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Euclidean Tesselations

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)
x∞o - aze
x∞x - aze

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.



---- 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 lattica 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

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

o o∞o o4o4/3o o4/3o4/3o
x x∞o - azip °
x x∞x - azip °

s s∞o - azap °
s s∞s - azap °
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 - 3that (?)
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 - 3that (?)
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 o4o4/3o∞*a o6o6/5o∞*a
x3o3/2o∞*a - 
o3x3/2o∞*a - 
o3o3/2x∞*a - 
x3x3/2o∞*a - chata
x3o3/2x∞*a - tha
o3x3/2x∞*a - [Grünbaumian]
x3x3/2x∞*a - [Grünbaumian]
x4o4/3o∞*a - 
o4x4/3o∞*a - 
o4o4/3x∞*a - 
x4x4/3o∞*a - gossa
x4o4/3x∞*a - sha
o4x4/3x∞*a - sossa
x4x4/3x∞*a - satsa

s4s4/3s∞*a - snassa
x6o6/5o∞*a - 
o6x6/5o∞*a - 
o6o6/5x∞*a - 
x6x6/5o∞*a - ghaha
x6o6/5x∞*a - hoha
o6x6/5x∞*a - shaha
x6x6/5x∞*a - hatha
other
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.

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 tosquat-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       - x6o6/5x∞*a, hoha
                      (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

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
= 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∞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"). 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 derived from A3 or B3, and alternatively described as mod 2 of the sum of the vertex coordinates of a (smaller) primitive cubical lattice. Its Voronoi cell is 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 srichnone 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).



---- 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
 omnitruncated tesseractic TC

s4o3o3o4o  - hext      - O104
s4o3x3o4o  - thext     - O105
s4o3o3o4x  - siphatit  - O108
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
 great cycloprismated pentachoric TC,
 grand prismatodispentachoric TC
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
 great birhombated icositetrachoric TC,
 great grand prismatodisicositetrachoric TC
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.



---- 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 - 
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
= 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 - 
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
= 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)
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 C8 (or the vertex set of of {4,36,4}) 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 {35,32,1}) 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.



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