3o3o3o4o - troch o3o3o *b3o3x3o3o3o4o - teroch o3o3o *b3o3o3x3o3o4o - troch o3o3o *b3o3o3o3x3o4o - broch o3o3o *b3o3o3o3o3x4o - rocth o3o3o *b3o3o3o3o3o4x - octh ``` ```x3o3o o3o3o *b3o3o3o3*e - hocth o3x3o o3o3o *b3o3o3o3*e - broch o3o3o o3o3o *b3x3o3o3*e - troch o3o3o o3o3o *b3o3x3o3*e - teroch ``` ```x3o3o3o3o3o3o3o3o3*a - cyenoh ``` ```x3o3o3o3o3o3o3o *c3o - bayoh o3x3o3o3o3o3o3o *c3o - robyah o3o3x3o3o3o3o3o *c3o - ribfoh o3o3o3x3o3o3o3o *c3o - tergoh o3o3o3o3x3o3o3o *c3o - trigoh o3o3o3o3o3x3o3o *c3o - bargoh o3o3o3o3o3o3x3o *c3o - rigoh o3o3o3o3o3o3o3x *c3o - goh o3o3o3o3o3o3o3o *c3x - bifoh ```

The lattice C8 (or the vertex set of of octh) can be represented by the Caylay-Graves integers Ocg = Z[i,j,e] = {a0 + a1i + a2j + a3k + a4e + a5ie + a6je + a7ke | a0,...,a7Z}.
The lattice E8 (or the vertex set of of goh) can be represented by (each of the 7 different types of) Coxeter-Dickson integers Ocd = Z[i,j,h] = {b0 + b1i + b2j + b3k + b4h + b5ih + b6jh + b7kh | b0,...,b7Z}, h = (i+j+k+e)/2.
The lattice F4×F4 (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] = {c0 + c1u + c2v + c3w + c4e + c5ue + c6ve + c7we | c0,...,c7Z}, u = (1-i-j+k)/2, v = (1+i-j-k)/2, w = (1-i+j-k)/2, uvw = 1.
The lattice A2×A2×A2×A2 (or the vertex set of of {3,6}4) can be represented by the compound Eisenstein integers Oce = Z[ω,j,e] = {d0 + d1ω + d2j + d3ωj + d4e + d5ωe + d6je + d7ωje | d0,...,d7Z}, ω = (-1+sqrt(-3))/2.

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

Just the quasiregular ones of the irreducible groups.

 o4o3o3o3o3o3o3o3o4o = C9 = R(10) o3o3o *b3o3o3o3o3o3o4o = B9 = S(10) o3o3o o3o3o *b3o3o3o3o3*e = D9 = Q(10) o3o3o3o3o3o3o3o3o3o3*a = A9 = P(10) ```x4o3o3o3o3o3o3o3o4o - enneh o4x3o3o3o3o3o3o3o4o - reneh o4o3x3o3o3o3o3o3o4o - breneh o4o3o3x3o3o3o3o3o4o - treneh o4o3o3o3x3o3o3o3o4o - terneh ``` ```x3o3o *b3o3o3o3o3o3o4o - heneh o3x3o *b3o3o3o3o3o3o4o - breneh o3o3o *b3x3o3o3o3o3o4o - treneh o3o3o *b3o3x3o3o3o3o4o - terneh o3o3o *b3o3o3x3o3o3o4o - terneh o3o3o *b3o3o3o3x3o3o4o - treneh o3o3o *b3o3o3o3o3x3o4o - breneh o3o3o *b3o3o3o3o3o3x4o - reneh o3o3o *b3o3o3o3o3o3o4x - enneh ``` ```x3o3o o3o3o *b3o3o3o3o3*e - heneh o3x3o o3o3o *b3o3o3o3o3*e - breneh o3o3o o3o3o *b3x3o3o3o3*e - treneh o3o3o o3o3o *b3o3x3o3o3*e - terneh ``` ```x3o3o3o3o3o3o3o3o3o3*a - cydoh ```

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