### Expanded Kaleido-Facetings

In February 2014 a noteworthy combination from

was brought up by W. Gevaert. In the sequel a vivid research for class members of according CRFs was initiated, which then provided several so far unknown "crown jewels".

Gevaert himself also elaborated some of the first known examples, but then he reduced to just outline his ideas and contribute to combinatorics. Quickfur, "student5", and Čtrnáct also found some further individual examples. A more systematical research with several more finds and esp. the concrete cell evaluation of all known cases then has been done by Klitzing. (Therefore the following content mostly is original work.)

For kaleidoscopical construction usually a seed point within the fundamental domain is used. Its reflection all over the mirror symmetry then constructs the vertex set of the to be designed polytope. Obviously an other such vertex outside the domain likewise could be used, again providing the same vertex set – but then other elements in the dimensions beyond. Generally these such derived polytopes then will be facetings of the former polytope. If esp. that point was taken to be a direct mirror image of that point within, then that edge, which is derived as the hull of these 2 points, gets just inverted.

If one wonders what these kaleido-faceted polytopes would look like in terms of their Dynkin diagram, then clearly some edges, i.e. node symbols which formerly had prograde unit size, now will have retrograde unite size. That is, some x just gets reflected into a (-x). But because faces, which stay within their face planes, keep their neighbouring vertices still being connected, the other edges accordingly will get elongated thereby by the amount of a vertex figure, i.e. by the corresponding shortchord.

Assume we start with any (sub)diagram x-n/m-y (where y is just any length edge, possibly zero), then by flipping that x into (-x) we elongate the neighbouring edge from size y into y+cos(π m/n)x. Moreover, if we start with some layered starting figure, then such an edge flip could be applied independently in any layer. We just have to check, that still any layer can be connected by unit-lacings to at least one of the other layers. For its converse cf. †).

Obviously retrograde edges would not be allowed in CRF polytopes. To that aim a corresponding Stott expansion will come in, which then adds one unit to the respective node position – uniformely within all layers. Sometimes longer edges also would occur within the symbols of individual layers. Here we have to check then that those "edges" become just false ones (pseudo edges), burried somewhere within. E.g. covered by inter-layer lacing elements. For its converse cf. ‡).

Besides of the former exclusions we still have to check for the regularity of all the polygonal faces (of the total figure).

One of the most common cases wrt. ‡) can already be filtered out a priori: When considering convex starting polytopes which become displayed as mere lace towers, i.e. when keeping the layer heights throughout all the transformations, then mixtures of prograde and retrograde edges at the same node would not be allowable. This is because we have to apply a Stott expansion to the retrograde edges, bringing these edges then back to zero size. But this same expansion would double up simultanuously the prograde edges at the same node. Generally speaking, an inner-layer edge of size u = 2x might be allowable, e.g. as the equator of an hexagon. But within the reach of the given preconditions the 2 adjoined "halves" would not be co-planar before the expansion, and so by mere parallel translation cannot become thereafter.

This same argument surely serves valid for full dimensional subsymmetries which are cartesian products, provided all transformations affect only one cartesian component, while the other keeps unchanged. (This then would be the generalization of unchanged heights.) Even changes in more than one component are compensated here, provided this change can be split into independent changes in either affected component each.

Wrt. mere lace towers we likewise have an a priori restriction: lacing triangles freely can be affected by edge reversals, so can squares only if both parallel edges are affected simultanuously, but any other 2D face can not. Nor could any face be elongated within its face plain by later partial Stott expansions, except of those which do respect the full symmetry of that polygon.

†) - Layers fall apart into subsets: at least one layer cannot be connected to none of the other ones by unit inter-layer lacings. Or dead ends will arise: vertices of at least one non-extremal layer wrt. some axial orientation do not allow for unit inter-lacings to any higher (or lower) vertex.
‡) - At least one non-unit layer edge survives at the outside.
°) - Asks for some non-regular polygonal faces.

#### 3D EKF:

EKF of the icosahedron
• subsymmetry o2o2o   (1 derived CRF)
• subsymmetry . o3o   (1 derived CRF)
• subsymmetry . o5o   (1 derived CRF)
EKF of the small rhombicuboctahedron
• subsymmetry o3o3o   (3 derived Wythoffians)
• subsymmetry o2o4o & . o4o   (2 derived CRFs  &  3 derived Wythoffians)
• subsymmetry . o3o   (none)
EKF of the cuboctahedron
• subsymmetry o3o3o   (4 derived Wythoffians)
• subsymmetry o2o4o & . o4o   (1 derived Wythoffian)
• subsymmetry o2o2o   (1 derived Wythoffian)
• subsymmetry . o3o   (1 derived Wythoffian)

#### 4D EKF:

EKF of the icosahedral pyramid
• subsymmetry . o2o2o   (1 derived CRF)
• subsymmetry . . o3o   (1 derived CRF)
• subsymmetry . . o5o   (1 derived CRF)
EKF of the hexacosachoron
EKF of the icositetrachoron
• subsymmetry o3o3o4o   (4 derived CRFs  &  7 derived Wythoffians)
• subsymmetry o2o3o4o & . o3o4o   (10 derived CRFs  &  5 derived Wythoffians)
• subsymmetry o3o3o *b3o   (4 derived CRFs  &  3 derived Wythoffians)
• subsymmetry o4o2o4o   (1 derived CRFs  &  1 derived Wythoffians)
• ...
• subsymmetry o2o3o4o & . o3o4o   (10 derived CRFs  &  5 derived Wythoffians)
• subsymmetry . o3o3o   (10 derived CRFs  &  2 derived Wythoffians)
• subsymmetry o4o2o4o   (5 derived CRFs  &  3 derived Wythoffians)
• subsymmetry o2o2o2o   (3 derived CRFs  &  1 derived Wythoffian)
• ...
EKF of the pentachoron
• subsymmetry . o3o3o   (8 derived CRFs  &  3 derived Wythoffians)
• ...
EKF of the rectified pentachoron
• subsymmetry . o3o3o   (8 derived CRFs  &  3 derived Wythoffians)
• ...
EKF of the small rhombated pentachoron
• subsymmetry . o3o3o   (2 derived CRFs  &  2 derived Wythoffians)
• ...
EKF of the decachoron
• subsymmetry . o3o3o   (3 derived CRFs  &  4 derived Wythoffians)
• subsymmetry . o2o3o   (2 derived Wythoffians)
• ...
EKF of the small prismatodecachoron
• subsymmetry . o3o3o   (2 derived CRFs  &  2 derived Wythoffians)
• subsymmetry . o2o3o   (1 derived CRF)
• ...
EKF of the rectified hexacosachoron
EKF of the small prismated icositetrachoron
EKF of the snub (dis)icositetrachoron
• subsymmetry o3o3o *b3o   (2 derived CRFs)
• subsymmetry o2o2o2o   (1 derived CRF)
• ...
EKF of the rectified tesseract
• subsymmetry o2o3o4o & . o3o4o   (4 derived CRFs  &  3 derived Wythoffians)
• subsymmetry . o3o3o   (1 derived CRF  &  2 derived Wythoffians)
• ...

#### EKF of the icosahedron (x3o5o)

in o2o2o subsymmetry   (up)
Representation:
```xof 2 fxo 2 ofx &#zx  (ike)
= xofox 2 ofxfo &#xt
```
All layers &
kaleido-facetings per layer:
```A:  x 2 f 2 o   →  A1:  (-x) 2   f  2   o
B:  o 2 x 2 f   →  B2:    o  2 (-x) 2   f
C:  f 2 o 2 x   →  C3:    f  2   o  2 (-x)
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```A1:     (-x)of 2 fxo 2 ofx &#zx
A1B2:   (-x)of 2 f(-x)o 2 ofx &#zx     → †)
A1B2C3: (-x)of 2 f(-x)o 2 of(-x) &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```oxF 2 fxo 2 ofx &#zx  (bilbiro, J91)
= oxFxo 2 ofxfo &#xt
```

in . o3o subsymmetry   (up)
Representation:
```xofo 3 ofox &#xt  (ike)
```
All layers &
kaleido-facetings per layer:
```A:  x 3 o   →  A1:  (-x) 3   x
B:  o 3 f
C:  f 3 o
D:  o 3 x   →  D2:    x  3 (-x)
```
A priori invalid combinations:
```A1 + D2  → ‡) (u in A, u in D)
```
Other layer-combinations:
```A1:    (-x)ofo 3 xfox &#xt
```
Stott expansion:
(derived potential CRFs)
```oxFx 3 xfox &#xt  (thawro, J92)
```

in . o5o subsymmetry   (up)
Representation:
```oxoo 5 ooxo &#xt  (ike)
```
All layers &
kaleido-facetings per layer:
```A:  o 5 o
B:  x 5 o   →  B1:  (-x) 5   f
C:  o 5 x   →  C2:    f  5 (-x)
D:  o 5 o
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```B1:    o(-x)oo 5 ofxo &#xt
B1C2:  o(-x)fo 5 of(-x)o &#xt  → †)
```
Stott expansion:
(derived potential CRFs)
```xoxx 5 ofxo &#xt  (pocuro, J32)

related: ..xx 5 ..xo &#xt  (pecu, J5)
related: xox. 5 ofx. &#xt  (pero, J6)
```

#### EKF of the small rhombicuboctahedron (x3o4x)

in o3o3o subsymmetry   (up)
Representation:
```qo 3 xx 3 oq &#zx  (sirco)
```
All layers &
kaleido-facetings per layer:
```A:  q 3 x 3 o   →  A2:    w  3 (-x) 3   x   →  A23:    w  3   o  3 (-x)
B:  o 3 x 3 q   →  B2:    x  3 (-x) 3   w   →  B21:  (-x) 3   o  3  w
```
A priori invalid combinations:
```A + B2  → ‡ (u in A)
A2 + B  → ‡ (u in B)
```
Other layer-combinations:
```B21:    q(-x) 3 xo 3 ow &#zx     → †)
A2B2:   wx 3 (-x)(-x) 3 xw &#zx
A2B21:  w(-x) 3 (-x)o 3 xw &#zx  → †)
A23B21: w(-x) 3 oo 3 (-x)w &#zx  → †)
```
Stott expansion:
(derived potential CRFs
& beyond)
```1:-: wx 3 xx 3 oq &#zx
→ °) (asks for non-regular hexagons: wx .. oq &#zx)
= non-Johnsonian (patex sirco)
```
```(-2):-: qo 3 oo 3 oq &#zx
= Wythoffian o3o4x (cube)
```
```13:-: wx 3 xx 3 xw &#zx
= Wythoffian x3x4x (girco)
```
```1(-2)3:-: wx 3 oo 3 xw &#zx
= Wythoffian o3x4x (tic)
```
```2:A2B2: wx 3 oo 3 xw &#zx
= Wythoffian o3x4x (tic)
```
```(-1)2:A2B2: qo 3 oo 3 xw &#zx
→ °) (asks for non-regular hexagons: qo .. xw &#zx)
= non-Johnsonian (patex cube)
```
```(-1)2(-3):A2B2: qo 3 oo 3 oq &#zx
= Wythoffian o3o4x (cube)
```

in o2o4o subsymmetry   (up)
Representation:
```wx 2 xx 4 ox &#zx  (sirco)
```
All layers &
kaleido-facetings per layer:
```A:  w 2 x 4 o   →  A2:    w  2 (-x) 4   q
B:  x 2 x 4 x   →  B1:  (-x) 2   x  4   x   →  B12:  (-x) 2 (-x) 4   w
↳  B13:  (-x) 2   w  4 (-x)
↳  B2:    x  2 (-x) 4   w   →  (B21 = B12)
↳  B3:    x  2   w  4 (-x)  →  (B31 = B13)
```
A priori invalid combinations:
```A + B2,B12  → ‡ (u in A)
A2  → ‡ (q or w in extremal layer, i.e. A)
```
Other layer-combinations:
```B1:  w(-x) 2 xx 4 ox &#zx     → †)
B3:  wx 2 xw 4 o(-x) &#zx
B13: w(-x) 2 xw 4 o(-x) &#zx  → †)
```
Stott expansion:
(derived potential CRFs
& beyond)
```(-1):-: qo 2 xx 4 ox &#zx   (squobcu, J28)
```
```(-2):-: wx 2 oo 4 ox &#zx   (esquidpy, J15)
```
```(-1)(-2):-: qo 2 oo 4 ox &#zx
= Wythoffian x3o4o (oct)
```
```3:B3: wx 2 xw 4 xo &#zx
= Wythoffian o3x4x (tic)
```
```(-1)3:B3: qo 2 xw 4 xo &#zx
→ °) (asks for non-regular hexagons: qo 2 xw &#zx)
= non-Johnsonian (pactic)
```
```(-2)3:B3: wx 2 oq 4 xo &#zx
→ °) (asks for non-regular hexagons: wx 2 oq &#zx)
= non-Johnsonian (pexco)
```
```(-1)(-2)3:B3: qo 2 oq 4 xo &#zx
= Wythoffian o3x4o (co)
```

in . o4o subsymmetry   (up)
Representation:
```xxxx 4 oxxo &#xt  (sirco)
```
not prismatically symmetric
combinations of formers:
```none, because A2 was ruled out already, and the possible combinations B1 + B3, B1 + B13, and B3 + B13
within the 2 medial layers B each would ask for u-edges there → ‡)
```

in . o3o subsymmetry   (up)
Representation:
```xxwoqo 3 oqowxx &#xt  (sirco)
```
All layers &
kaleido-facetings per layer:
```A:  x 3 o   →  A1:  (-x) 3   x   →  A12:    o  3 (-x)
B:  x 3 q   →  B1:  (-x) 3   w
C:  w 3 o
D:  o 3 w
E:  q 3 x   →  E2:    w  3 (-x)
F:  o 3 x   →  F2:    x  3 (-x)  →  F21:  (-x) 3   o
```
A priori invalid combinations:
```A   + B1,F21   → ‡ (u in A)
A1  + B,E2,F2  → ‡ (u in A)
A12 + E,F      → ‡ (u in A)
B   + F21      → ‡ (u in F)
B1  + F2       → ‡ (u in F)
E   + F2       → ‡ (u in F)
E2  + F        → ‡ (u in F)
```
Other layer-combinations:
```E2F2       xxwowx 3 oqow(-x)(-x) &#xt
A1B1F21    (-x)(-x)woq(-x) 3 xwowxo &#xt     → †)
A12B1E2F21 o(-x)wow(-x) 3 (-x)wow(-x)o &#xt  → †)
```
Stott expansion:
(derived potential CRFs
& beyond)
```1:E2F2: xxwowx 3 xwxQoo &#xt
→ °) (asks for non-unit edges DC, DE, DF)

related: xxw.wx 3 xwx.oo &#xt
→ °) (asks for non-regular hexagons: ... xwx &#xt)
```

#### EKF of the cuboctahedron (o3x4o)

in o3o3o subsymmetry   (up)
Representation:
```x3o3x  (co)
```
All layers &
kaleido-facetings per layer:
```A: x3o3x  →  A1: (-x)3 x 3  x   →  A12:   o 3(-x)3  u
↳  A13: (-x)3  u 3(-x)
↳  A3:   x 3 x 3(-x)  →  (A31 = A13)
↳  (A32 = A23)
```
A priori invalid combinations:
```A12, A13  → ‡ (u in A)
```
Stott expansion:
(derived potential CRFs)
```(-1):-: o3o3x
= Wythoffian (tet)
```
```2:-: x3x3x
= Wythoffian (toe)
```
```(-1)2:-: o3x3x
= Wythoffian (tut)
```
```(-1)2(-3):-: o3x3o
= Wythoffian (oct)
```
```1:A1: o3x3x
= Wythoffian (tut)
```
```1(-2):A1: o3o3x
= Wythoffian (tet)
```
```1(-3):A1: o3x3o
= Wythoffian (oct)
```

in o2o4o subsymmetry   (up)
Representation:
```qo 2 xo 4 oq &#zx  (co)
```
All layers &
kaleido-facetings per layer:
```A: q2x4o  →  A2: q 2(-x)4 q
B: o2o4q
```
A priori invalid combinations:
```A2  → ‡ (q in A)
```
Stott expansion:
(derived potential CRFs
& beyond)
```1:-: wx 2 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: wx .. oq &#zx)
= non-Johnsonian (pexco)
```
```3:-: qo 2 xo 4 xw &#zx
→ °) (asks for non-regular hexagons: qo .. xw &#zx)
= non-Johnsonian (pactic)
```
```13:-: wx 2 xo 4 xw &#zx
= Wythoffian o3x4x (tic)
```

in . o4o subsymmetry   (up)
Representation:
```xox 4 oqo &#xt  (co)
```
not prismatically symmetric
combinations of formers:
```none, because A2 was disallowed and thus no independent changes in extremal layers A remain possible.
```

in o2o2o subsymmetry   (up)
Representation:
```qoq 2 qqo 2 oqq &#zx  (co)
```
All layers &
kaleido-facetings per layer:
```A: q2q2o
B: o2q2q
C: q2o2q

(As this representation does not show up (true / x-) edges within the layers, those thus cannot be inverted either.
Therefore at most pure partial Stott expansions wrt. this subsymmetry remain possible.)
```
Stott expansion:
(derived potential CRFs)
```1:-: wxw 2 qqo 2 oqq &#zx
→ °) (asks for non-regular hexagons: wx .. oq &#zx)
= non-Johnsonian (pexco)
```
```12:-: wxw 2 wwx 2 oqq &#zx
→ °) (asks for non-regular hexagons: wx .. oq &#zx)
= non-Johnsonian (pactic)
```
```123:-: wxw 2 wwx 2 xww &#zx
= Wythoffian o3x4x (tic)
```

in . o3o subsymmetry   (up)
Representation:
```xxo 3 oxx &#xt  (co)
```
All layers &
kaleido-facetings per layer:
```A: x3o  →  A1: (-x)3x  →  A12: o3(-x)
B: x3x  →  B1: (-x)3u
↳  B2: u3(-x)
C: o3x  →  C2: x3(-x)  →  C21: (-x)3o
```
A priori invalid combinations:
```A   + B1,C21  → ‡ (u in A)
A1  + B2,C2   → ‡ (u in A)
A12 + B,C     → ‡ (u in A)
B   + C2,C21  → ‡ (u in B)
B1  + C2      → ‡ (u in C)
B2  + C       → ‡ (u in C)
```
Other layer-combinations:
```C2       xxx 3 ox(-x) &#xt  → †)
B2C2     xux 3 o(-x)(-x) &#xt
A1C21    (-x)x(-x) 3 xxo &#xt  → †)
A1B1C21  (-x)(-x)(-x) 3 xuo &#xt  → †)
A12B1C21 o(-x)(-x) 3 (-x)uo &#xt  → †)
```
Stott expansion:
(derived potential CRFs)
```2:AB2C2: xux 3 xoo &#xt
= Wythoffian o3x3x (tut)
```

#### EKF of the icosahedral pyramid (ox3oo5oo&#x)

Obviously here at most those subsymmetries can be applied, which do respect the possibilities of the base polyhedron, i.e. of the icosahedron (cf. above). As it turns out, all three types become positively applicable.

in . o2o2o subsymmetry   (up)
Representation:
```xof 2 fxo 2 ofx &#zx || o2o2o  (ikepy)
= xofox 2 ofxfo &#xt || o2o
```
` ©`
All layers &
kaleido-facetings per layer:
```A:  x 2 f 2 o   →  A1:  (-x) 2   f  2   o
B:  o 2 x 2 f   →  B2:    o  2 (-x) 2   f
C:  f 2 o 2 x   →  C3:    f  2   o  2 (-x)
D:  o 2 o 2 o
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```A1:     (-x)of 2 fxo 2 ofx &#zx || o2o2o
A1B2:   (-x)of 2 f(-x)o 2 ofx &#zx || o2o2o     → †)
A1B2C3: (-x)of 2 f(-x)o 2 of(-x) &#zx || o2o2o  → †)
```
Stott expansion:
(derived potential CRFs)
```oxF 2 fxo 2 ofx &#zx || x2o2o
= oxFxo 2 ofxfo &#xt || x2o
= bilbiro || line

→ CRF with cell list:
1 bilbiro (J91)
4 peppies (J2)
4 squippies (J1)
4 tets
2 trips
```

in . . o3o subsymmetry   (up)
Representation:
```xofo 3 ofox &#xt || o3o  (ikepy)
```
` ©`
All layers &
kaleido-facetings per layer:
```A:  x 3 o   →  A1:  (-x) 3   x
B:  o 3 f
C:  f 3 o
D:  o 3 x   →  D2:    x  3 (-x)
E:  o 3 o
```
A priori invalid combinations:
```A1 + D2  → ‡) (u in A, u in D)
```
Other layer-combinations:
```A1:    (-x)ofo 3 xfox &#xt || o3o
```
Stott expansion:
(derived potential CRFs)
```oxFx 3 xfox &#xt || x3o
= thawro || {3}

→ CRF with cell list:
1 oct
3 peppies (J2)
3 squippies (J1)
9 tets
1 thawro (J92)
1 tricu (J3)
3 trips
```

in . . o5o subsymmetry   (up)
Representation:
```oxoo 5 ooxo &#xt || o5o  (ikepy)
```
` ©`
All layers &
kaleido-facetings per layer:
```A:  o 5 o
B:  x 5 o   →  B1:  (-x) 5   f
C:  o 5 x   →  C2:    f  5 (-x)
D:  o 5 o
E:  o 5 o
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```B1:    o(-x)oo 5 ofxo &#xt || o5o
B1C2:  o(-x)fo 5 of(-x)o &#xt || o5o  → †)
```
Stott expansion:
(derived potential CRFs)
```xoxx 5 ofxo &#xt || x5o
= pocuro || {5}

→ CRF with cell list:
5  peppies (J2)
2  pips
1  pocuro (J32)
10 squippies (J1)
5  tets
5  trips

related: xox. 5 ofx. &#xt || x5o
= pero || {5}

→ CRF (segmentochoron) with cell list:
1  pecu (J5)
5  peppies (J2)
1  pero (J6)
1  pip
10 squippies (J1)
```

#### EKF of the hexacosachoron (x3o3o5o)

in o2o3o5o subsymmetry   (up)
Representation:
```VFfxo 2 oxofo 3 oooox 5 ooxoo &#zx  =  oxofofoxo 3 ooooxoooo 5 ooxoooxoo &#xt  (ex)
```
All layers &
kaleido-facetings per layer:
```A:  V2o3o5o
B:  F2x3o5o   →  B2:   F 2(-x)3  x 5  o   →  B23:   F 2  o 3(-x)5  f
C:  f2o3o5x   →  C4:   f 2  o 3  f 5(-x)
D:  x2f3o5o   →  D1: (-x)2  f 3  o 5  o
E:  o2o3x5o   →  E3:   o 2  x 3(-x)5  f   →  E32:   o 2(-x)3  o 5  f
```
A priori invalid combinations:
```when component 1 (o2.3.5.) remains unchanged (no D1):
B2 + E3  → ‡) (u in B, u in E)
E32 + B  → ‡) (u in B)
B23 + E  → ‡) (u in E)

when component 2 (.2o3o5o) remains unchanged (neither B2, B23, C4, E3, E32):
none

As changes in (o2.3.5.) do not affect (.2o3o5o) and vice versa,
the above exclusions remain valid too, when D1 is activ.
```
Other layer-combinations:
```D1:         VFf(-x)o 2 oxofo 3 oooox 5 ooxoo &#zx
B2:         VFfxo 2 o(-x)ofo 3 oxoox 5 ooxoo &#zx  → †)
B2E32:      VFfxo 2 o(-x)of(-x) 3 oxooo 5 ooxof &#zx
E3:         VFfxo 2 oxofx 3 oooo(-x) 5 ooxof &#zx
B23E3:      VFfxo 2 ooofx 3 o(-x)oo(-x) 5 ofxof &#zx
C4:         VFfxo 2 oxofo 3 oofox 5 oo(-x)oo &#zx
B2D1:       VFf(-x)o 2 o(-x)ofo 3 oxoox 5 ooxoo &#zx
B2D1E32:    VFf(-x)o 2 o(-x)of(-x) 3 oxooo 5 ooxof &#zx  → †)
D1E3:       VFf(-x)o 2 oxofx 3 oooo(-x) 5 ooxof &#zx
B23D1E3:    VFf(-x)o 2 ooofx 3 o(-x)oo(-x) 5 ofxof &#zx
C4D1:       VFf(-x)o 2 oxofo 3 oofox 5 oo(-x)oo &#zx
B23E32:     VFfxo 2 ooof(-x) 3 o(-x)ooo 5 ofxof &#zx  → †)
B2C4:       VFfxo 2 o(-x)ofo 3 oxfox 5 oo(-x)oo &#zx
B2C4E32:    VFfxo 2 o(-x)of(-x) 3 oxfoo 5 oo(-x)of &#zx  → †)
C4E3:       VFfxo 2 oxofx 3 oofo(-x) 5 oo(-x)of &#zx
B23D1E32:   VFf(-x)o 2 ooof(-x) 3 o(-x)ooo 5 ofxof &#zx  → †)
B2C4D1:     VFf(-x)o 2 o(-x)ofo 3 oxfox 5 oo(-x)oo &#zx
B2C4D1E32:  VFf(-x)o 2 o(-x)of(-x) 3 oxfoo 5 oo(-x)of &#zx  → †)
C4D1E3:     VFf(-x)o 2 oxofx 3 oofo(-x) 5 oo(-x)of &#zx  → †)
B23C4D1E3:  VFf(-x)o 2 ooofx 3 o(-x)fo(-x) 5 of(-x)of &#zx  → †)
B23C4E32:   VFfxo 2 ooof(-x) 3 o(-x)foo 5 of(-x)of &#zx  → †)
B23C4D1E32: VFf(-x)o 2 ooof(-x) 3 o(-x)foo 5 of(-x)of &#zx   → †)
```
Stott expansion:
(derived potential CRFs)
```1:D1: BAFox 2 oxofo 3 oooox 5 ooxoo &#zx
= oxoofooxo 3 oooxoxooo 5 ooxoooxoo &#xt (telex)
→ CRF with cell list:
24  ikes
60  squippies (J1)
180 tets
20  trips

related: ..Fox 2 ..ofo 3 ..oox 5 ..xoo &#zx
= ..oofoo.. 3 ..oxoxo.. 5 ..xooox.. &#xt
→ CRF with cell list:
2  does
24 gyepips (J11)
60 squippies (J1)
40 tets
20 trips

related: ...ox 2 ...fo 3 ...ox 5 ...oo &#zx
= ...ofo... 3 ...xox... 5 ...ooo... &#xt (twau iddip)
→ CRF with cell list:
2  ids
24 peppies (J2)
60 squippies (J1)
20 trips
```
```2:B2E32: VFfxo 2 xoxFo 3 oxooo 5 ooxof &#zx
= xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt
→ CRF with cell list:
30 bilbiroes (J91)
26 ikes
80 octs
60 squippies (J1)
40 tets

related: ..fxo 2 ..xFo 3 ..ooo 5 ..xof &#zx
= ..xFoFx.. 3 ..ooooo.. 5 ..xofox.. &#xt
→ CRF with cell list:
30 bilbiroes (J91)
24 peppies (J2)
2  srids
40 tets
```
```3:E3: VFfxo 2 oxofx 3 xxxxo 5 ooxof &#zx
= oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt
→ CRF with cell list:
2   ids
30  ikes
40  octs
60  pips
180 squippies (J1)
180 tets
80  tricues (J3)
120 trips

related: ..fxo 2 ..ofx 3 ..xxo 5 ..xof &#zx
= ..ofxfo.. 3 ..xxoxx.. 5 ..xofox.. &#xt
→ CRF with cell list:
30  ikes
40  octs
24  pecues (J5)
12  pips
180 squippies (J1)
2   tids
```
```3:B23E3: VFfxo 2 ooofx 3 xoxxo 5 ofxof &#zx
= ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt
→ CRF with cell list:
2   ids
40  ikes
40  octs
12  pips
24  pocuroes (J32)
180 squippies (J1)
80  tets

related: ..fxo 2 ..ofx 3 ..xxo 5 ..xof &#zx
= ..ofxfo.. 3 ..xxoxx.. 5 ..xofox.. &#xt
→ CRF with cell list:
30  ikes
40  octs
24  pecues (J5)
12  pips
180 squippies (J1)
2   tids
```
```4:C4: VFfxo 2 oxofo 3 oofox 5 xxoxx &#zx
= oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt
→ CRF with cell list:
2   does
40  ikes
60  pips
300 squippies (J1)
100 tets
120 trips

related: .Ffxo 2 .xofo 3 .ofox 5 .xoxx &#zx
= .xofofox. 3 .ofoxofo. 5 .xoxxxox. &#xt
→ CRF with cell list:
40  ikes
36  pips
300 squippies (J1)
2   srids
60  tets
60  trips
```
```12:B2D1: BAFox 2 xoxFx 3 oxoox 5 ooxoo &#zx
= xoxxFxxox 3 oxoxoxoxo 5 ooxoooxoo &#xt
→ CRF with cell list:
48  gyepips (J11)
20  hips
2   ikes
80  octs
120 squippies (J1)
40  tricues (J3)
60  trips

related: ...ox 2 ...Fx 3 ...ox 5 ...oo &#zx
= ...xFx... 3 ...xox... 5 ...ooo... &#xt (twau tipe)
→ CRF with cell list:
20 hips
24 peppies (J2)
60 squippies (J1)
2  ties
```
```13:D1E3: BAFox 2 oxofx 3 xxxxo 5 ooxof &#zx
= oxoxfxoxo 3 xxxoxoxxx 5 ooxfofxoo &#xt
→ CRF with cell list:
30  bilbiroes (J91)
2   ids
40  octs
24  pips
24  pocuroes (J32)
120 tets
80  tricues (J3)
20  trips

related: ..Fox 2 ..ofx 3 ..xxo 5 ..xof &#zx
= ..oxfxo.. 3 ..xoxox.. 5 ..xfofx.. &#xt
→ CRF with cell list:
30 bilbiroes (J91)
40 octs
24 peroes (J6)
2  tids
60 tets
20 trips
```
```13:B23D1E3: BAFox 2 ooofx 3 xoxxo 5 ofxof &#zx
= oooxfxooo 3 xoxoxoxox 5 ofxfofxfo &#xt
→ CRF with cell list:
30  bilbiroes (J91)
2   ids
40  octs
48  peroes (J6)
140 tets
20  trips

related: ..Fox 2 ..ofx 3 ..xxo 5 ..xof &#zx
= ..oxfxo.. 3 ..xoxox.. 5 ..xfofx.. &#xt
→ CRF with cell list:
30 bilbiroes (J91)
40 octs
24 peroes (J6)
2  tids
60 tets
20 trips
```
```14:C4D1: BAFox 2 oxofo 3 oofox 5 xxoxx &#zx
= oxoofooxo 3 oofxoxfoo 5 xxoxxxoxx &#xt
→ CRF with cell list:
2   does
24  pips
24  pocuroes (J32)
120 squippies (J1)
40  teddies (J63)
40  tets
140 trips

related: .AFox 2 .xofo 3 .ofox 5 .xoxx &#zx
= .xoofoox. 3 .ofxoxfo. 5 .xoxxxox. &#xt
→ CRF with cell list:
24  pocuroes (J32)
120 squippies (J1)
2   srids
40  teddies (J63)
80  trips

related: ...ox 2 ...fo 3 ...ox 5 ...xx &#zx
= ...ofo... 3 ...xox... 5 ...xxx... &#xt (twau tiddip)
→ CRF with cell list:
24 pecues (J5)
60 squippies (J1)
2  tids
80 trips
```
```24:B2C4: VFfxo 2 xoxFx 3 oxfox 5 xxoxx &#zx
= xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt
→ CRF with cell list:
40  octs
12  pips
24  pocuroes (J32)
120 squippies (J1)
2   srids
120 tets
40  thawroes (J92)
180 trips

related: .Ffxo 2 .oxFx 3 .xfox 5 .xoxx &#zx
= .oxFxFxo. 3 .xfoxofx. 5 .xoxxxox. &#xt
→ CRF with cell list:
24  peroes (J6)
12  pips
120 squippies (J1)
120 tets
40  thawroes (J92)
2   tids
120 trips
```
```34:C4E3: VFfxo 2 oxofx 3 xxFxo 5 xxoxF &#zx
= oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt
→ CRF with cell list:
30  bilbiroes (J91)
60  dips
240 squippies (J1)
40  thawroes (J92)
2   tids
40  tricues (J3)
60  trips

related: .Ffxo 2 .xofx 3 .xFxo 5 .xoxF &#zx
= .xofxfox. 3 .xFxoxFx. 5 .xoxFxox. &#xt
→ CRF with cell list:
30  bilbiroes (J91)
36  dips
2   grids
240 squippies (J1)
40  thawroes (J92)
```
```124:B2C4D1: BAFox 2 xoxFx 3 oxfox 5 xxoxx &#zx
→ ‡) (f in C)
```
in . o3o5o subsymmetry   (up)
not prismatically symmetric
combinations of formers:
```3:(B23)E3:
oxo|fxf|ooo 3 xxx|xox|xox 5 oox|ofo|xfo &#xt
→ CRF with cell list:
2   ids
30  ikes
40  octs
36  pips
12  pocuroes (J32)
180 squippies (J1)
130 tets
40  tricues (J3)
60  trips
```
```13:(B23)D1E3:
oxo|xfx|ooo 3 xxx|oxo|xox 5 oox|fof|xfo &#xt
→ CRF with cell list:
30  bilbiroes (J91)
2   ids
40  octs
24  peroes (J6)
12  pips
12  pocuroes (J32)
130 tets
40  tricues (J3)
20  trips
```
using here node marks / (pseudo) edge lengths: F=f+x, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x

in o3o3o3o subsymmetry   (up)
Representation:
```xffoo 3 oxoof 3 fooxo 3 ooffx &#zx  (ex)
```
All layers &
kaleido-facetings per layer:
```A:  x3o3f3o   →  A1: (-x)3  x 3  f 3  o   →  A12:  o 3(-x)3  F 3 o
B:  f3x3o3o   →  B2:   F 3(-x)3  x 3  o   →  B23:  F 3  o 3(-x)3 x   →  B234:   F 3 o 3 o 3(-x)
C:  f3o3o3f
D:  o3o3x3f   →  D3:   o 3  x 3(-x)3  F   →  D32:  x 3(-x)3  o 3 F   →  D321: (-x)3 o 3 o 3  F
E:  o3f3o3x   →  E4:   o 3  f 3  x 3(-x)  →  E43:  o 3  F 3(-x)3 o
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```A1:             (-x)ffoo 3 xxoof 3 fooxo 3 ooffx &#zx
D321:           xff(-x)o 3 oxoof 3 foooo 3 oofFx &#zx  → †)
A1D321:         (-x)ff(-x)o 3 xxoof 3 foooo 3 oofFx &#zx  → †)
A12:            offoo 3 (-x)xoof 3 Fooxo 3 ooffx &#zx
B2:             xFfoo 3 o(-x)oof 3 fxoxo 3 ooffx &#zx
D32:            xffxo 3 oxo(-x)f 3 foooo 3 oofFx &#zx
A12B2:          oFfoo 3 (-x)(-x)oof 3 Fxoxo 3 ooffx &#zx
B2D32:          xFfxo 3 o(-x)o(-x)f 3 fxooo 3 oofFx &#zx
A12D32:         offxo 3 (-x)xo(-x)f 3 Foooo 3 oofFx &#zx  → †)
A12B2D32:       oFfxo 3 (-x)(-x)o(-x)f 3 Fxooo 3 oofFx &#zx  → †)
A1B2:           (-x)Ffoo 3 x(-x)oof 3 fxoxo 3 ooffx &#zx
A1D32:          (-x)ffxo 3 xxo(-x)f 3 foooo 3 oofFx &#zx
B2D321:         xFf(-x)o 3 o(-x)oof 3 fxooo 3 oofFx &#zx
A12D321:        off(-x)o 3 (-x)xoof 3 Foooo 3 oofFx &#zx  → †)
A1B2D32:        (-x)Ffxo 3 x(-x)o(-x)f 3 fxooo 3 oofFx &#zx
A1B2D321:       (-x)Ff(-x)o 3 x(-x)oof 3 fxooo 3 oofFx &#zx  → †)
A12B2D321:      oFf(-x)o 3 (-x)(-x)oof 3 Fxooo 3 oofFx &#zx  → †)
A1D3:           (-x)ffoo 3 xxoxf 3 foo(-x)o 3 oofFx &#zx
A1B23:          (-x)Ffoo 3 xooof 3 f(-x)oxo 3 oxffx &#zx
A1E43:          (-x)ffoo 3 xxooF 3 foox(-x) 3 ooffo &#zx
B23D321:        xFf(-x)o 3 oooof 3 f(-x)ooo 3 oxfFx &#zx  → †)
D321E43:        xff(-x)o 3 oxooF 3 fooo(-x) 3 oofFo &#zx  → †)
A1B23D3:        (-x)Ffoo 3 xooxf 3 f(-x)o(-x)o 3 oxfFx &#zx
A1D3E43:        (-x)ffoo 3 xxoxF 3 foo(-x)(-x) 3 oofFo &#zx  → †)
A1B23D321:      (-x)Ff(-x)o 3 xooof 3 f(-x)ooo 3 oxfFx &#zx  → †)
A1B23E43:       (-x)Ffoo 3 xoooF 3 f(-x)ox(-x) 3 oxffo &#zx  → †)
A1D321E43:      (-x)ff(-x)o 3 xxooF 3 fooo(-x) 3 oofFo &#zx  → †)
B23D321E43:     xFf(-x)o 3 ooooF 3 f(-x)oo(-x) 3 oxfFo &#zx  → †)
A1B23D3E43:     (-x)Ffoo 3 xooxF 3 f(-x)o(-x)(-x) 3 oxfFo &#zx  → †)
A1B23D321E43:   (-x)Ff(-x)o 3 xoooF 3 f(-x)oo(-x) 3 oxfFo &#zx  → †)
A1E4:           (-x)ffoo 3 xxoof 3 fooxx 3 ooff(-x) &#zx
A1B234:         (-x)Ffoo 3 xooof 3 fooxo 3 o(-x)ffx &#zx  → †)
B234D321:       xFf(-x)o 3 oooof 3 foooo 3 o(-x)fFx &#zx  → †)
A1B234D321:     (-x)Ff(-x)o 3 xooof 3 foooo 3 o(-x)fFx &#zx  → †)
A1B234E4:       (-x)Ffoo 3 xooof 3 fooxx 3 o(-x)ff(-x) &#zx  → †)
A1B234D321E4:   (-x)Ff(-x)o 3 xooof 3 fooox 3 o(-x)fF(-x) &#zx  → †)
B2D3:           xFfoo 3 o(-x)oxf 3 fxo(-x)o 3 oofFx &#zx
A12D3:          offoo 3 (-x)xoxf 3 Foo(-x)o 3 oofFx &#zx  → †)
A12B23:         oFfoo 3 (-x)ooof 3 F(-x)oxo 3 oxffx &#zx
A12E43:         offoo 3 (-x)xooF 3 Foox(-x) 3 ooffo &#zx
B23D32:         xFfxo 3 ooo(-x)f 3 f(-x)ooo 3 oxfFx &#zx
A12B2D3:        oFfoo 3 (-x)(-x)oxf 3 Fxo(-x)o 3 oofFx &#zx  → †)
A12B23D3:       oFfoo 3 (-x)ooxf 3 F(-x)o(-x)o 3 oxfFx &#zx  → †)
A12B2E43:       oFfoo 3 (-x)(-x)ooF 3 Fxox(-x) 3 ooffo &#zx  → †)
A12B23D32:      oFfxo 3 (-x)oo(-x)f 3 F(-x)ooo 3 oxfFx &#zx  → †)
A12B2E43:       oFfoo 3 (-x)(-x)ooF 3 Fxox(-x) 3 ooffo &#zx  → †)
A12B23E43:      oFfoo 3 (-x)oooF 3 F(-x)ox(-x) 3 oxffo &#zx  → †)
A12B2D3E43:     oFfoo 3 (-x)(-x)oxF 3 Fxo(-x)(-x) 3 oofFo &#zx  → †)
A12B23D3E43:    oFfoo 3 (-x)ooxF 3 F(-x)o(-x)(-x) 3 oxfFo &#zx  → †)
A12B23D32E43:   oFfxo 3 (-x)oo(-x)F 3 F(-x)oo(-x) 3 oxfFo &#zx  → †)
A1B2D3:         (-x)Ffoo 3 x(-x)oxf 3 fxo(-x)o 3 oofFx &#zx
A1B23D32:       (-x)Ffxo 3 xoo(-x)f 3 f(-x)ooo 3 oxfFx &#zx
A1B2E43:        (-x)Ffoo 3 x(-x)ooF 3 fxox(-x) 3 ooffo &#zx  → †)
A1D32E43:       (-x)ffxo 3 xxo(-x)F 3 fooo(-x) 3 oofFo &#zx  → †)
A12B23D321:     oFf(-x)o 3 (-x)ooof 3 F(-x)ooo 3 oxfFx &#zx  → †)
A12D321E43:     off(-x)o 3 (-x)xooF 3 Fooo(-x) 3 oofFo &#zx  → †)
B2D321E43:      xFf(-x)o 3 o(-x)ooF 3 fxoo(-x) 3 oofFo &#zx  → †)
A1B2D3E43:      (-x)Ffoo 3 x(-x)oxF 3 fxo(-x)(-x) 3 oofFo &#zx  → †)
A1B2D32E43:     (-x)Ffxo 3 x(-x)o(-x)F 3 fxoo(-x) 3 oofFo &#zx  → †)
A1B2D321E43:    (-x)Ff(-x)o 3 x(-x)ooF 3 fxoo(-x) 3 oofFo &#zx  → †)
A1B23D32E43:    (-x)Ffxo 3 xoo(-x)F 3 f(-x)oo(-x) 3 oxfFo &#zx  → †)
A12B2D321E43:   oFf(-x)o 3 (-x)(-x)ooF 3 Fxoo(-x) 3 oofFo &#zx  → †)
A12B23D321E43:  oFf(-x)o 3 (-x)oooF 3 F(-x)oo(-x) 3 oxfFo &#zx  → †)
A1B2E4:         (-x)Ffoo 3 x(-x)oof 3 fxoxx 3 ooff(-x) &#zx
A1B234D32:      (-x)Ffxo 3 xoo(-x)f 3 foooo 3 o(-x)fFx &#zx  → †)
A1D32E4:        (-x)ffxo 3 xxo(-x)f 3 fooox 3 oofF(-x) &#zx
B2D321E4:       xFf(-x)o 3 o(-x)oof 3 fxoox 3 oofF(-x) &#zx  → †)
A12B234D321:    oFf(-x)o 3 (-x)ooof 3 Foooo 3 o(-x)fFx &#zx  → †)
A12D321E4:      off(-x)o 3 (-x)xoof 3 Fooox 3 oofF(-x) &#zx  → †)
A1B2D32E4:      (-x)Ffxo 3 x(-x)o(-x)f 3 fxoox 3 oofF(-x) &#zx  → †)
A1B2D321E4:     (-x)Ff(-x)o 3 x(-x)oof 3 fxoox 3 oofF(-x) &#zx  → †)
A12B2D321E4     oFf(-x)o 3 (-x)(-x)oof 3 Fxoox 3 oofF(-x) &#zx  → †)
A1B234D32E4:    (-x)Ffxo 3 xoo(-x)f 3 fooox 3 o(-x)fF(-x) &#zx  → †)
A12B234D321E4:  oFf(-x)o 3 (-x)ooof 3 Fooox 3 o(-x)fF(-x) &#zx  → †)
A1B2D3E4:       (-x)Ffoo 3 x(-x)oxf 3 fxo(-x)x 3 oofF(-x) &#zx  → †)
A1B23D32E4:     (-x)Ffxo 3 xoo(-x)f 3 f(-x)oox 3 oxfF(-x) &#zx  → †)
A1B234D32E43:   (-x)Ffxo 3 xoo(-x)F 3 fooo(-x) 3 o(-x)fFo &#zx  → †)
A12B234D321E43: oFf(-x)o 3 (-x)oooF 3 Fooo(-x) 3 o(-x)fFo &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```1:A1: oFFxx 3 xxoof 3 fooxo 3 ooffx &#zx
→ CRF with cell list:
5   coes
30  ikes
20  octs
90  squippies (J1)
125 tets
40  trips

related: oF.xx 3 xx.of 3 fo.xo 3 oo.fx &#zx
→ CRF with cell list:
5  coes
30 mibdies (J62)
20 octs
90 squippies (J1)
20 teddies (J63)
25 tets
40 trips
```
```2:A12: offoo 3 ouxxF 3 Fooxo 3 ooffx &#zx
→ ‡) (u in B)
```
```2:B2: xFfoo 3 xoxxF 3 fxoxo 3 ooffx &#zx
→ CRF with cell list:
20  ikes
25  octs
60  squippies (J1)
270 tets
40  tricues (J3)
60  trips
5   tuts
```
```2:D32: xffxo 3 oxo(-x)f 3 foooo 3 oofFx &#zx
→ ‡) (u in B)
```
```2:A12B2: oFfoo 3 ooxxF 3 Fxoxo 3 ooffx &#zx
→ CRF with cell list:
30 bilbiroes (J91)
25 octs
20 teddies (J63)
80 tets
20 tricues (J3)
5  tuts
```
```12:A12B2: xAFxx 3 ooxxF 3 Fxoxo 3 ooffx &#zx
→ CRF with cell list:
30 bilbiroes (J91)
25 octs
30 pips
20 teddies (J63)
60 tets
5  toes
40 tricues (J3)
40 trips
```
```2:B2D32: xFfxo 3 xoxoF 3 fxooo 3 oofFx &#zx
→ CRF with cell list:
20 ikes
25 octs
60 squippies (J1)
55 tets
20 thawroes (J92)
```
```12:A1B2: oAFxx 3 uoxxF 3 fxoxo 3 ooffx &#zx
→ ‡) (u in A)
```
```12:A1D32: oFFux 3 uuxoF 3 foooo 3 oofFx &#zx
→ ‡) (u in A, u in B, u in D)
```
```12: B2D321: uAFox 3 xoxxF 3 fxooo 3 oofFx &#zx
→ ‡) (u in A)
```
```12:A1B2D32: oAFux 3 uoxoF 3 fxooo 3 oofFx &#zx
→ ‡) (u in A, u in D)
```
```13:A1D3: oFFxx 3 xxoxf 3 Fxxox 3 oofFx &#zx
→ CRF with cell list:
10 hips
20 octs
30 mibdies (J62)
90 squippies (J1)
20 thawroes (J92)
40 tricues (J3)
90 trips
10 tuts
```
```13:A1B23: oAFxx 3 xooof 3 Foxux 3 oxffx &#zx
→ ‡) (u in D)
```
```13:A1E43: oFFxx 3 xxooF 3 Fxxuo 3 ooffo &#zx
→ ‡) (u in D)
```
```13:A1B23D3: oAFxx 3 xooxf 3 Foxox 3 oxfFx &#zx
→ ‡) (f in C)
```
```14:A1E4: oFFxx 3 xxoof 3 fooxx 3 xxFFo &#zx
→ CRF with cell list:
60 bilbiroes (J91)
10 coes
40 octs
70 tets
20 trips
```
```23:A12E43: offoo 3 ouxxA 3 Axxuo 3 ooffo &#zx
→ ‡) (u in B, u in D)
```
```23:B2D3: xFfoo 3 xoxuF 3 Fuxox 3 oofFx &#zx
→ ‡) (u in B, u in D)
```
```23:A12B23: oFfoo 3 oxxxF 3 Aoxux 3 oxffx &#zx
→ ‡) (u in D)
```
```23:B23D32: xFfxo 3 xxxoF 3 Foxxx 3 oxfFx &#zx
→ CRF with cell list:
10 coes
60 pips
30 tets
40 thawroes (J92)
40 tricues (J3)
```
```123:A1B2D3: oAFxx 3 uoxuF 3 Fuxox 3 oofFx &#zx
→ ‡) (u in A, u in B, u in D)
```
```123:A1B23D32: oAFux 3 uxxoF 3 Foxxx 3 oxfFx &#zx
→ ‡) (u in A, u in D)
```
```124:A1B2E4: oAFxx 3 uoxxF 3 fxoxx 3 xxFFo &#zx
→ ‡) (u in A)
```
```124:A1D32E4: oFFux 3 uuxoF 3 fooox 3 xxFAo &#zx
→ ‡) (u in A, u in B, u in D)
```
using here node marks / (pseudo) edge lengths: F=f+x, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x

in o3o3o *b3o subsymmetry   (up)
Representation:
```foxo 3 ooof 3 xfoo *b3 oxfo &#zx  (ex)

with cyclical layer symmetry: A(134) → B(341) → C(413) → A(134)
```
All layers &
kaleido-facetings per layer:
```A:  f3o3x *b3o  →  A1:   f 3 x 3(-x)*b3  o   →  A12:  F 3(-x)3 o *b3  x  →  A123:   F 3 o 3  o *b3(-x)
B:  o3o3f *b3x  →  B1:   o 3 x 3  f *b3(-x)  →  B12:  x 3(-x)3 F *b3  o  →  B123: (-x)3 o 3  F *b3  o
C:  x3o3o *b3f  →  C1: (-x)3 x 3  o *b3  f   →  C12:  o 3(-x)3 x *b3  F  →  C123:   o 3 o 3(-x)*b3  F
D:  o3f3o *b3o
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```C1:           fo(-x)o 3 ooxf 3 xfoo *b3 oxfo &#zx
B123:         f(-x)xo 3 ooof 3 xFoo *b3 oofo &#zx  → †)
B123C1:       f(-x)(-x)o 3 ooxf 3 xFoo *b3 oofo &#zx  → †)
A12:          Foxo 3 (-x)oof 3 ofoo *b3 xxfo &#zx
A12B12:       Fxxo 3 (-x)(-x)of 3 oFoo *b3 xofo &#zx  → †)
A12B12C12:    Fxoo 3 (-x)(-x)(-x)f 3 oFxo *b3 xoFo &#zx  → †)
A12C1:        Fo(-x)o 3 (-x)oxf 3 ofoo *b3 xxfo &#zx  → †)
A12B123:      F(-x)xo 3 (-x)oof 3 oFoo *b3 xofo &#zx  → †)
A12B12C1:     Fx(-x)o 3 (-x)(-x)xf 3 oFoo *b3 xofo &#zx  → †)
A12B123C1:    F(-x)(-x)o 3 (-x)oxf 3 oFoo *b3 xofo &#zx  → †)
A1C1:         fo(-x)o 3 xoxf 3 (-x)foo *b3 oxfo &#zx
A1B123:       f(-x)xo 3 xoof 3 (-x)Foo *b3 oofo &#zx  → †)
B123C123:     f(-x)oo 3 ooof 3 xF(-x)o *b3 ooFo &#zx  → †)
A1B123C1:     f(-x)(-x)o 3 xoxf 3 (-x)Foo *b3 oofo &#zx  → †)
A1B123C123:   f(-x)oo 3 xoof 3 (-x)F(-x)o *b3 ooFo &#zx  → †)
A1B12C1:      fx(-x)o 3 x(-x)xf 3 (-x)Foo *b3 oofo &#zx  → †)
A1B123C12:    f(-x)oo 3 xo(-x)f 3 (-x)Fxo *b3 ooFo &#zx  → †)
A12B123C123:  F(-x)oo 3 (-x)oof 3 oF(-x)o *b3 xoFo &#zx  → †)
A1B1C1:       fo(-x)o 3 xxxf 3 (-x)foo *b3 o(-x)fo &#zx
A123B123C123: F(-x)oo 3 ooof 3 oF(-x)o *b3 (-x)oFo &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```2:0: foxo 3 xxxF 3 xfoo *b3 oxfo &#zx (icau prissi)
→ CRF with cell list:
480 tets
96  tricues (J3)
96  trips
24  tuts

related: fox. 3 xxx. 3 xfo. *b3 oxf. &#zx (prissi)
→ CRF with cell list:
24 ikes
96 tricues (J3)
96 trips
24 tuts
```
```1:C1: Fxox 3 ooxf 3 xfoo *b3 oxfo &#zx (icau pretasto)
→ CRF with cell list:
8   coes
32  ikes
40  octs
96  squippies (J1)
136 tets
48  trips

related: Fxo. 3 oox. 3 xfo. *b3 oxf. &#zx (pretasto)
→ CRF with cell list:
24 bilbiroes (J91)
8  coes
40 octs
32 teddies (J63)
40 tets
```
```2:A12: Foxo 3 oxxF 3 ofoo *b3 xxfo &#zx
→ CRF with cell list:
136 tets
32  thawroes (J92)
32  tricues (J3)
16  tuts
```
```23:A12: Foxo 3 oxxF 3 xFxx *b3 xxfo &#zx
→ CRF with cell list:
8  coes
48 pips
96 tets
32 thawroes (J92)
8  toes
32 tricues (J3)
48 trips
8  tuts
```
```13:A1C1: Fxox 3 xoxf 3 oFxx *b3 oxfo &#zx
→ CRF with cell list:
8  coes
40 octs
96 squippies (J1)
32 teddies (J63)
32 thawroes (J92)
96 trips
8  tuts
```
```134:A1B1C1: Fxox 3 xxxf 3 oFxx *b3 xoFx &#zx
→ ‡) (f in D)
```
using here node marks / (pseudo) edge lengths: F=f+x

in o5o2o5o subsymmetry   (up)
Representation:
```xfooxo 5 xofxoo 2 oxofox 5 ooxofx &#zx  (ex)

with layer symmetry:
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234) ↔
A(2143),C(2143),B(2143),E(2143),D(2143),F(2143)
and cycle:
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234) →
F(3421),D(3421),E(3421),C(3421),B(3421),A(3421) →
A(2143),C(2143),B(2143),E(2143),D(2143),F(2143) →
F(4312),E(4312),D(4312),B(4312),C(4312),A(4312) →
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234)
```
All layers &
kaleido-facetings per layer:
```A:  x5x o5o   →   A1: (-x)5  F   o 5  o    or    A2:   F 5(-x)  o 5  o
B:  f5o x5o   →   B3:   f 5  o (-x)5  f
C:  o5f o5x   →   C4:   o 5  f   f 5(-x)
D:  o5x f5o   →   D2:   f 5(-x)  f 5  o
E:  x5o o5f   →   E1: (-x)5  f   o 5  f
F:  o5o x5x   →   F3:   o 5  o (-x)5  F    or    F4:   o 5  o   F 5(-x)
```
A priori invalid combinations:
```when component 1 (o5o2.5.) remains unchanged (neither A1, A2, D2, E1):
B3 + neither F3 nor F4  → ‡) (u in F)
C4 + neither F3 nor F4  → ‡) (u in F)
F3 + no B3              → ‡) (u in B)
F4 + no C4              → ‡) (u in C)

when component 2 (.5.2o5o) remains unchanged (neither B3, C4, F3, F4):
E1 + neither A1 nor A2  → ‡) (u in A)
D2 + neither A1 nor A2  → ‡) (u in A)
A1 + no E1              → ‡) (u in E)
A2 + no D2              → ‡) (u in D)

As changes in (o5o2.5.) do not affect (.5.2o5o) and vice versa,
the above exclusions become valid generally.
```
Other layer-combinations:
```A1E1:         (-x)foo(-x)o 5 Fofxfo 2 oxofox 5 ooxofx &#zx
A1D2E1:       (-x)fof(-x)o 5 Fof(-x)fo 2 oxofox 5 ooxofx &#zx
A1B3E1F3:     (-x)foo(-x)o 5 Fofxfo 2 o(-x)ofo(-x) 5 ofxofF &#zx
A1B3D2E1F3:   (-x)fof(-x)o 5 Fof(-x)fo 2 o(-x)ofo(-x) 5 ofxofF &#zx  → †)
A2B3D2E1F3:   Ffof(-x)o 5 (-x)of(-x)fo 2 o(-x)ofo(-x) 5 ofxofF &#zx  → †)
A1B3C4D2E1F3: (-x)fof(-x)o 5 Fof(-x)fo 2 o(-x)ffo(-x) 5 of(-x)ofF &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```-:0: xfooxo 5 xofxoo 2 oxofox 5 ooxofx &#zx
→ regular (ex itself) with cell list:
600 tets

related: xfoo.o 5 xofx.o 2 oxof.x 5 ooxo.x &#zx
→ CRF with cell list:
25  mibdies (J62)
225 tets

related: .foo.o 5 .ofx.o 2 .xof.x 5 .oxo.x &#zx
→ CRF with cell list:
25 mibdies (J62)
10 paps
75 tets

related: .foox. 5 .ofxo. 2 .xofo. 5 .oxof. &#zx
→ uniform (gap) with cell list:
20  paps
300 tets
```
```1:A1E1: oFxxox 5 Fofxfo 2 oxofox 5 ooxofx &#zx
→ CRF with cell list:
10  gyepips (J11)
25  ikes
10  pips
150 squippies (J1)
75  tets
50  trips

related: .Fxxox 5 .ofxfo 2 .xofox 5 .oxofx &#zx
→ CRF with cell list:
10  paps
15  pips
25  ikes
75  tets
50  trips
125 squippies (J1)

related: oFxx.x 5 Fofx.o 2 oxof.x 5 ooxo.x &#zx
→ CRF with cell list:
10 gyepips (J11)
25 mibdies (J62)
35 pips
25 squippies (J1)
75 tets
50 trips

related: .Fxx.x 5 .ofx.o 2 .xof.x 5 .oxo.x &#zx
→ CRF with cell list:
25 mibdies (J62)
10 paps
40 pips
75 tets
50 trips
```
```12:A1D2E1: oFxFox 5 AxFoFx 2 oxofox 5 ooxofx &#zx
→ CRF with cell list:
50 bilbiroes (J91)
10 dips
10 gyepips (J11)
5  pips
75 squippies (J1)
```
```13:A1B3E1F3: oFxxox 5 Fofxfo 2 xoxFxo 5 ofxofF &#zx
→ ‡) (f in B)
```
using here node marks / (pseudo) edge lengths: F=f+x, A=F+x=f+2x

in o3o2o3o subsymmetry   (up)
Representation:
```fFoxffooxo 3 foFfxofxoo 2 oxofofxFof 3 ooxofxfoFf &#zx  (ex)

with layer symmetry:
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234),G(1234),H(1234),I(1234),J(1234) ↔
A(2143),C(2143),B(2143),E(2143),D(2143),G(2143),F(2143),I(2143),H(2143),J(2143)
and cycle:
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234),G(1234),H(1234),I(1234),J(1234) →
J(3421),H(3421),I(3421),G(3421),F(3421),D(3421),E(3421),C(3421),B(3421),A(3421) →
A(2143),C(2143),B(2143),E(2143),D(2143),G(2143),F(2143),I(2143),H(2143),J(2143) →
J(4312),I(4312),H(4312),F(4312),G(4312),E(4312),D(4312),B(4312),C(4312),A(4312) →
A(1234),B(1234),C(1234),D(1234),E(1234),F(1234),G(1234),H(1234),I(1234),J(1234)
```
All layers &
kaleido-facetings per layer:
```A:  f3f o3o
B:  F3o x3o   →   B3:   F 3  o (-x)3  x   →   B34:   F 3  o   o 3(-x)
C:  o3F o3x   →   C4:   o 3  F   x 3(-x)  →   C43:   o 3  F (-x)3  o
D:  x3f f3o   →   D1: (-x)3  F   f 3  o
E:  f3x o3f   →   E2:   F 3(-x)  o 3  f
F:  f3o f3x   →   F4:   f 3  o   F 3(-x)
G:  o3f x3f   →   G3:   o 3  f (-x)3  F
H:  o3x F3o   →   H2:   x 3(-x)  F 3  o   →   H21: (-x)3  o   F 3  o
I:  x3o o3F   →   I1: (-x)3  x   o 3  F   →   I12:   o 3(-x)  o 3  F
J:  o3o f3f
```
A priori invalid combinations:
```when component 1 (o3o2.3.) remains unchanged (neither D1, E2, H2, H21, I1, I12):
B3 + C4                   → ‡) (u in B, u in C)
B3 + F4                   → ‡) (u in B)
B3 + no G3                → ‡) (u in G)
B34 + neither C4 nor C43  → ‡) (u in C)
B34 + no F4               → ‡) (u in F)
C4 + no F4                → ‡) (u in F)
C4 + G3                   → ‡) (u in C)
C43 + neither B3 nor B34  → ‡) (u in B)
C43 + no G3               → ‡) (u in G)
F4 + neither C4 nor C43   → ‡) (u in C)
G3 + neither B3 nor B34   → ‡) (u in B)

when component 2 (.3.2o3o) remains unchanged (neither B3, B34, C4, C43, F4, G3):
D1 + H2                   → ‡) (u in H)
D1 + neither I1 nor I12   → ‡) (u in I)
E2 + neither H2 nor H21   → ‡) (u in H)
E2 + I1                   → ‡) (u in I)
H2 + no E2                → ‡) (u in E)
H2 + I1                   → ‡) (u in H, u in I)
H21 + no D1               → ‡) (u in D)
H21 + neither I1 nor I12  → ‡) (u in I)
I1 + no D1                → ‡) (u in D)
I12 + no E2               → ‡) (u in E)
I12 + neither H2 nor H21  → ‡) (u in H)

As changes in (o3o2.3.) do not affect (.3.2o3o) and vice versa,
the above exclusions become valid generally.
```
Other layer-combinations:
```D1I1:                 fFo(-x)ffoo(-x)o 3 foFFxofxxo 2 oxofofxFof 3 ooxofxfoFf &#zx
D1H21I1:              fFo(-x)ffo(-x)(-x)o 3 foFFxofoxo 2 oxofofxFof 3 ooxofxfoFf &#zx
D1E2H21I12:           fFo(-x)Ffo(-x)oo 3 foFF(-x)ofo(-x)o 2 oxofofxFof 3 ooxofxfoFf &#zx
B3D1G3I1:             fFo(-x)ffoo(-x)o 3 foFFxofxxo 2 o(-x)ofof(-x)Fof 3 oxxofxFoFf &#zx
B3C43D1G3I1:          fFo(-x)ffoo(-x)o 3 foFFxofxxo 2 o(-x)(-x)fof(-x)Fof 3 oxoofxFoFf &#zx
B3D1G3H21I1:          fFo(-x)ffo(-x)(-x)o 3 foFFxofoxo 2 o(-x)ofof(-x)Fof 3 oxxofxFoFf &#zx
B3C43D1G3H21I1:       fFo(-x)ffo(-x)(-x)o 3 foFFxofoxo 2 o(-x)(-x)fof(-x)Fof 3 oxoofxFoFf &#zx  → †)
B3D1E2G3H21I12:       fFo(-x)Ffo(-x)oo 3 foFF(-x)ofo(-x)o 2 o(-x)ofof(-x)Fof 3 oxxofxFoFf &#zx
B3C43D1E2G3H21I12:    fFo(-x)Ffo(-x)oo 3 foFF(-x)ofo(-x)o 2 o(-x)(-x)fof(-x)Fof 3 oxoofxFoFf &#zx  → †)
B34C43D1E2F4G3H21I12: fFo(-x)Ffo(-x)oo 3 foFF(-x)ofo(-x)o 2 oo(-x)foF(-x)Fof 3 o(-x)oof(-x)FoFf &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```1:D1I1: FAxoFFxxox 3 foFFxofxxo 2 oxofofxFof 3 ooxofxfoFf &#zx
→ CRF with cell list:
27  ikes
6   octs
72  squippies (J1)
138 tets
6   tricues (J3)
30  trips
```
```1:D1H21I1: FAxoFFxoox 3 foFFxofoxo 2 oxofofxFof 3 ooxofxfoFf &#zx
→ CRF with cell list:
9  bilbiroes (J91)
18 gyepips (J11)
9  ikes
54 squippies (J1)
87 tets
12 trips
```
```12:D1E2H21I12: FAxoAFxoxx 3 FxAAoxFxox 2 oxofofxFof 3 ooxofxfoFf &#zx
→ CRF with cell list:
18 bilbiroes (J91)
18 pips
54 squippies (J1)
36 teddies (J63)
12 tets
12 tricues (J3)
24 trips
```
```13:B3D1G3I1: FAxoFFxxox 3 foFFxofxxo 2 xoxFxFoAxF 3 oxxofxFoFf &#zx
→ CRF with cell list:
27 bilbiroes (J91)
3  hips
12 octs
9  mibdies (J62)
54 squippies (J1)
18 teddies (J63)
36 tets
12 tricues (J3)
30 trips
```
```13:B3C43D1G3I1: FAxoFFxxox 3 foFFxofxxo 2 xooFxFoAxF 3 oxoofxFoFf &#zx
→ †) (dead end at D)
```
```13:B3D1G3H21I1: FAxoFFxoox 3 foFFxofoxo 2 xoxFxFoAxF 3 oxxofxFoFf &#xz
→ †) (dead end at D)
```
```123:B3D1E2G3H21I12: FAxoAFxoxx 3 FxAAoxFxox 2 xoxFxFoAxF 3 oxxofxFoFf &#zx
→ †) (dead end at D)
```

```
```
using here node marks / (pseudo) edge lengths: F=f+x, A=F+x=f+2x

in . o3o3o subsymmetry   (up)
Representation:
```xoo|fox|Ffo|ofx|ofo 3 oof|oxf|ooo|fxo|foo 3 ofo|xfo|ofF|xof|oox &#xt  (ex)
```
All layers &
kaleido-facetings per layer:
```A:  x3o3o  →  A1: (-x)3  x 3  o   →  A12:   o 3(-x)3 x  →  A123:  o 3 o 3(-x)
B:  o3o3f
C:  o3f3o
D:  f3o3x  →  D3:   f 3  x 3(-x)  →  D32:   F 3(-x)3 o
E:  o3x3f  →  E2:   x 3(-x)3  F   →  E21: (-x)3  o 3 F
F:  x3f3o  →  F1: (-x)3  F 3  o
G:  F3o3o
H:  f3o3f
g:  o3o3F
f:  o3f3x  →  f3:   o 3  F 3(-x)
etc. antisymmetrically
```
A priori invalid combinations:
```A1 + D32                      → ‡) (u in A)
A1 + E2                       → ‡) (u in A, u in E)
A1 + no F1                    → ‡) (u in F)
A1 + e2                       → ‡) (u in A)
A1 + d12                      → ‡) (u in A)
A1 + neither d1 nor d12       → ‡) (u in d)
A1 + a32                      → ‡) (u in A, u in a)
A12 + D3                      → ‡) (u in A, u in D)
A12 + d1                      → ‡) (u in A, u in d)
A12 + neither E2 nor E21      → ‡) (u in E)
A12 + f3                      → ‡) (u in A)
A12 + e23                     → ‡) (u in A)
A12 + neither e2 nor e23      → ‡) (u in e)
A12 + a3                      → ‡) (u in A, u in a)
A123 + neither D3 nor D32     → ‡) (u in D)
A123 + no f3                  → ‡) (u in f)
A123 + e2                     → ‡) (u in e)
A123 + neither a3, a32, a321  → ‡) (u in a)
D3 + E2                       → ‡) (u in D)
D3 + no f3                    → ‡) (u in f)
D3 + e2                       → ‡) (u in D, u in e)
D3 + d12                      → ‡) (u in D)
D3 + a32                      → ‡) (u in D, u in a)
D3 + neither a3, a32, a321    → ‡) (u in a)
D32 + neither E2 nor E21      → ‡) (u in E)
D32 + neither e2 nor e23      → ‡) (u in e)
D32 + d1                      → ‡) (u in d)
D32 + a3                      → ‡) (u in a)
E2 + F1                       → ‡) (u in E)
E2 + neither e2 nor e23       → ‡) (u in e)
E2 + d1                       → ‡) (u in E, u in d)
```
```E2 + a3                       → ‡) (u in E)
E2 + a321                     → ‡) (u in E)
E21 + neither A1, A12, A123   → ‡) (u in A)
E21 + no F1                   → ‡) (u in F)
E21 + neither d1 nor d12      → ‡) (u in d)
E21 + a32                     → ‡) (u in a)
F1 + neither A1, A12, A123    → ‡) (u in A)
F1 + neither d1 nor d12       → ‡) (u in d)
F1 + a32                      → ‡) (u in a)
f3 + neither D3 nor D32       → ‡) (u in D)
f3 + e2                       → ‡) (u in e)
f3 + neither a3, a32, a321    → ‡) (u in a)
e2 + neither E2 nor E21       → ‡) (u in E)
e2 + d1                       → ‡) (u in d)
e2 + a3                       → ‡) (u in e, u in a)
e23 + neither D3 nor D32      → ‡) (u in D)
e23 + no f3                   → ‡) (u in f)
e23 + neither a3, a32, a321   → ‡) (u in a)
d1 + neither A1, A12, A123    → ‡) (u in A)
d1 + no F1                    → ‡) (u in F)
d1 + a32                      → ‡) (u in d, u in a)
d12 + neither E2 nor E21      → ‡) (u in E)
d12 + neither e2 nor e23      → ‡) (u in e)
d12 + a3                      → ‡) (u in a)
a3 + neither D3 nor D32       → ‡) (u in D)
a3 + no f3                    → ‡) (u in f)
a32 + neither E2 nor E21      → ‡) (u in E)
a32 + neither e2 nor e23      → ‡) (u in e)
a321 + neither A1, A12, A123  → ‡) (u in A)
a321 + no F1                  → ‡) (u in F)
a321 + neither d1 nor d12     → ‡) (u in d)
```
Other layer-combinations:
```A1F1d1:                   (-x)oo|fo(-x)|Ffo|of(-x)|ofo 3 xof|oxF|ooo|fxx|foo 3 ofo|xfo|ofF|xof|oox &#xt
A1E21F1d1:                (-x)oo|f(-x)(-x)|Ffo|of(-x)|ofo 3 xof|ooF|ooo|fxx|foo 3 ofo|xFo|ofF|xof|oox &#xt
A1F1d1a321:               (-x)oo|fo(-x)|Ffo|of(-x)|of(-x) 3 xof|oxF|ooo|fxx|foo 3 ofo|xfo|ofF|xof|ooo &#xt  → †)
A1E21F1d1a321:            (-x)oo|f(-x)(-x)|Ffo|of(-x)|of(-x) 3 xof|ooF|ooo|fxx|foo 3 ofo|xFo|ofF|xof|ooo &#xt  → †)
E2e2:                     xoo|fxx|Ffo|oFx|ofo 3 oof|o(-x)f|ooo|f(-x)o|foo 3 ofo|xFo|ofF|xxf|oox &#xt
A12E2e2:                  ooo|fxx|Ffo|oFx|ofo 3 (-x)of|o(-x)f|ooo|f(-x)o|foo 3 xfo|xFo|ofF|xxf|oox &#xt
D32E2e2:                  xoo|Fxx|Ffo|oFx|ofo 3 oof|(-x)(-x)f|ooo|f(-x)o|foo 3 ofo|oFo|ofF|xxf|oox &#xt
A12D32E2e2:               ooo|Fxx|Ffo|oFx|ofo 3 (-x)of|(-x)(-x)f|ooo|f(-x)o|foo 3 xfo|oFo|ofF|xxf|oox &#xt  → †) (dead end at D)
A12E2e2d12:               ooo|fxx|Ffo|oFo|ofo 3 (-x)of|o(-x)f|ooo|f(-x)(-x)|foo 3 xfo|xFo|ofF|xxF|oox &#xt
A12E2e2a32:               ooo|fxx|Ffo|oFx|ofx 3 (-x)of|o(-x)f|ooo|f(-x)o|fo(-x) 3 xfo|xFo|ofF|xxf|ooo &#xt
D32E2e2d12:               xoo|Fxx|Ffo|oFo|ofo 3 oof|(-x)(-x)f|ooo|f(-x)(-x)|foo 3 ofo|oFo|ofF|xxF|oox &#xt
A12D32E2e2d12:            ooo|Fxx|Ffo|oFo|ofo 3 (-x)of|(-x)(-x)f|ooo|f(-x)(-x)|foo 3 xfo|oFo|ofF|xxF|oox &#xt  → †) (dead end at D)
A12D32E2e2a32:            ooo|Fxx|Ffo|oFx|ofx 3 (-x)of|(-x)(-x)f|ooo|f(-x)o|fo(-x) 3 xfo|oFo|ofF|xxf|ooo &#xt  → †) (dead end at D)
A12D32E2e2d12a32:         ooo|Fxx|Ffo|oFo|ofx 3 (-x)of|(-x)(-x)f|ooo|f(-x)(-x)|fo(-x) 3 xfo|oFo|ofF|xxF|ooo &#xt  → †) (dead end at D)
A12E21F1e2d12:            ooo|f(-x)(-x)|Ffo|oFo|ofo 3 (-x)of|ooF|ooo|f(-x)(-x)|foo 3 xfo|xFo|ofF|xxF|oox &#xt
A12D32E21F1e2d12:         ooo|F(-x)(-x)|Ffo|oFo|ofo 3 (-x)of|(-x)oF|ooo|f(-x)(-x)|foo 3 xfo|oFo|ofF|xxF|oox &#xt  → †) (dead end at D)
A12E21F1e2d12a321:        ooo|f(-x)(-x)|Ffo|oFo|of(-x) 3 (-x)of|ooF|ooo|f(-x)(-x)|foo 3 xfo|xFo|ofF|xxF|ooo &#xt  → †)
A12D32E21F1e2d12a321:     ooo|F(-x)(-x)|Ffo|oFo|of(-x) 3 (-x)of|(-x)oF|ooo|f(-x)(-x)|foo 3 xfo|oFo|ofF|xxF|ooo &#xt  → †)
A1D3F1f3d1a3:             (-x)oo|fo(-x)|Ffo|of(-x)|ofo 3 xof|xxF|ooo|Fxx|fox 3 ofo|(-x)fo|ofF|(-x)of|oo(-x) &#xt
A1D3F1f3d1a321:           (-x)oo|fo(-x)|Ffo|of(-x)|of(-x) 3 xof|xxF|ooo|Fxx|foo 3 ofo|(-x)fo|ofF|(-x)of|ooo &#xt  → †)
A123D3F1f3d1a321:         ooo|fo(-x)|Ffo|of(-x)|of(-x) 3 oof|xxF|ooo|Fxx|foo 3 (-x)fo|(-x)fo|ofF|(-x)of|ooo &#xt  → †)
A1D3E21F1f3d1a3:          (-x)oo|f(-x)(-x)|Ffo|of(-x)|ofo 3 xof|xoF|ooo|Fxx|fox 3 ofo|(-x)Fo|ofF|(-x)of|oo(-x) &#xt
A1D3E21F1f3d1a321:        (-x)oo|f(-x)(-x)|Ffo|of(-x)|of(-x) 3 xof|xoF|ooo|Fxx|foo 3 ofo|(-x)Fo|ofF|(-x)of|ooo &#xt  → †)
A123D3E21F1f3d1a321:      ooo|f(-x)(-x)|Ffo|of(-x)|of(-x) 3 oof|xoF|ooo|Fxx|foo 3 (-x)fo|(-x)Fo|ofF|(-x)of|ooo &#xt  → †)
A1D3F1f3e23d1a321:        (-x)oo|fo(-x)|Ffo|oF(-x)|of(-x) 3 xof|xxF|ooo|Fox|foo 3 ofo|(-x)fo|ofF|(-x)(-x)f|ooo &#xt  → †)
A1D3E21F1f3e23d1a3:       (-x)oo|f(-x)(-x)|Ffo|oF(-x)|ofo 3 xof|xoF|ooo|Fox|fox 3 ofo|(-x)Fo|ofF|(-x)(-x)f|oo(-x) &#xt
A1D3E21F1f3e23d1a321:     (-x)oo|f(-x)(-x)|Ffo|oF(-x)|of(-x) 3 xof|xoF|ooo|Fox|foo 3 ofo|(-x)Fo|ofF|(-x)(-x)f|ooo &#xt  → †)
A123D3E21F1f3e23d1a321:   ooo|f(-x)(-x)|Ffo|oF(-x)|of(-x) 3 oof|xoF|ooo|Fox|foo 3 (-x)fo|(-x)Fo|ofF|(-x)(-x)f|ooo &#xt  → †)
A123D32E21F1f3e23d12a321: ooo|F(-x)(-x)|Ffo|oFo|of(-x) 3 oof|(-x)oF|ooo|Fo(-x)|foo 3 (-x)fo|oFo|ofF|(-x)(-x)F|ooo &#xt  → †)
```
Stott expansion:
(derived potential CRFs)
```1:A1F1d1: oxxFxoAFxxFoxFx 3 xofoxFooofxxfoo 3 ofoxfoofFxofoox &#xt
...
```
```1:A1E21F1d1: oxxFooAFxxFoxFx 3 xofooFooofxxfoo 3 ofoxFoofFxofoox &#xt
...
```
```2:E2e2: xoofxxFfooFxofo 3 xxFxoFxxxFoxFxx 3 ofoxFoofFxxfoox &#xt
→ CRF with cell list:
24  ikes
16  octs
72  squippies (J1)
198 tets
24  tricues (J3)
48  trips
2   tuts

related: xo.fxxFfooFx.fo 3 xx.xoFxxxFox.xx 3 of.xFoofFxxf.ox &#xt
→ CRF with cell list:
24 gyepips (J11)
12 mibdies (J62)
16 octs
72 squippies (J1)
78 tets
24 tricues (J3)
48 trips
2  tuts
```
```2:A12E2|e2: ooofxx|Ffo|oFxofo 3 oxFxoF|xxx|FoxFxx 3 xfoxFo|ofF|xxfoox &#xt
→ CRF with cell list:
24  ikes
16  octs
72  squippies (J1)
155 tets
4   thawroes (J92)
16  tricues (J3)
36  trips
1   tut
(asymmetric hemiglomal combination of others)
```
```2:D32E2|e2: xooFxx|Ffo|oFxofo 3 xxFooF|xxx|FoxFxx 3 ofooFo|ofF|xxfoox &#xt
→ CRF with cell list:
12  bilbiroes (J91)
12  ikes
12  mibdies (J62)
16  octs
48  squippies (J1)
122 tets
16  tricues (J3)
24  trips
2   tuts
(asymmetric hemiglomal combination of others)
```
```2:A12E2|e2d12: ooofxx|Ffo|oFoofo 3 oxFxoF|xxx|FooFxx 3 xfoxFo|ofF|xxFoox &#xt
→ CRF with cell list:
12 bilbiroes (J91)
12 ikes
12 mibdies (J62)
16 octs
48 squippies (J1)
79 tets
4  thawroes (J92)
8  tricues (J3)
12 trips
1  tut
(asymmetric hemiglomal combination of others)
```
```2:A12E2e2a32: ooofxxFfooFxofx 3 oxFxoFxxxFoxFxo 3 xfoxFoofFxxfooo &#xt
→ CRF with cell list:
24  ikes
16  octs
72  squippies (J1)
112 tets
8   thawroes (J92)
8   tricues (J3)
24  trips
```
```2:D32E2e2d12: xooFxxFfooFoofo 3 xxFooFxxxFooFxx 3 ofooFoofFxxFoox &#xt
→ CRF with cell list:
24 bilbiroes (J91)
24 mibdies (J62)
16 octs
24 squippies (J1)
46 tets
8  tricues (J3)
2  tuts
```
```12:A12E21F1e2d12: xxxFooAFxxAxxFx 3 oxFxxAxxxFooFxx 3 xfoxFoofFxxFoox &#xt
→ CRF with cell list:
24 bilbiroes (J91)
1  co
8  octs
30 pips
18 squippies (J1)
24 teddies (J63)
12 tets
4  thawroes (J92)
1  toe
28 tricues (J3)
24 trips
```
```13:A1D3F1f3d1a3: oxxFxoAFxxFoxFx 3 xofxxFoooFxxfox 3 xFxoFxxFAoxFxxo &#xt
→ CRF with cell list:
24 bilbiroes (J91)
16 octs
48 squippies (J1)
24 teddies (J63)
30 tets
8  thawroes (J92)
16 tricues (J3)
48 trips
2  tuts
```
```13:A1D3E21F1f3d1a3: oxxFooAFxxFoxFx 3 xofxoFoooFxxfox 3 xFxoAxxFAoxFxxo &#xt
...
```
```13:A1D3E21F1f3e23d1a3: oxxFooAFxxAoxFx 3 xofxoFoooFoxfox 3 xFxoAxxFAooFxxo &#xt
...
```
using here node marks / (pseudo) edge lengths: F=f+x, A=F+x=f+2x

in o2o2o2o subsymmetry   (up)
Representation:
```ooo|xxx|fff|FFF|Vooo|f 2 Fxf|oFf|xFo|fxo|oVoo|f 2 xfF|Ffo|Fox|xof|ooVo|f 2 fFx|foF|oxF|ofx|oooV|f &#zx  (ex)
```
All layers &
kaleido-facetings per layer:
```A:  o2F2x2f  →  A3:   o 2  F 2(-x)2  f
B:  o2x2f2F  →  B2:   o 2(-x)2  f 2  F
C:  o2f2F2x  →  C4:   o 2  f 2  F 2(-x)

D:  x2o2F2f  →  D1: (-x)2  o 2  F 2  f
E:  x2F2f2o  →  E1: (-x)2  F 2  f 2  o
F:  x2f2o2F  →  F1: (-x)2  f 2  o 2  F

G:  f2x2F2o  →  G2:   f 2(-x)2  F 2  o
H:  f2F2o2x  →  H4:   f 2  F 2  o 2(-x)
I:  f2o2x2F  →  I3:   f 2  o 2(-x)2  F
```
```J:  F2f2x2o  →  J3:   F 2  f 2(-x)2  o
K:  F2x2o2f  →  K2:   F 2(-x)2  o 2  f
L:  F2o2f2x  →  L4:   F 2  o 2  f 2(-x)

M:  V2o2o2o
N:  o2V2o2o
O:  o2o2V2o
P:  o2o2o2V

Q:  f2f2f2f
```
A priori invalid combinations:
```A  + I3,J3
A3 + I,J
B  + G2,K2
B2 + G,K
C  + H4,L4
C4 + H,L
D  + E1,F1
D1 + E,F
```
```E  + F1
E1 + F
G  + K2
G2 + K
H  + L4
H4 + L
I  + J3
I3 + J	(all giving rise to u edges)
```
Other layer-combinations:
```ABCD1E1F1GHIJKL:          ooo(-x)(-x)(-x)fffFFFVooof 2 FxfoFfxFofxooVoof 2 xfFFfoFoxxofooVof 2 fFxfoFoxFofxoooVf &#zx
ABC4D1E1F1GH4IJKL4:       ooo(-x)(-x)(-x)fffFFFVooof 2 FxfoFfxFofxooVoof 2 xfFFfoFoxxofooVof 2 fF(-x)foFo(-x)Fof(-x)oooVf &#zx
AB2C4D1E1F1G2H4IJK2L4:    ooo(-x)(-x)(-x)fffFFFVooof 2 F(-x)foFf(-x)Fof(-x)ooVoof 2 xfFFfoFoxxofooVof 2 fF(-x)foFo(-x)Fof(-x)oooVf &#zx
A3B2C4D1E1F1G2H4I3J3K2L4: ooo(-x)(-x)(-x)fffFFFVooof 2 F(-x)foFf(-x)Fof(-x)ooVoof 2 (-x)fFFfoFo(-x)(-x)ofooVof 2 fF(-x)foFo(-x)Fof(-x)oooVf &#zx  → †)
```
Stott expansion:
(derived potential CRFs)
```1:ABCD1E1F1GHIJKL: xxxoooFFFAAABxxxF 2 FxfoFfxFofxooVoof 2 xfFFfoFoxxofooVof 2 fFxfoFoxFofxoooVf &#zx (telex)
→ CRF with cell list:
24  ikes
60  squippies (J1)
180 tets
20  trips
(as this is just an axial change, the orthogonal symmetry remains;
thus it also can be described as: BAFox 2 oxofo 3 oooox 5 ooxoo &#zx)

related: xxxoooFFF.......F 2 FxfoFfxFo.......f 2 xfFFfoFox.......f 2 fFxfoFoxF.......f &#zx
→ CRF with cell list:
6  bilbiroes (J91)
2  does
24 mibdies (J62)
36 squippies (J1)
16 tets
8  trips
```
```14:ABC4D1E1F1GH4IJKL4: xxxoooFFFAAABxxxF 2 FxfoFfxFofxooVoof 2 xfFFfoFoxxofooVof 2 FAoFxAxoAxFoxxxBF &#zx
→ CRF with cell list:
16 bilbiroes (J91)
16 gyepips (J11)
64 squippies (J1)
16 teddies (J63)
24 tets
24 trips

related: xxxoooFFFAAA.xx.F FxfoFfxFofxo.Vo.f xfFFfoFoxxof.oV.f FAoFxAxoAxFo.xx.F&#zx
→ CRF with cell list:
20 bilbiroes (J91)
16 paps
48 squippies (J1)
16 teddies (J63)
8  tets
16 trips
```
```124:AB2C4D1E1F1G2H4IJK2L4: xxxoooFFFAAABxxxF 2 AoFxAFoAxFoxxBxxF 2 xfFFfoFoxxofooVof 2 FAoFxAxoAxFoxxxBF &#zx  → °)
```
using here node marks / (pseudo) edge lengths: F=f+x=ff, V=2f, A=F+x=f+2x, B=V+x=2f+x=fff

#### EKF of the icositetrachoron (x3o4o3o)

in o3o3o4o subsymmetry   (up)
Representation:
```qo 3 oo 3 oo 4 ox &#zx  (ico)
```
All layers &
kaleido-facetings per layer:
```A: q3o3o4o
B: o3o3o4x  →  B4: o3o3q4(-x)
```
A priori invalid combinations:
```none
```
Other layer-combinations:
```B4:  qo 3 oo 3 oq 4 o(-x) &#zx
```
Stott expansion:
(derived potential CRFs)
```1:-: wx 3 oo 3 oo 4 ox &#zx (poxic)
→ CRF with cell list:
24 esquidpies (J15)
16 tets
32 trips
```
```2:-: qo 3 xx 3 oo 4 ox &#zx
= Wythoffian x3o4o3x (spic) with cell list:
48  octs
192 trips
```
```3:-: qo 3 oo 3 xx 4 ox &#zx (pocsric)
→ CRF with cell list:
8  coes
24 squobcues (J28)
16 tets
64 trips
```
```4:B4: qo 3 oo 3 oq 4 xo &#zx
= Wythoffian o3x4o3o (rico) with cell list:
24 coes
24 cubes
```
```12:-: wx 3 xx 3 oo 4 ox &#zx (owauprit)
→ CRF with cell list:
24  esquidpies (J15)
32  hips
8   octs
160 trips
16  tuts
```
```13:-: wx 3 oo 3 xx 4 ox &#zx
= Wythoffian x3o4x3o (srico) with cell list:
24 coes
24 sircoes
96 trips
```
```14:B4: wx 3 oo 3 oq 4 xo &#zx
→ °) (asks for non-regular hexagons: wx .. oq .. &#zx)
```
```23:-: qo 3 xx 3 xx 4 ox &#zx (pocprico)
→ CRF with cell list:
64  hips
24  squobcues (J28)
8   toes
128 trips
16  tuts
```
```24:B4: qo 3 xx 3 oq 4 xo &#zx
= Wythoffian x3o4x3o (srico) with cell list:
24 coes
24 sircoes
96 trips
```
```34:B4: qo 3 oo 3 xw 4 xo &#zx
→ °) (asks for non-regular hexagons: qo .. xw .. &#zx)
```
```123:-: wx 3 xx 3 xx 4 ox &#zx
= Wythoffian x3x4o3x (prico) with cell list:
96 hips
24 sircoes
24 toes
96 trips
```
```124:B4: wx 3 xx 3 oq 4 xo &#zx
→ °) (asks for non-regular hexagons: wx .. oq .. &#zx)
```
```134:B4: wx 3 oo 3 xw 4 xo &#zx
= Wythoffian o3x4x3o (cont) with cell list:
48 tics
```
```234:B4: qo 3 xx 3 xw 4 xo &#zx
→ °) (asks for non-regular hexagons: qo .. xw .. &#zx)
```
```1234:B4: wx 3 xx 3 xw 4 xo &#zx
= Wythoffian x3x4x3o (grico) with cell list:
24 gircoes
24 tics
96 trips
```

(As it will turn out, the set of resulting polychora in o2o3o4o resp. . o3o4o here is identical to those which turn up from the corresponding consideration of hex wrt. these subsymmetries.)

in o2o3o4o subsymmetry   (up)
Representation:
```qo 2 xo 3 ox 4 oo &#zx  (ico)
```
All layers &
kaleido-facetings per layer:
```A: q2x3o4o   →   A2:  q 2(-x)3  x 4 o   →   A23:  q 2  o 3(-x)4 q
B: o2o3x4o   →   B3:  o 2  x 3(-x)4 q   →   B32:  o 2(-x)3  o 4 q
```
A priori invalid combinations:
```A23                       → ‡) (q or w in extremal layers, i.e. A)

B32 + neither A2 nor A23  → ‡) (u in A)
A2 + B3                   → ‡) (u in A, u in B)
```
Other layer-combinations:
```A2:     qo 2 (-x)o 3 xx 4 oo &#zx
A2B32:  qo 2 (-x)(-x) 3 xo 4 oq &#zx
B3:     qo 2 xx 3 o(-x) 4 oq &#zx
```
Stott expansion:
(derived potential CRFs)
```1:-: wx 2 xo 3 ox 4 oo &#zx (pexic)
→ CRF with cell list:
6  esquidpies (J15)
18 octs
8  trips
```
```2:A2: qo 2 ox 3 xx 4 oo &#zx (coatobcu)
→ CRF with cell list:
2  coes
12 cubes
16 tricues (J3)
```
```2:A2B32: qo 2 oo 3 xo 4 oq &#zx
= Wythoffian o3o3x4o (rit) with cell list:
8  coes
16 tets
```
```3:B3: qo 2 xx 3 xo 4 oq &#zx (pabdirico)
→ CRF with cell list:
6  coes
12 cubes
2  toes
16 tricues (J3)
```
```4:-: qo 2 xo 3 ox 4 xx &#zx (pacsrit)
→ CRF with cell list:
16 octs
2  sircoes
6  squobcues (J28)
24 trips
```
```12:A2: wx 2 ox 3 xx 4 oo &#zx
→ CRF with cell list:
2  coes
18 cubes
8  hips
16 tricues (J3)
```
```12:A2B32: wx 2 oo 3 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: wx .. .. oq &#zx)
```
```13:B3: wx 2 xx 3 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: wx .. .. oq &#zx)
```
```14:-: wx 2 xo 3 ox 4 xx &:#zx
= Wythoffian o3x3o4x (srit) with cell list:
16 octs
8  sircoes
32 trips
```
```2(-3):A2: qo 2 ox 3 oo 4 oo &#zx
= Wythoffian x3o3o4o (hex) with cell list:
16 tets
```
```24:A2: qo 2 ox 3 xx 4 xx &#zx (tica gircobcu)
→ CRF with cell list:
12 ops
2  tics
16 tricues (J3)
24 trips
```
```24:A2B32: qo 2 oo 3 xo 4 xw &#zx
→ °) (asks for non-regular hexagons: qo .. .. xw &#zx)
```
```34:B3: qo 2 xx 3 xo 4 xw &#zx
→ °) (asks for non-regular hexagons: qo .. .. xw &#zx)
```
```12(-3):A2: wx 2 ox 3 oo 4 oo &#zx (pex hex)
→ CRF with cell list:
16 tets
8  trips
```
```124:A2: wx 2 ox 3 xx 4 xx &#zx
→ CRF with cell list:
12 cubes
8  hips
18 ops
2  tics
16 tricues (J3)
24 trips
```
```124: A2B32: wx 2 oo 3 xo 4 xw &#zx
= Wythoffian o3o3x4x (tat) with cell list:
16 tets
8  tics
```
```134:B3: wx 2 xx 3 xo 4 xw &#zx (pabdiproh)
→ CRF with cell list:
2  gircoes
12 ops
6  tics
16 tricues (J3)
8  trips
```
```2(-3)4:A2: qo 2 ox 3 oo 4 xx &#zx (pacsid pith)
→ CRF with cell list:
14 cubes
16 tets
24 trips
```
```12(-3)4:A2: wx 2 ox 3 oo 4 xx &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```

in . o3o4o subsymmetry   (up)
not prismatically symmetric
combinations of formers:
```none

(As A23 already was ruled out a priori this would ask for a local A and A2 at the top resp. bottom layer.
But that combination would suffer again from producing an u=2x sized edge in A.)
```

in o3o3o *b3o subsymmetry   (up)
Representation:
```qoo 3 ooo 3 oqo *b3 ooq &#zx  (ico)
```
All layers &
kaleido-facetings per layer:
```A: q3o3o *b3o
B: o3o3q *b3o
C: o3o3o *b3q

(As this representation does not show up (true / x-) edges within the layers, those thus cannot be inverted either.
Therefore at most pure partial Stott expansions wrt. this subsymmetry remain possible.)
```
Stott expansion:
(derived potential CRFs)
```1:-: wxx 3 ooo 3 oqo *b3 ooq &#zx (poxic)
→ CRF with cell list:
24 esquidpies (J15)
16 tets
32 trips

related: .xx 3 .oo 3 .qo *b3 .oq &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```
```2:-: qoo 3 xxx 3 oqo *b3 ooq &#zx
= Wythoffian x3o4o3x (spic) with cell list:
48  octs
192 trips

related: .oo 3 .xx 3 .qo *b3 .oq &#zx
= Wythoffian o3x3o4x (srit) with cell list:
16 octs
8  sircoes
32 trips
```
```12:-: wxx 3 xxx 3 oqo *b3 ooq &#zx (owau prit)
→ CRF with cell list:
24  esquidpies (J15)
32  hips
8   octs
160 trips
16  tuts

related: .xx 3 .xx 3 .qo *b3 .oq &#zx
= Wythoffian x3x3o4x (prit) with cell list:
24 cubes
32 hips
8  sircoes
16 tuts
```
```13:-: wxx 3 ooo 3 xwx *b3 ooq &#zx (pocsric)
→ CRF with cell list:
8  coes
24 squobcues (J28)
16 tets
64 trips

related: wx. 3 oo. 3 xw. *b3 oo. &#zx
= Wythoffian x3o3x4x (tat) with cell list:
16 tets
8  tics
```
```123:-: wxx 3 xxx 3 xwx *b3 ooq &#zx (poc prico)
→ CRF with cell list:
64  hips
24  squobcues (J28)
8   toes
128 trips
16  tuts

related: wx. 3 xx. 3 xw. *b3 oo. &#zx
= Wythoffian o3x3x4x (grit) with cell list:
8  gircoes
32 trips
16 tuts
```
```134:-: wxx 3 ooo 3 xwx *b3 xxw &#zx
= Wythoffian x3o4x3o (srico) with cell list:
24 coes
24 sircoes
96 trips

related: wx. 3 oo. 3 xw. *b3 xx. &#zx
= Wythoffian x3o3x4x (proh) with cell list:
16 coes
24 ops
8  tics
32 trips
```
```1234:-: wxx 3 xxx 3 xwx *b3 xxw &#zx
= Wythoffian x3x4o3x (prico) with cell list:
96 hips
24 sircoes
24 toes
96 trips

related: .xx 3 .xx 3 .wx *b3 .xw &#zx
= Wythoffian x3x3x4x (gidpith) with cell list:
8  gircoes
32 hips
24 ops
16 toes
```

in o4o2o4o subsymmetry   (up)
Representation:
```oxo 4 ooq 2 oxo 4 qoo &#zx  (ico)
```
All layers &
kaleido-facetings per layer:
```A: o4o o4q
B: x4o x4o   →  B1: (-x)4q x4o   →  B13: (-x)4q (-x)4q
↳  B3: x4o (-x)4q   →  (B31 = B13)
C: o4q o4o
```
Stott expansion:
(derived potential CRFs)
```1:B1: xox 4 oqq 2 oxo 4 qoo &#zx
→ ‡) (q in C)
```
```2:-: oxo 4 xxw 2 oxo 4 qoo &#zx (bicyte ausodip)
→ CRF with cell list:
4  esquidpies (J15)
16 octs
4  squobcues (J28)
16 trips

related: .xo 4 .xw 2 .xo 4 .oo &#zx (cyte cubau sodip)
→ CRF with cell list:
4  esquidpies (J15)
4  ops
16 squippies (J1)
```
```12:B1: xox 4 xww 2 oxo 4 qoo &#zx
→ ‡) (w in C)
```
```13:B13: xox 4 oqq 2 xox 4 qqo &#zx
→ ‡) (q in A, q in C)
```
```24:-: oxo 4 xxw 2 oxo 4 wxx &#zx
= Wythoffian o3x3o4x (srit) with cell list:
16 octs
8  sircoes
32 trips

related: .xo 4 .xw 2 .xo 4 .xx &#zx (cyted srit)
→ CRF with cell list:
8  ops
4  sircoes
16 squippies (J1)
16 trips
```
```123:B13: xox 4 xww 2 xox 4 qqo &#zx
→ ‡) (q in A, w in C)
```
```124:B1: xox 4 xww 2 oxo 4 wxx &#zx
→ ‡) (w in C)
```
```1234:B13: xox 4 xww 2 xox 4 wwx &#zx
→ ‡) (w in A, w in C)
```

#### EKF of the hexadecachoron (x3o3o4o)

(As it will turn out, the set of resulting polychora in o2o3o4o resp. . o3o4o here is identical to those which turn up from the corresponding consideration of ico wrt. these subsymmetries.)

in o2o3o4o subsymmetry   (up)
Representation:
```qo 2 ox 3 oo 4 oo &#zx  (hex)
```
All layers &
kaleido-facetings per layer:
```A: q2o3o4o
B: o2x3o4o   →  B2:  o 2(-x)3  x 4 o   →  B23:  o 2  o 3(-x)4 q
```
A priori invalid combinations:
```none
```
Stott expansion:
(derived potential CRFs)
```1:-: wx 2 ox 3 oo 4 oo &#zx (pex hex)
= oxxo 3 oooo 4 oooo &#xt
→ CRF with cell list:
16 tets
8  trips
```
```2:B2: qo 2 xo 3 ox 4 oo &#zx
= xox 3 oxo 4 ooo &#xt
= Wythoffian x3o4o3o (ico) with cell list:
24 octs
```
```3:-: qo 2 ox 3 xx 4 oo &#zx (coatobcu)
= oxo 3 xxx 4 ooo &#xt
→ CRF with cell list:
2  coes
12 cubes
16 tricues (J3)
```
```3:B23: qo 2 oo 3 xo 4 oq &#zx
= ooo 3 xox 4 oqo &#xt
= Wythoffian o3o3x4o (rit) with cell list:
8  coes
16 tets
```
```4:-: qo 2 ox 3 oo 4 xx &#zx (pacsid pith)
= oxo 3 ooo 4 xxx &#xt
→ CRF with cell list:
14 cubes
16 tets
24 trips
```
```12:B2: wx 2 xo 3 ox 4 oo &#zx (pexic)
= xoox 3 oxxo 4 oooo &#xt
→ CRF with cell list:
6  esquidpies (J15)
18 octs
8  trips
```
```13:-: wx 2 ox 3 xx 4 oo &#zx
= oxxo 3 xxxx 4 oooo &#xt
→ CRF with cell list:
2  coes
18 cubes
8  hips
16 tricues (J3)
```
```13:B23: wx 2 oo 3 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: wx .. .. oq &#zx)
```
```14:-: wx 2 ox 3 oo 4 xx &#zx
= oxxo 3 oooo 4 xxxx &#xt
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```
```23:B23: qo 2 xx 3 xo 4 oq &#zx (pabdirico)
= xxx 3 xox 4 oqo &#xt
→ CRF with cell list:
6  coes
12 cubes
2  toes
16 tricues (J3)
```
```24:B2: qo 2 xo 3 ox 4 xx &#zx (pacsrit)
= xox 3 oxo 4 xxx &#xt
→ CRF with cell list:
16 octs
2  sircoes
6  squobcues (J28)
24 trips
```
```34:-: qo 2 ox 3 xx 4 xx &#zx (tica gircobcu)
= oxo 3 xxx 4 xxx &#xt
→ CRF with cell list:
12 ops
2  tics
16 tricues (J3)
24 trips
```
```34:B23: qo 2 oo 3 xo 4 xw &#zx
→ °) (asks for non-regular hexagons: qo .. .. xw &#zx)
```
```123:B23: wx 2 xx 3 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: wx .. .. oq &#zx)
```
```124:B2: wx 2 xo 3 ox 4 xx &#zx
= xoox 3 oxxo 4 xxxx &#xt
= Wythoffian o3x3o4x (srit) with cell list:
16 octs
8  sircoes
32 trips
```
```134:-: wx 2 ox 3 xx 4 xx &#zx
= oxxo 3 xxxx 4 xxxx &#xt
→ CRF with cell list:
12 cubes
8  hips
18 ops
2  tics
16 tricues (J3)
24 trips
```
```134:B23: wx 2 oo 3 xo 4 xw &#zx
= oooo 3 xoox 4 xwwx &#xt
= Wythoffian o3o3x4x (tat) with cell list:
16 tets
8  tics
```
```234:B23: qo 2 xx 3 xo 4 xw &#zx
→ °) (asks for non-regular hexagons: qo .. .. xw &#zx)
```
```1234:B23: wx 2 xx 3 xo 4 xw &#zx (pabdiproh)
= xxxx 3 xoox 4 xwwx &#xt
→ CRF with cell list:
2  gircoes
12 ops
6  tics
16 tricues (J3)
8  trips
```

in . o3o4o subsymmetry   (up)
not prismatically symmetric
combinations of formers:
```none
```

in . o3o3o subsymmetry   (up)
Representation:
```xo 3 oo 3 ox &#x  (hex)
```
All layers &
kaleido-facetings per layer:
```A: x3o3o   →  A1:(-x)3  x 3  o   →  A12:  o 3(-x)3  x   →  A123:  o 3  o 3(-x)
B: o3o3x   →  B3:  o 3  x 3(-x)  →  B32:  x 3(-x)3  o   →  B321:(-x)3  o 3  o
```
A priori invalid combinations:
```A + B321  → ‡) (u in A)
A1 + B32  → ‡) (u in A, u in B)
A12 + B3  → ‡) (u in A, u in B)
A123 + B  → ‡) (u in B)
```
Stott expansion:
(derived potential CRFs)
```1:A1: ox 3 xo 3 ox &#x (octaco)
→ CRF (segmentochoron) with cell list:
1 co
9 octs
6 squippies (J1)
```
```1:A1B321: oo 3 xo 3 oo &#x (octpy)
→ CRF (segmentochoron) with cell list:
1 oct
8 tets
```
```2:-: xo 3 xx 3 ox &#x (tuta)
→ CRF (segmentochoron) with cell list:
6 tets
8 tricues (J3)
2 tuts
```
```2:A12: oo 3 ox 3 xx &#x (tetatut)
→ CRF (segmentochoron) with cell list:
5 tets
4 tricues (J3)
1 tut
```
```2:A12B32: ox 3 oo 3 xo &#x
= Wythoffian x3o3o4o (hex) with cell list:
16 tets
(axially dual orientation)
```
```12:A12: xx 3 ox 3 xx &#x (coatoe)
→ CRF (segmentochoron) with cell list:
1 co
6 cubes
1 toe
8 tricues (J3)
```
```12:A12B321: xo 3 ox 3 xo &#x (octaco)
→ CRF (segmentochoron) with cell list:
1 co
9 octs
6 squippies (J1)
```
```13:A1B3: ox 3 xx 3 xo &#x (tuta)
→ CRF (segmentochoron) with cell list:
6 tets
8 tricues (J3)
2 tuts
```
```13:A1B321: oo 3 xo 3 xx &#x (tetatut)
→ CRF (segmentochoron) with cell list:
5 tets
4 tricues (J3)
1 tut
```
```13:A123B321: xo 3 oo 3 ox &#x
= Wythoffian x3o3o4o (hex) with cell list:
16 tets
(identical orientation)
```
```123:A123B321: xo 3 xx 3 ox &#x (tuta)
→ CRF (segmentochoron) with cell list:
6 tets
8 tricues (J3)
2 tuts
```

in o4o2o4o subsymmetry   (up)
Representation:
```xo 4 oo 2 ox 4 oo &#zx  (hex)
```
All layers &
kaleido-facetings per layer:
```A: x4o o4o   →  A1: (-x)4 q   o 4 o
B: o4o x4o   →  B3:   o 4 o (-x)4 q
```
Stott expansion:
(derived potential CRFs)
```1:A1: ox 4 qo 2 ox 4 oo &#zx (cytau tes)
→ CRF with cell list:
4  cubes
4  octs
16 squippies (J1)
```
```2:-: xo 4 xx 2 ox 4 oo &#zx (quawros)
→ CRF with cell list:
4  cubes
16 tets
16 trips
```
```12:A1: ox 4 wx 2 ox 4 oo &#zx (cyte cubau sodip)
→ CRF with cell list:
4  esquidpies (J15)
4  ops
16 squippies (J1)
```
```13:A1B3: ox 4 qo 2 xo 4 oq &#zx
= Wythoffian o3o3x4o (rit) with cell list:
8  coes
16 tets
```
```14:A1: ox 4 qo 2 ox 4 xx &#zx (cyte opau sodip)
→ CRF with cell list:
8  cubes
16 squippies (J1)
4  squobcues (J28)
16 trips
```
```24:-: xo 4 xx 2 ox 4 xx &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```
```123:A1B3: ox 4 wx 2 xo 4 oq &#zx
→ °) (asks for non-regular hexagons: .. wx .. oq &#zx)
```
```124:A1: ox 4 wx 2 ox 4 xx &#zx (cyted srit)
→ CRF with cell list:
8  ops
4  sircoes
16 squippies (J1)
16 trips
```
```1234:A1B3: ox 4 wx 2 xo 4 xw &#zxx
= Wythoffian o3o3x4x (tat) with cell list:
16 tets
8  tics
```

in o2o2o2o subsymmetry   (up)
Representation:
```qooo 2 oqoo 2 ooqo 2 oooq &#zx  (hex)
```
All layers &
kaleido-facetings per layer:
```A: q2o2o2o
B: o2q2o2o
C: o2o2q2o
D: o2o2o2q

(As this representation does not show up (true / x-) edges within the layers, those thus cannot be inverted either.
Therefore at most pure partial Stott expansions wrt. this subsymmetry remain possible.)
```
Stott expansion:
(derived potential CRFs)
```1:A1: wxxx 2 oqoo 2 ooqo 2 oooq &#zx (pexhex)
→ CRF with cell list:
16 tets
8  trips

related: .xxx 2 .qoo 2 .oqo 2 .ooq &#zx
= Wythoffian x x3o4o (ope) with cell list:
2 octs
8 trips
```
```12:-: wxxx 2 xwxx 2 ooqo 2 oooq &#zx (quawros)
→ CRF with cell list:
4  cubes
16 tets
16 trips
```
```123:-: wxxx 2 xwxx 2 xxwx 2 oooq &#zx (pacsid pith)
→ CRF with cell list:
14 cubes
16 tets
24 trips
```
```1234:-: wxxx 2 xwxx 2 xxwx 2 xxxw &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips

related: .xxx 2 .wxx 2 .xwx 2 .xxw &#zx
= Wythoffian x x3o4x (sircope) with cell list:
18 cubes
2  sircoes
8  trips
```

#### EKF of the pentachoron (x3o3o3o)

(As it will turn out, the set of resulting polychora in . o3o3o here is identical to those which turn up from the corresponding consideration of rap wrt. this subsymmetry.)

in . o3o3o subsymmetry   (up)
Representation:
```ox 3 oo 3 oo &#x  (pen)
```
All layers &
kaleido-facetings per layer:
```A: o3o3o
B: x3o3o   →  B1:(-x)3  x 3  o   →  B12:  o 3(-x)3  x   →  B123:  o 3  o 3(-x)
```
A priori invalid combinations:
```none
```
Stott expansion:
(derived potential CRFs)
```1:B1: xo 3 ox 3 oo &#x
= Wythoffian o3x3o4o (rap) with cell list:
5 octs
5 tets
```
```2:-: ox 3 xx 3 oo &#x (octatut)
→ CRF (segmentochoron) with cell list:
1 oct
4 tricues (J3)
4 trips
1 tut
```
```2:B12: oo 3 xo 3 ox &#x
= Wythoffian o3x3o3o (rap) with cell list:
5 octs
5 tets
(inverted orientation)
```
```3:-: ox 3 oo 3 xx &#x (tetaco)
→ CRF (segmentochoron) with cell list:
1  co
5  tets
10 trips
```
```3:B123: oo 3 oo 3 xo &#x
= Wythoffian o3o3o3x (pen) with cell list:
5 tets
(dual orientation)
```
```12:B12: xx 3 xo 3 ox &#x (coatut)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
```
```13:B1: xo 3 ox 3 xx &#x (coatut)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
(inverted orientation)
```
```13:B123: xx 3 oo 3 xo &#x (tetaco)
→ CRF (segmentochoron) with cell list:
1  co
5  tets
10 trips
(inverted orientation)
```
```23:-: ox 3 xx 3 xx &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
```
```23:B123: oo 3 xx 3 xo &#x (octatut)
→ CRF (segmentochoron) with cell list:
1 oct
4 tricues (J3)
4 trips
1 tut
```
```123:B123: xx 3 xx 3 xo &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
(inverted orientation)
```

#### EKF of the rectified pentachoron (o3x3o3o)

(As it will turn out, the set of resulting polychora in . o3o3o here is identical to those which turn up from the corresponding consideration of pen wrt. this subsymmetry.)

in . o3o3o subsymmetry   (up)
Representation:
```xo 3 ox 3 oo &#x  (rap)
```
All layers &
kaleido-facetings per layer:
```A: x3o3o   →  A1: (-x)3x3o   →  A12: o3(-x)3x   →  A123: o3o3(-x)
B: o3x3o   →  B2: x3(-x)3x   →  B21: (-x)3o3x   →  B213: (-x)3x(-x)   →  B2132: o3(-x)3o
↳  B23: x3o3(-x)   →  B231 = B213
```
A priori invalid combinations:
```A    + B21,B213
A1   + B2,B2132,B23
A12  + B,B213,B23
A123 + B2,B21
```
Other layer-combinations:
```B2:         xx3o(-x)3ox&#x
B2132:      xo3o(-x)3oo&#x  → ‡
B23:        xx3oo3o(-x)&#x
A1:         (-x)o3xx3oo&#x
A1B21:      (-x)(-x)3xo3ox&#x
A1B213:     (-x)(-x)3xx3o(-x)&#x
A12B2:      ox3(-x)(-x)3xx&#x
A12B2132:   oo3(-x)(-x)3xo&#x
A123:       oo3ox3(-x)o&#x  → ‡
A123B213:   o(-x)3ox3(-x)(-x)&#x
A123B2132:  oo3o(-x)3(-x)o&#x
A123B23:    ox3oo3(-x)(-x)&#x
```
Stott expansion:
(derived potential CRFs)
```1:A1 = 23:A123B23: ox 3 xx 3 oo &#x (octatut)
→ CRF (segmentochoron) with cell list:
1 oct
4 tricues (J3)
4 trips
1 tut
```
```1:A1B21 = 23:A123B2132: oo 3 xo 3 ox &#x
= Wythoffian o3x3o3o (rap) with cell list:
5 octs
5 tets
(inverted orientation)
```
```2:B2 = 123:A123B2132: xx 3 xo 3 ox &#x (coatut)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
```
```2:A12B2: ox 3 oo 3 xx &#x (tetaco)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
```
```2:A12B2132: oo 3 oo 3 xo &#x
= Wythoffian x3o3o3o (pen) with cell list:
5 tets
```
```3:-: xo 3 ox 3 xx &#x (coatut)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
(inverted orientation)
```
```3:A123B23: ox 3 oo 3 oo &#x
= Wythoffian o3o3o3x (pen) with cell list:
5 tets
(dual orientation)
```
```12:A12B2132: xx 3 oo 3 xo &#x (tetaco)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut
(inverted orientation)
```
```13:A1: ox 3 xx 3 xx &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
```
```13:A1B213: oo 3 xx 3 xo &#x (octatut)
→ CRF (segmentochoron) with cell list:
1 oct
4 tricues (J3)
4 trips
1 tut
(inverted orientation)
```
```13:A123B213: xo 3 ox 3 oo &#x
= Wythoffian o3x3o3o (rap) with cell list:
5 octs
5 tets
(itself again)
```
```23:B23: xx 3 xx 3 xo &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
(inverted orientation)
```

#### EKF of the small rhombated pentachoron (x3o3x3o)

(seen oct first   –     seen co first)

(As it will turn out, the set of resulting polychora in . o3o3o here is identical to those which turn up from the corresponding consideration of spid wrt. this subsymmetry.)

in . o3o3o subsymmetry   (up)
Representation:
```oxx 3 xxo 3 oox &#xt  (srip)
```
All layers &
kaleido-facetings per layer:
```A: o3x3o   →  A2:  x 3(-x)3  x   →  A21: (-x)3  o 3  x   →  A213: (-x)3  x 3(-x)   →  A2132:  o 3(-x)3 o
↳  A23:   x 3  o 3(-x)  →  (A231 = A213)          →  (A2312 = A2132)
B: x3x3o   →  B1:(-x)3  u 3  o
↳  B2:  u 3(-x)3  x   →  B23:   u 3  o 3(-x)
C: x3o3x   →  C1:(-x)3  x 3  x   →  C12:   o 3(-x)3  u
↳  C13: (-x)3  u 3(-x)
↳  C3:  x 3  x 3(-x)  →  (C31 = C13)
↳  C32:   u 3(-x)3  o
```
A priori invalid combinations:
```A     + B2,B23
A     + C12,C32
A2    + B,B1,B23
A2    + C1,C3,C13
A21   + B,B1,B2,B23 → generally
A23   + B1,B2
A23   + C,C1,C12
A213  + B,B2,B23
A213  + C,C1,C3,C12,C32
A2132 + B,B1,B2,B23 → generally
B     + C1,C12,C13,C32
B1    + C,C3,C12,C32
B2    + C1,C3,C12,C32
B23   + C,C1,C12,C13
C12   generally (u in extremal layer)
C13   generally (u in extremal layer)
C32   generally (u in extremal layer)
```
Other layer-combinations:
```B1C1:     o(-x)(-x) 3 xux 3 oox &#xt
A2B2:     xux 3 (-x)(-x)o 3 xxx &#xt
C3:       oxx 3 xxx 3 oo(-x) &#xt
A23C3:    xxx 3 oxx 3 (-x)o(-x) &#xt  → †)
A23B23C3: xux 3 oox 3 (-x)(-x)(-x) &#xt
```
Stott expansion:
(derived potential CRFs)
```1:B1C1: xoo 3 xux 3 oox &#xt
= Wythoffian o3x3x3o (deca) with cell list:
10 tuts
```
```2:A2B2: xux 3 oox 3 xxx &#xt (coatotum)
→ CRF with cell list:
1 co
6 hips
1 toe
4 tricues (J3)
4 trips
4 tuts
```
```3:C3: oxx 3 xxx 3 xxo &#xt (tutatoe gybcu)
→ CRF with cell list:
8  hips
8  tricues (J3)
12 trips
2  tuts
(bistratic segmentochoral stack)

related: ox. 3 xx. 3 xx. &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
```
```3:A23B23C3: xux 3 oox 3 ooo &#xt
= Wythoffian x3x3o3o (tip) with cell list:
5 tets
5 tuts
```

#### EKF of the decachoron (o3x3x3o)

in . o3o3o subsymmetry   (up)
Representation:
```oox 3 xux 3 xoo &#xt  (deca)
```
All layers &
kaleido-facetings per layer:
```A: o3x3x   →  A2: x3(-x)3u   →  A21: (-x)3o3u   →  A213: (-x)3u3(-u)   →  A2132: x3(-u)3o   →  A21321: (-x)3(-x)3o
↳  A23: x3x3(-u)   →  A231 = A213
↳  A232: u3(-x)3(-x)   →  A2321: (-u)3x3(-x)   →  A23212 = A21321
↳  A3: o3u3(-x)   →  A32: u3(-u)3x   →  A321: (-u)3o3x   →  A3213 = A2321
↳  A323 = A232
B: o3u3o   →  B2: u3(-u)3u   →  B21: (-u)3o3u   →  B213: (-u)3u3(-u)   →  B2132: o3(-u)3o
↳  B23: u3o3(-u)   →  B231 = B213
C: x3x3o   →  C1: (-x)3u3o   →  C12: x3(-u)3u   →  C121: (-x)3(-x)3u   →  C1213: (-x)3x3(-u)   →  C12132: o3(-x)3(-x)
↳  C123: x3o3(-u)   →  C1231 = C1213
↳  C2: u3(-x)3x   →  C21: (-u)3x3x   →  C212 = C121
↳  C213: (-u)3u3(-x)   →  C2132: o3(-u)3x   →  C21323 = C12132
↳  C23: u3o3(-x)   →  C231 = C213
```
A priori invalid combinations:
```B      + A2,A21,A213,A2132,A21321,A23,A232,A2321,A32,A321 - i.e. only A,A3
+ C12,C121,C1213,C12132,C123,C2,C21,C213,C2132,C23 - i.e. only C,C1
B2     + A,A21,A213,A2132,A21321,A23,A232,A2321,A3,A321 - i.e. only A2,A32
+ C,C1,C121,C1213,C12132,C123,C21,C213,C2132,C23 - i.e. only C12,C2
B21    + A,A2,A213,A2132,A21321,A23,A232,A2321,A3,A32 - i.e. only A21,A321
+ C,C1,C12,C1213,C12132,C123,C2,C213,C2132,C23 - i.e. only C121,C21
B213   + A,A2,A21,A2132,A21321,A23,A232,A3,A32,A321 - i.e. only A213,A2321
+ C,C1,C12,C121,C12132,C123,C2,C21,C2132,C23 - i.e. only C1213,C213
B2132  + A,A2,A21,A213,A23,A232,A2321,A3,A32,A321 - i.e. only A2132,A21321
+ C,C1,C12,C121,C1213,C123,C2,C21,C213,C23 - i.e. only C12132,C2132
B23    + A,A2,A21,A213,A2132,A21321,A2321,A3,A32,A321 - i.e. only A23,A232
+ C,C1,C12,C121,C1213,C12132,C2,C21,C213,C2132 - i.e. only C123,C23
```
Other layer-combinations:
```ABC               oox 3 xux 3 xoo &#xt                   (self-inv) : itself
ABC1              oo(-x) 3 xuu 3 xoo &#xt                (inv = A3BC)
A3BC1             oo(-x) 3 uuu 3 (-x)oo &#xt             (self-inv)
A2B2C12           xux 3 (-x)(-u)(-u) 3 uuu &#xt          (inv = A32B2C2)
A2B2C2            xuu 3 (-x)(-u)(-x) 3 uux &#xt          (self-inv) : asks for corealmic cells with non-convex exterior blend
A32B2C12          uux 3 (-u)(-u)(-u) 3 xuu &#xt          (self-inv)
A21B21C121        (-x)(-u)(-x) 3 oo(-x) 3 uuu &#xt       (inv = A232B23C23) : asks for non-convex cell-join at inner layer
A21B21C21         (-x)(-u)(-u) 3 oox 3 uux &#xt          (inv = A23B23C23) : asks for non-convex cell-join at inner layer
A321B21C121       (-u)(-u)(-x) 3 oo(-x) 3 xuu &#xt       (inv = A232B23C123)
A321B21C21        (-u)(-u)(-u) 3 oox 3 xux &#xt          (inv = A23B23C123)
A213B213C1213     (-x)(-u)(-x) 3 uux 3 (-u)(-u)(-u) &#xt (inv = A2321B213C213) : asks for non-convex cell-join at inner layer
A213B213C213      (-x)(-u)(-u) 3 uuu 3 (-u)(-u)(-x) &#xt (self-inv) : asks for non-convex cell-join at inner layer
A2321B213C1213    (-u)(-u)(-x) 3 xux 3 (-x)(-u)(-u) &#xt (self-inv)
A2132B2132C12132  xoo 3 (-u)(-u)(-u) 3 oo(-x) &#xt       (inv = A21321B2132C2132) : asks for non-convex cell-join at inner layer
A2132B2132C2132   xoo 3 (-u)(-u)(-u) 3 oox &#xt          (self-inv) : asks for non-convex cell-join at inner layer
A21321B2132C12132 (-x)oo 3 (-x)(-u)(-u) 3 oo(-x) &#xt    (self-inv)
```
Stott expansion:
(derived potential CRFs)
```-:ABC = 1133:A2321B213C1213: oox 3 xux 3 xoo &#xt
= Wythoffian o3x3x3o (deca) with cell list:
10 tuts
(itself)
```
```1(-2):ABC1 = inv 112(-3):A321B21C121: xxo 3 oxx 3 xoo &#xt
= Wythoffian x3o3x3o (srip) with cell list:
5  coes
5  octs
10 trips
```
```1(-2)3:A3BC1 = (-1)222(-3):A32B2C12: xxo 3 xxx 3 oxx &#xt (tutato gybcu)
→ CRF with cell list:
8  hips
8  tricues (J3)
12 trips
2  tuts
```
```1(-2)(-2)3:A3BC1 = (-1)22(-3):A32B2C12: xxo 3 ooo 3 oxx &#xt
= Wythoffian x3o3o3x (spid) with cell list:
10 tets
20 trips
```
```22(-3):A2B2C12 = inv 111:A321B21C21: xux 3 xoo 3 xxx &#xt (coatotum)
→ CRF with cell list:
1 co
6 hips
1 toe
4 tricues (J3)
4 trips
4 tuts
```
```22(-3)(-3):A2B2C12 = inv 11:A321B21C21: xux 3 xoo 3 ooo &#xt
= Wythoffian x3x3o3o (tip) with cell list:
5 tets
5 tuts
```
```1221:A21321B2132C12132: oxx 3 xoo 3 xxo &#xt (tetaco altut)
→ CRF with cell list:
6 gybefs (J26)
4 octs
5 tets
4 tricues (J3)
4 trips
1 tut
```

in . o2o3o subsymmetry   (up)
Representation:
```oxuxo 2 xuxoo 3 ooxux &#xt  (deca)
```
All layers &
kaleido-facetings per layer:
```A: o x3o  →  A2: o (-x)3x  →  A23: o o3(-x)
B: x u3o  →  B1: (-x) u3o  →  B12: (-x) (-u)3u  →  B123: (-x) o3(-u)
↳  B2: x (-u)3u  →  B21 = B12
↳  B23: x o3(-u)  →  B231 = B123
C: u x3x  →  C1: (-u) x3x  →  C12: (-u) (-x)3u  →  C123: (-u) x3(-u)  →  C1232: (-u) (-x)3(-x)
↳  C2: u (-x)3u  →  C21 = C12
↳  C23 = u x3(-u)  →  C231 = C123
↳  C232: u (-x)3(-x)  →  C2321 = C1232
↳  C3: u u3(-x)  →  C31: (-u) u3(-x)  →  C312: (-u) (-u)3x  →  C3123 = C1232
↳  C32 = u (-u)3x  →  C321 = C312
↳  C323 = C232
D: x o3u  →  D1: (-x) o3u  →  D13: (-x) u3(-u)  →  D132: (-x) (-u)3o
↳  D3: x u3(-u)  →  D31= D13
↳  D32: x (-u)3o  →  D321 = D132
E: o o3x  →  E3: o x3(-x)  →  E32: o (-x)3o
```
A priori invalid combinations:
```A     + B12,B123,B2,B23, i.e. only B,B1
A2    + B,B1,B123,B23, i.e. only B12,B2
A23   + B,B1,B12,B2, i.e. only B123,B23
B     + C1,C12,C123,C1232,C2,C23,C232,C31,C312,C32, i.e. only C,C3
B1    + C,C12,C123,C1232,C2,C23,C232,C3,C312,C32, i.e. only C1,C31
B12   + C,C1,C123,C1232,C2,C23,C232,C3,C31,C32, i.e. only C12,C312
B123  + C,C1,C12,C2,C23,C232,C3,C31,C312,C32, i.e. only C123,C1232
B2    + C,C1,C12,C123,C1232,C23,C232,C3,C31,C312, i.e. only C2,C32
B23   + C,C1,C12,C123,C1232,C2,C3,C31,C312,C32, i.e. only C23,C232
C     + D1,D13,D132,D3,D32, i.e. only D
C1    + D,D13,D132,D3,D32, i.e. only D1
C12   + D,D13,D132,D3,D32, i.e. only D1
C123  + D,D1,D132,D3,D32, i.e. only D13
C1232 + D,D1,D13,D3,D32, i.e. only D132
C2    + D1,D13,D132,D3,D32, i.e. only D
C23   + D,D1,D13,D132,D32, i.e. only D3
C232  + D,D1,D13,D132,D3, i.e. only D32
C3    + D,D1,D13,D132,D32, i.e. only D3
C31   + D,D1,D132,D3,D32, i.e. only D13
C312  + D,D1,D13,D3,D32, i.e. only D132
C32   + D,D1,D13,D132,D3, i.e. only D32
D     + E3,E32, i.e. only E
D1    + E3,E32, i.e. only E
D13   + E,E32, i.e. only E3
D132  + E,E3, i.e. only E32
D3    + E,E32, i.e. only E3
D32   + E,E3, i.e. only E32
```
Other layer-combinations:
```ABCDE               oxuxo 2 xuxoo 3 ooxux &#xt : (self-inv)  →  itself
ABC3D3E3            oxuxo 2 xuuux 3 oo(-x)(-u)(-x) &#xt : (inv = A2B2C2DE)  →  ‡
AB1C1D1E            o(-x)(-u)(-x)o 2 xuxoo 3 ooxux &#xt : (self-inv)  →  ‡
AB1C31D13E3         o(-x)(-u)(-x)o 2 xuuux 3 oo(-x)(-u)(-x) &#xt : (inv = A2B12C12D1E)  →  ‡
A2B12C312D132E32    o(-x)(-u)(-x)o 2 (-x)(-u)(-u)(-u)(-x) 3 xuxoo &#xt : (inv = A23B123C123D13E3)  →  ‡
A23B123C1232D132E32 o(-x)(-u)(-x)o 2 oo(-x)(-u)(-x) 3 (-x)(-u)(-x)oo &#xt : (self-inv)  →  ‡
A23B23C23D3E3       oxuxo 2 ooxux 3 (-x)(-u)(-u)(-u)(-x) &#xt : (inv = A2B2C32D32E32)
A23B23C232D32E32    oxuxo 2 oo(-x)(-u)(-x) 3 (-x)(-u)(-x)oo &#xt : (self-inv)  →  ‡
```
Stott expansion:
(derived potential CRFs)
```22:A23B23C23D3E3: oxuxo 2 ooxux 3 xooox &#xt
: itself non-convex, but dissectable into

related: ox... 2 oo... 3 xo... &#x
= Wythoffian x3o3o3o (pen) with cell list:
5 tet

related: .xuxo 2 .oxux 3 .ooox &#xt
= Wythoffian x3x3o3o (tip) with cell list:
5 tet
5 tut
```

#### EKF of the small prismatodecachoron (x3o3o3x)

(As it will turn out, the set of resulting polychora in . o3o3o here is identical to those which turn up from the corresponding consideration of srip wrt. this subsymmetry.)

in . o3o3o subsymmetry   (up)
Representation:
```xxo 3 ooo 3 oxx &#xt  (spid)
```
All layers &
kaleido-facetings per layer:
```A: x3o3o   →  A1: (-x)3  x 3  o   →  A12:   o 3(-x)3  x   →  A123:   o 3  o 3(-x)
B: x3o3x   →  B1: (-x)3  x 3  x   →  B12:   o 3(-x)3  u
↳  B13: (-x)3  u 3(-x)
↳  B3:   x 3  x 3(-x)  →  (B31 = B13)
↳  B32:   u 3(-x)3  o
C: o3o3x   →  C3:   o 3  x 3(-x)  →  C32:   x 3(-x)3  o   →  C321: (-x)3  o 3  o
```
A priori invalid combinations:
```A    + B1,B12,B13
A1   + B,B3,B12,B32
A1   + C32
A12  + B1,B3,B13,B32
A12  + C3
A123 + B,B1,B12,B13,B32
A123 + C
B    + C3,C321
B1   + C3,C32
B3   + C,C32,C321
B12  + C3,C32,C321
B13  + C,C32,C321
B32  + C,C3,C321
```
Other layer-combinations:
```A1B1:     (-x)(-x)o 3 xxo 3 oxx &#xt
A12:      oxo 3 (-x)oo 3 xxx &#xt  → †)
A12B12:   ooo 3 (-x)(-x)o 3 xux &#xt
A12C32:   oxx 3 (-x)o(-x) 3 xxo &#xt  → †)
A123B3C3: oxo 3 oxx 3 (-x)(-x)(-x) &#xt  → †)
A1B13C3:  (-x)(-x)o 3 xux 3 o(-x)(-x) &#xt
```
Stott expansion:
(derived potential CRFs)
```1:A1B1: oox 3 xxo 3 oxx &#xt
= Wythoffian x3o3x3o (srip) with cell list:
5  coes
5  octs
10 trips

related: .ox 3 .xo 3 .xx &#x (coatut)
→ CRF (segmentochoron) with cell list:
1 co
4 octs
4 tricues (J3)
6 trips
1 tut

related: oo. 3 xx. 3 ox. &#x (octatut)
→ CRF (segmentochoron) with cell list:
1 oct
4 tricues (J3)
4 trips
1 tut
```
```2:-: xxo 3 xxx 3 oxx &#xt (tutatoe gybcu)
→ CRF with cell list:
8  hips
8  tricues (J3)
12 trips
2  tuts
(bistratic segmentochoral stack)

related: xx. 3 xx. 3 ox. &#x (tutatoe)
→ CRF (segmentochoron) with cell list:
4 hips
1 toe
4 tricues (J3)
6 trips
1 tut
```
```2:A12B12: ooo 3 oox 3 xux &#xt
= Wythoffian x3x3o3o (tip) with cell list:
5 tets
5 tuts
```
```13:A1B13C3: xoo 3 xux 3 oox &#xt
= Wythoffian o3x3x3o (deca) with cell list:
10 tuts
```

in . o2o3o subsymmetry   (up)
Representation:
```x(ou)x 2 x(xo)o 3 o(xo)x &#xt  (spid)
```
All layers &
kaleido-facetings per layer:
```A: x2x3o   →  A1: (-x)2  x 3  o   →  A12: (-x)2(-x)3  x   →  A123: (-x)2  o 3(-x)
↳  A2:   x 2(-x)3  x   →  (A21 = A12)
↳  A23:   x 2  o 3(-x)  →  (A231 = A123)
B: o2x3x   →  B2:   o 2(-x)3  u
↳  B3:   o 2  u 3(-x)
C: u2o3o
D: x2o3x   →  D1: (-x)2  o 3  x   →  D13: (-x)2  x 3(-x)  →  D132: (-x)2(-x)3  o
↳  D3:   x 2  x 3(-x)  →  (D31 = D13)
↳  D32:   x 2(-x)3  o   →  (D321 = D132)
```
A priori invalid combinations:
```A    + B2,D1,D13,D32,D132
A1   + B2,D,D3,D32,D132
A2   + B,B3,D1,D3,D13,D132
A12  + B,B3,D,D3,D13,D32
A23  + B,D,D1,D13,D132
A123 + B,D,D1,D3,D32
B    + D3,D13,D32,D132
B2   + D3,D13
B3   + D,D1
```
Other layer-combinations:
```B3D3       x(ou)x 2 x(uo)x 3 o((-x)o)(-x) &#xt
A1D1       (-x)(ou)(-x) 2 x(xo)o 3 o(xo)x &#xt  → †
A1B3D13    (-x)(ou)(-x) 2 x(uo)x 3 o((-x)o)(-x) &#xt  → †
A2B2D32    x(ou)x 2 (-x)((-x)o)(-x) 3 x(uo)o &#xt  → †
A12B2D132  (-x)(ou)(-x) 2 (-x)((-x)o)(-x) 3 x(uo)o &#xt  → †
A23B2D32   x(ou)x 2 o((-x)o)(-x) 3 (-x)(uo)o &#xt  → †
A123B2D132 (-x)(ou)(-x) 2 o((-x)o)(-x) 3 (-x)(uo)o &#xt  → †
```
Stott expansion:
(derived potential CRFs)
```3:B3D3: x(ou)x 2 x(uo)x 3 x(ox)o &#xt (biscsrip)
→ CRF with cell list:
3 coes
1 hip
2 oct
3 squippies (J1)
2 tricues (J3)
7 trips
```

#### EKF of the rectified hexacosachoron (o3x3o5o)

in o2o3o5o subsymmetry   (up)
Representation:
```DCBAVFfxoo 2 xoxFofofVx 3 oxoofoxxoo 5 ooxooxxoof &#zx  (rox)
```
All layers &
kaleido-facetings per layer:
```A: D2x3o5o   →  A2: D2(-x)3x5o   →  A23: D2o3(-x)5f
B: C2o3x5o   →  B3: C2x3(-x)5f   →  B32: C2(-x)3o5f
C: B2x3o5x   →  C2: B2(-x)3x5x   →  C23: B2o3(-x)5F
↳  C24: B2(-x)3F5(-x)
↳  C4: B2x3f5(-x)   →  (C42 = C24)
D: A2F3o5o
E: V2o3f5o
F: F2f3o5x   →  F4: F2f3f5(-x)
G: f2o3x5x   →  G3: f2x3(-x)5F   →  G32: f2(-x)3o5F
↳  G4: f2o3F5(-x)
H: x2f3x5o   →  H1: (-x)2f3x5o   →  H13: (-x)2F3(-x)5f
↳  H3: x2F3(-x)5f   →  (H31 = H13)
I: o2V3o5o
i: o2x3o5f   →  i2: o2(-x)3x5f   →  i23: o2o3(-x)5V

(Note, I and i both belong to the same hyperplane o2... .)
```
A priori invalid combinations:
```A   + B32, C2, C24, G32, i2
A2  + B3, C, C4, C23, G3, H3, H13, i, i23
A23 + B, C2, G, H, H1, i2
B   + C23, G3, H3, H13, i23
B3  + C2, C24, G, G32, H, H1, i2
B32 + C, C4, G3, i
C   + F4, G4, G32, i2
C2  + F4, G3, G4, H3, H13, i, i23
C4  + F, G, G32, i2
C23 + G, H, H1, i2
C24 + F, G, G3, i
F   + G4
F4  + G
G   + H3, H13, i23
G3  + H, H1, i2
G32 + i
H   + i23
H1  + i23
H3  + i2
H13 + i2

(all these would provide u edges (→ ‡); thus reducing from 4320 to 70 potential combinations only)
```
Other layer-combinations:
```H1:                   DCBAVFf(-x)oo 2 xoxFofofVx 3 oxoofoxxoo 5 ooxooxxoof &#zx  → †) (dead end at I)
C4F4G4:               DCBAVFfxoo 2 xoxFofofVx 3 oxfoffFxoo 5 oo(-x)oo(-x)(-x)oof &#zx  → ‡ (f in C, E, F)
C4F4G4H1:             DCBAVFf(-x)oo 2 xoxFofofVx 3 oxfoffFxoo 5 oo(-x)oo(-x)(-x)oof &#zx  → †)
B3G3H3:               DCBAVFfxoo 2 xxxFofxFVx 3 o(-x)oofo(-x)(-x)oo 5 ofxooxFfof &#zx  → ‡ (f in G, i)
B3G3H3i23:            DCBAVFfxoo 2 xxxFofxFVo 3 o(-x)oofo(-x)(-x)o(-x) 5 ofxooxFfoV &#zx
B3G3H13:              DCBAVFf(-x)oo 2 xxxFofxFVx 3 o(-x)oofo(-x)(-x)oo 5 ofxooxFfof &#zx  → †) (dead end at I)
B3G3H13i23:           DCBAVFf(-x)oo 2 xxxFofxFVo 3 o(-x)oofo(-x)(-x)o(-x) 5 ofxooxFfoV &#zx  → †) (dead end at I)
B3C4F4G3H3:           DCBAVFfxoo 2 xxxFofxFVx 3 o(-x)foff(-x)(-x)oo 5 of(-x)oo(-x)Ffof &#zx  → †)
B3C4F4G3H3i23:        DCBAVFfxoo 2 xxxFofxFVo 3 o(-x)foff(-x)(-x)o(-x) 5 of(-x)oo(-x)FfoV &#zx  → †)
B3C4F4G3H13:          DCBAVFf(-x)oo 2 xxxFofxFVx 3 o(-x)foff(-x)(-x)oo 5 of(-x)oo(-x)Ffof &#zx  → †)
B3C4F4G3H13i23:       DCBAVFf(-x)oo 2 xxxFofxFVo 3 o(-x)foff(-x)(-x)o(-x) 5 of(-x)oo(-x)FfoV &#zx  → †)
B3C4F4G4H3:           DCBAVFfxoo 2 xxxFofoFVx 3 o(-x)foffF(-x)oo 5 of(-x)oo(-x)(-x)fof &#zx  → †)
B3C4F4G4H3i23:        DCBAVFfxoo 2 xxxFofoFVo 3 o(-x)foffF(-x)o(-x) 5 of(-x)oo(-x)(-x)foV &#zx  → †)
B3C4F4G4H13:          DCBAVFf(-x)oo 2 xxxFofoFVx 3 o(-x)foffF(-x)oo 5 of(-x)oo(-x)(-x)fof &#zx  → †)
B3C4F4G4H13i23:       DCBAVFf(-x)oo 2 xxxFofoFVo 3 o(-x)foffF(-x)o(-x) 5 of(-x)oo(-x)(-x)foV &#zx  → †)
B3C23G3H3:            DCBAVFfxoo 2 xxoFofxFVx 3 o(-x)(-x)ofo(-x)(-x)oo 5 ofFooxFfof &#zx  → †)
B3C23G3H3i23:         DCBAVFfxoo 2 xxoFofxFVo 3 o(-x)(-x)ofo(-x)(-x)o(-x) 5 ofFooxFfoV &#zx  → †)
B3C23G3H13:           DCBAVFf(-x)oo 2 xxoFofxFVx 3 o(-x)(-x)ofo(-x)(-x)oo 5 ofFooxFfof &#zx  → †)
B3C23G3H13i23:        DCBAVFf(-x)oo 2 xxoFofxFVo 3 o(-x)(-x)ofo(-x)(-x)o(-x) 5 ofFooxFfoV &#zx  → †)
A2C2i2:               DCBAVFfxoo 2 (-x)o(-x)FofofV(-x) 3 xxxofoxxox 5 ooxooxxoof &#zx  → †) (dead end at D)
A2C2H1i2:             DCBAVFf(-x)oo 2 (-x)o(-x)FofofV(-x) 3 xxxofoxxox 5 ooxooxxoof &#zx  → †)
A2C2G32i2:            DCBAVFfxoo 2 (-x)o(-x)Fof(-x)fV(-x) 3 xxxofooxox 5 ooxooxFoof &#zx  → †)
A2C2G32H1i2:          DCBAVFf(-x)oo 2 (-x)o(-x)Fof(-x)fV(-x) 3 xxxofooxox 5 ooxooxFoof &#zx  → †)
A2C24F4G4i2:          DCBAVFfxoo 2 (-x)o(-x)FofofV(-x) 3 xxFoffFxox 5 oo(-x)oo(-x)(-x)oof &#zx  → †) (dead end at D)
A2C24F4G4H1i2:        DCBAVFf(-x)oo 2 (-x)o(-x)FofofV(-x) 3 xxFoffFxox 5 oo(-x)oo(-x)(-x)oof &#zx  → †) (dead end at D)
A2C24F4G32i2:         DCBAVFfxoo 2 (-x)o(-x)Fof(-x)fV(-x) 3 xxFoffoxox 5 oo(-x)oo(-x)Foof &#zx  → †) (dead end at D)
A2C24F4G32H1i2:       DCBAVFf(-x)oo 2 (-x)o(-x)Fof(-x)fV(-x) 3 xxFoffoxox 5 oo(-x)oo(-x)Foof &#zx  → †) (dead end at D)
A2B32C2i2:            DCBAVFfxoo 2 (-x)(-x)(-x)FofofV(-x) 3 xoxofoxxox 5 ofxooxxoof &#zx  → †) (dead end at D)
A2B32C2H1i2:          DCBAVFf(-x)oo 2 (-x)(-x)(-x)FofofV(-x) 3 xoxofoxxox 5 ofxooxxoof &#zx  → †) (dead end at D)
A2B32C2G32i2:         DCBAVFfxoo 2 (-x)(-x)(-x)Fof(-x)fV(-x) 3 xoxofooxox 5 ofxooxFoof &#zx  → †) (dead end at D)
A2B32C2G32H1i2:       DCBAVFf(-x)oo 2 (-x)(-x)(-x)Fof(-x)fV(-x) 3 xoxofooxox 5 ofxooxFoof &#zx  → †) (dead end at D)
A2B32C24F4G4i2:       DCBAVFfxoo 2 (-x)(-x)(-x)FofofV(-x) 3 xoFoffFxox 5 of(-x)oo(-x)(-x)oof &#zx  → †)
A2B32C24F4G4H1i2:     DCBAVFf(-x)oo 2 (-x)(-x)(-x)FofofV(-x) 3 xoFoffFxox 5 of(-x)oo(-x)(-x)oof &#zx  → †)
A2B32C24F4G32i2:      DCBAVFfxoo 2 (-x)(-x)(-x)Fof(-x)fV(-x) 3 xoFoffoxox 5 of(-x)oo(-x)Foof &#zx  → †)
A2B32C24F4G32H1i2:    DCBAVFf(-x)oo 2 (-x)(-x)(-x)Fof(-x)fV(-x) 3 xoFoffoxox 5 of(-x)oo(-x)Foof &#zx  → †)
A23B3G3H3:            DCBAVFfxoo 2 oxxFofxFVx 3 (-x)(-x)oofo(-x)(-x)oo 5 ffxooxFfof &#zx  → ‡ (f in A)
A23B3G3H3i23:         DCBAVFfxoo 2 oxxFofxFVo 3 (-x)(-x)oofo(-x)(-x)o(-x) 5 ffxooxFfoV &#zx  → ‡ (f in A)
A23B3G3H13:           DCBAVFf(-x)oo 2 oxxFofxFVx 3 (-x)(-x)oofo(-x)(-x)oo 5 ffxooxFfof &#zx  → ‡ (f in A)
A23B3G3H13i23:        DCBAVFf(-x)oo 2 oxxFofxFVo 3 (-x)(-x)oofo(-x)(-x)o(-x) 5 ffxooxFfoV &#zx  → ‡ (f in A)
A23B3C4F4G3H3:        DCBAVFfxoo 2 oxxFofxFVx 3 (-x)(-x)foff(-x)(-x)oo 5 ff(-x)oo(-x)Ffof &#zx  → ‡ (F in A)
A23B3C4F4G3H3i23:     DCBAVFfxoo 2 oxxFofxFVo 3 (-x)(-x)foff(-x)(-x)o(-x) 5 ff(-x)oo(-x)FfoV &#zx  → ‡ (F in A)
A23B3C4F4G3H13:       DCBAVFf(-x)oo 2 oxxFofxFVx 3 (-x)(-x)foff(-x)(-x)oo 5 ff(-x)oo(-x)Ffof &#zx  → ‡ (F in A)
A23B3C4F4G3H13i23:    DCBAVFf(-x)oo 2 oxxFofxFVo 3 (-x)(-x)foff(-x)(-x)o(-x) 5 ff(-x)oo(-x)FfoV &#zx  → ‡ (F in A)
A23B3C4F4G4H3:        DCBAVFfxoo 2 oxxFofoFVx 3 (-x)(-x)foffF(-x)oo 5 ff(-x)oo(-x)(-x)fof &#zx  → ‡ (F in A)
A23B3C4F4G4H3i23:     DCBAVFfxoo 2 oxxFofoFVo 3 (-x)(-x)foffF(-x)o(-x) 5 ff(-x)oo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B3C4F4G4H13:       DCBAVFf(-x)oo 2 oxxFofoFVx 3 (-x)(-x)foffF(-x)oo 5 ff(-x)oo(-x)(-x)fof &#zx  → ‡ (F in A)
A23B3C4F4G4H13i23:    DCBAVFf(-x)oo 2 oxxFofoFVo 3 (-x)(-x)foffF(-x)o(-x) 5 ff(-x)oo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B3C23G3H3:         DCBAVFfxoo 2 oxoFofxFVx 3 (-x)(-x)(-x)ofo(-x)(-x)oo 5 ffFooxFfof &#zx  → ‡ (f in A)
A23B3C23G3H3i23:      DCBAVFfxoo 2 oxoFofxFVo 3 (-x)(-x)(-x)ofo(-x)(-x)o(-x) 5 ffFooxFfoV &#zx  → ‡ (f in A)
A23B3C23G3H13:        DCBAVFf(-x)oo 2 oxoFofxFVx 3 (-x)(-x)(-x)ofo(-x)(-x)oo 5 ffFooxFfof &#zx  → ‡ (f in A)
A23B3C23G3H13i23:     DCBAVFf(-x)oo 2 oxoFofxFVo 3 (-x)(-x)(-x)ofo(-x)(-x)o(-x) 5 ffFooxFfoV &#zx  → ‡ (f in A)
A23B3C23F4G3H3:       DCBAVFfxoo 2 oxoFofxFVx 3 (-x)(-x)(-x)off(-x)(-x)oo 5 ffFoo(-x)Ffof &#zx  → ‡ (F in A)
A23B3C23F4G3H3i23:    DCBAVFfxoo 2 oxoFofxFVo 3 (-x)(-x)(-x)off(-x)(-x)o(-x) 5 ffFoo(-x)FfoV &#zx  → ‡ (F in A)
A23B3C23F4G3H13:      DCBAVFf(-x)oo 2 oxoFofxFVx 3 (-x)(-x)(-x)off(-x)(-x)oo 5 ffFoo(-x)Ffof &#zx  → ‡ (F in A)
A23B3C23F4G3H13i23:   DCBAVFf(-x)oo 2 oxoFofxFVo 3 (-x)(-x)(-x)off(-x)(-x)o(-x) 5 ffFoo(-x)FfoV &#zx  → ‡ (F in A)
A23B3C23F4G4H3:       DCBAVFfxoo 2 oxoFofoFVx 3 (-x)(-x)(-x)offF(-x)oo 5 ffFoo(-x)(-x)fof &#zx  → ‡ (F in A)
A23B3C23F4G4H3i23:    DCBAVFfxoo 2 oxoFofoFVo 3 (-x)(-x)(-x)offF(-x)o(-x) 5 ffFoo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B3C23F4G4H13:      DCBAVFf(-x)oo 2 oxoFofoFVx 3 (-x)(-x)(-x)offF(-x)oo 5 ffFoo(-x)(-x)fof &#zx  → ‡ (F in A)
A23B3C23F4G4H13i23:   DCBAVFf(-x)oo 2 oxoFofoFVo 3 (-x)(-x)(-x)offF(-x)o(-x) 5 ffFoo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B32C23G32H3i23:    DCBAVFfxoo 2 o(-x)oFof(-x)FVo 3 (-x)o(-x)ofoo(-x)o(-x) 5 ffFooxFfoV &#zx  → ‡ (f in A)
A23B32C23G32H13i23:   DCBAVFf(-x)oo 2 o(-x)oFof(-x)FVo 3 (-x)o(-x)ofoo(-x)o(-x) 5 ffFooxFfoV &#zx  → ‡ (f in A)
A23B32C23F4G4H3i23:   DCBAVFfxoo 2 o(-x)oFofoFVo 3 (-x)o(-x)offF(-x)o(-x) 5 ffFoo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B32C23F4G4H13i23:  DCBAVFf(-x)oo 2 o(-x)oFofoFVo 3 (-x)o(-x)offF(-x)o(-x) 5 ffFoo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B32C23F4G32H3i23:  DCBAVFfxoo 2 o(-x)oFof(-x)FVo 3 (-x)o(-x)offo(-x)o(-x) 5 ffFoo(-x)FfoV &#zx  → ‡ (F in A)
A23B32C23F4G32H13i23: DCBAVFf(-x)oo 2 o(-x)oFof(-x)FVo 3 (-x)o(-x)offo(-x)o(-x) 5 ffFoo(-x)FfoV &#zx  → ‡ (F in A)
A23B32C24F4G4H3i23:   DCBAVFfxoo 2 o(-x)(-x)FofoFVo 3 (-x)oFoffF(-x)o(-x) 5 ff(-x)oo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B32C24F4G4H13i23:  DCBAVFf(-x)oo 2 o(-x)(-x)FofoFVo 3 (-x)oFoffF(-x)o(-x) 5 ff(-x)oo(-x)(-x)foV &#zx  → ‡ (F in A)
A23B32C24F4G32H3i23:  DCBAVFfxoo 2 o(-x)(-x)Fof(-x)FVo 3 (-x)oFoffo(-x)o(-x) 5 ff(-x)oo(-x)FfoV &#zx  → ‡ (F in A)
A23B32C24F4G32H13i23: DCBAVFf(-x)oo 2 o(-x)(-x)Fof(-x)FVo 3 (-x)oFoffo(-x)o(-x) 5 ff(-x)oo(-x)FfoV &#zx  → ‡ (F in A)
```
Stott expansion:
(derived CRFs)
```3:B3G3H3i23: DCBAVFfxoo 2 xxxFofxFVo 3 xoxxFxooxo 5 ofxooxFfoV &#zx
→ CRF with cell list:
30  ids
120 octs
60  pips
48  pocuroes (J32)
60  squippies (J1)
220 tets
40  thawroes (J92)
2   ties
80  tricues (J3)
120 trips

related: ..BAVFfxoo 2 ..xFofxFVo 3 ..xxFxooxo 5 ..xooxFfoV &#zx
→ CRF with cell list:
2   grids
30  ids
120 octs
24  pecues (J5)
24  pocuroes (J32)
60  squippies (J1)
220 tets
40  thawroes (J92)
120 trips
```

in . o3o5o subsymmetry   (up)
not prismatically symmetric
combinations of formers:
```none

(As there are just 2 symmetrical combinations - rox and that single CRF - which differ in their equatorial sections
there is no further combination of resp. hemiglomes either.)
```
using here node marks / (pseudo) edge lengths: F=f+x=ff, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x=fff, C=2F=2f+2x, D=F+V=3f+x

in o3o3o *b3o subsymmetry   (up)
Representation:
```Voo|Fxf|ofx 3 xxx|ooo|fff 3 oVo|fFx|xof *b3 ooV|xfF|fxo &#zx  (rox)

with cyclical layer symmetries: A(134) → B(341) → C(413) → A(134)
C(134) → D(341) → E(413) → C(134)
G(134) → H(341) → I(413) → G(134)
```
All layers &
kaleido-facetings per layer:
```A: V3x3o *b3o   →  A2: B3(-x)3x *b3x   →  A23: B3o3(-x) *b3x   →  A234: B3x3(-x) *b3(-x)   →  A2342: C3(-x)3o *b3o
↳  A24: B3o3x *b3(-x)   →  (A243.. = A234..)
B: o3x3V *b3o   →  B2: x3(-x)3B *b3x   →  B21: (-x)3o3B *b3x   →  B214: (-x)3x3B *b3(-x)   →  B2142: o3(-x)3C *b3o
↳  B24: x3o3B *b3(-x)   →  (B241.. = B214..)
C: o3x3o *b3V   →  C2: x3(-x)3x *b3B   →  C21: (-x)3o3x *b3B   →  C213: (-x)3x3(-x) *b3B   →  C2132: o3(-x)3o *b3C
↳  C23: x3o3(-x) *b3B   →  (C231.. = C213..)
D: F3o3f *b3x   →  D4: F3x3f *b3(-x)   →  D42: A3(-x)3F *b3o
E: x3o3F *b3f   →  E1: (-x)3x3F *b3f   →  E12: o3(-x)3A *b3F
F: f3o3x *b3F   →  F3: f3x3(-x) *b3F   →  F32: F3(-x)3o *b3A
G: o3f3x *b3f   →  G3: o3F3(-x) *b3f
H: f3f3o *b3x   →  H4: f3F3o *b3(-x)
I: x3f3f *b3o   →  I1: (-x)3F3f *b3o
```
A priori invalid combinations:
```A     + B2, B2142, C2, C2132, D42, E12, F32
A2    + B, B214, B24, C, C213, C23, D4, E1, F3, G3, H4
A23   + B214, B24, C2, C21, D4, F, G, H4
A234  + B2, B21, B2142, C2, C21, C2132, D, D42, E12, F, G, H
A2342 + B, B214, C, C213, D4, E1, F3
A24   + B2, B21, C213, C23, D, F3, G3, H
B     + C2, C2132, D42, E12, F32
B2    + C, C21, C213, D4, E1, F3, H4, I1
B21   + C2, C23, D4, E, H4, I
B214  + C2, C2132, C23, D, D42, E, E12, F32, H, I
B2142 + C, C213, D4, E1, F3
B24   + C21, C213, D, E1, H, I1
C     + D42, E12, F32
C2    + D4, E1, F3, G3, I1
C21   + E, F3, G3, I
C213  + D42, E, E12, F, F32, G, I
C2132 + D4, E1, F3
C23   + E1, F, G, I1
```
```D   + H4
D4  + E12, F32, H
D42 + E1, F3
E   + I1
E1  + F32, I
E12 + F3
F   + G3
F3  + G

(all these would provide u edges (→ ‡); thus reducing
from 46655 to 175 other potential combinations only)
```
Other layer-combinations:
```ABCDEF3G3HI:                    VooFxfofx3xxxooxFff3oVofF(-x)(-x)of *b3ooVxfFfxo&#zx → ‡ (f in D)
ABCDE1F3G3HI1:                  VooF(-x)fof(-x)3xxxxoxFfF3oVofF(-x)(-x)of *b3ooVxfFfxo&#zx → †
ABCD4E1F3G3H4I1:                VooF(-x)fof(-x)3xxxxxxFFF3oVofF(-x)(-x)of *b3ooV(-x)fFf(-x)o&#zx → †
ABC21DE1FGHI1:                  Vo(-x)F(-x)fof(-x)3xxooxoffF3oVxfFxxof *b3ooBxfFfxo&#zx → †
ABC21D4E1FGH4I1:                Vo(-x)F(-x)fof(-x)3xxoxxofFF3oVxfFxxof *b3ooB(-x)fFf(-x)o&#zx → †
ABC213DE1F3G3HI1:               Vo(-x)F(-x)fof(-x)3xxxoxxFfF3oV(-x)fF(-x)(-x)of *b3ooBxfFfxo&#zx → †
ABC213D4E1F3G3H4I1:             Vo(-x)F(-x)fof(-x)3xxxxxxFFF3oV(-x)fF(-x)(-x)of *b3ooB(-x)fFf(-x)o&#zx → †
ABC23DEF3G3HI:                  VoxFxfofx3xxoooxFff3oV(-x)fF(-x)(-x)of *b3ooBxfFfxo&#zx → ‡ (f in D)
ABC23D4EF3G3H4I:                VoxFxfofx3xxoxoxFFf3oV(-x)fF(-x)(-x)of *b3ooB(-x)fFf(-x)o&#zx → †
AB21C21DE1FGHI1:                V(-x)(-x)F(-x)fof(-x)3xoooxoffF3oBxfFxxof *b3oxBxfFfxo&#zx → †
AB21C213DE1F3G3HI1:             V(-x)(-x)F(-x)fof(-x)3xoxoxxFfF3oB(-x)fF(-x)(-x)of *b3oxBxfFfxo&#zx → †
AB214CD4E1FGH4I1:               V(-x)oF(-x)fof(-x)3xxxxxofFF3oBofFxxof *b3o(-x)V(-x)fFf(-x)o&#zx → †
AB214CD4E1F3G3H4I1:             V(-x)oF(-x)fof(-x)3xxxxxxFFF3oBofF(-x)(-x)of *b3o(-x)V(-x)fFf(-x)o&#zx → †
AB214C21D4E1FGH4I1:             V(-x)(-x)F(-x)fof(-x)3xxoxxofFF3oBxfFxxof *b3o(-x)B(-x)fFf(-x)o&#zx → †
AB214C213D4E1F3G3H4I1:          V(-x)(-x)F(-x)fof(-x)3xxxxxxFFF3oB(-x)fF(-x)(-x)of *b3o(-x)B(-x)fFf(-x)o&#zx → †
AB24C23D4EF3G3H4I:              VxxFxfofx3xooxoxFFf3oB(-x)fF(-x)(-x)of *b3o(-x)B(-x)fFf(-x)o&#zx → †
A2B2C2DEFGHI:                   BxxFxfofx3(-x)(-x)(-x)ooofff3xBxfFxxof *b3xxBxfFfxo&#zx
A2B2C2DEF32GHI:                 BxxFxFofx3(-x)(-x)(-x)oo(-x)fff3xBxfFoxof *b3xxBxfAfxo&#zx → †
A2B2C2DE12F32GHI:               BxxFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3xBxfAoxof *b3xxBxFAfxo&#zx → †
A2B2C2D42E12F32GHI:             BxxAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3xBxFAoxof *b3xxBoFAfxo&#zx → †
A2B2C2132DEFGHI:                BxoFxfofx3(-x)(-x)(-x)ooofff3xBofFxxof *b3xxCxfFfxo&#zx → †
A2B2C2132DEF32GHI:              BxoFxFofx3(-x)(-x)(-x)oo(-x)fff3xBofFoxof *b3xxCxfAfxo&#zx → †
A2B2C2132DE12FGHI:              BxoFofofx3(-x)(-x)(-x)o(-x)offf3xBofAxxof *b3xxCxFFfxo&#zx → †
A2B2C2132DE12F32GHI:            BxoFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3xBofAoxof *b3xxCxFAfxo&#zx → †
A2B2C2132D42EFGHI:              BxoAxfofx3(-x)(-x)(-x)(-x)oofff3xBoFFxxof *b3xxCofFfxo&#zx → †
A2B2C2132D42EF32GHI:            BxoAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3xBoFFoxof *b3xxCofAfxo&#zx → †
A2B2C2132D42E12FGHI:            BxoAofofx3(-x)(-x)(-x)(-x)(-x)offf3xBoFAxxof *b3xxCoFFfxo&#zx → †
A2B2C2132D42E12F32GHI:          BxoAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3xBoFAoxof *b3xxCoFAfxo&#zx → †
A2B21C21DE12FGHI1:              B(-x)(-x)Fofof(-x)3(-x)ooo(-x)offF3xBxfAxxof *b3xxBxFFfxo&#zx → †
A2B21C21DE12F32GHI1:            B(-x)(-x)FoFof(-x)3(-x)ooo(-x)(-x)ffF3xBxfAoxof *b3xxBxFAfxo&#zx → †
A2B21C21D42E12FGHI1:            B(-x)(-x)Aofof(-x)3(-x)oo(-x)(-x)offF3xBxFAxxof *b3xxBoFFfxo&#zx → †
A2B21C21D42E12F32GHI1:          B(-x)(-x)AoFof(-x)3(-x)oo(-x)(-x)(-x)ffF3xBxFAoxof *b3xxBoFAfxo&#zx → †
A2B21C2132DE12FGHI1:            B(-x)oFofof(-x)3(-x)o(-x)o(-x)offF3xBofAxxof *b3xxCxFFfxo&#zx → †
A2B21C2132DE12F32GHI1:          B(-x)oFoFof(-x)3(-x)o(-x)o(-x)(-x)ffF3xBofAoxof *b3xxCxFAfxo&#zx → †
A2B21C2132D42E12FGHI1:          B(-x)oAofof(-x)3(-x)o(-x)(-x)(-x)offF3xBoFAxxof *b3xxCoFFfxo&#zx → †
A2B21C2132D42E12F32GHI1:        B(-x)oAoFof(-x)3(-x)o(-x)(-x)(-x)(-x)ffF3xBoFAoxof *b3xxCoFAfxo&#zx → †
A2B2142C2DEFGHI:                BoxFxfofx3(-x)(-x)(-x)ooofff3xCxfFxxof *b3xoBxfFfxo&#zx → †
A2B2142C2DEF32GHI:              BoxFxFofx3(-x)(-x)(-x)oo(-x)fff3xCxfFoxof *b3xoBxfAfxo&#zx → †
A2B2142C2DE12FGHI:              BoxFofofx3(-x)(-x)(-x)o(-x)offf3xCxfAxxof *b3xoBxFFfxo&#zx → †
A2B2142C2DE12F32GHI:            BoxFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3xCxfAoxof *b3xoBxFAfxo&#zx → †
A2B2142C2D42EFGHI:              BoxAxfofx3(-x)(-x)(-x)(-x)oofff3xCxFFxxof *b3xoBofFfxo&#zx → †
A2B2142C2D42EF32GHI:            BoxAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3xCxFFoxof *b3xoBofAfxo&#zx → †
A2B2142C2D42E12FGHI:            BoxAofofx3(-x)(-x)(-x)(-x)(-x)offf3xCxFAxxof *b3xoBoFFfxo&#zx → †
A2B2142C2D42E12F32GHI:          BoxAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3xCxFAoxof *b3xoBoFAfxo&#zx → †
A2B2142C21DE12FGHI1:            Bo(-x)Fofof(-x)3(-x)(-x)oo(-x)offF3xCxfAxxof *b3xoBxFFfxo&#zx → †
A2B2142C21DE12F32GHI1:          Bo(-x)FoFof(-x)3(-x)(-x)oo(-x)(-x)ffF3xCxfAoxof *b3xoBxFAfxo&#zx → †
A2B2142C21D42E12FGHI1:          Bo(-x)Aofof(-x)3(-x)(-x)o(-x)(-x)offF3xCxFAxxof *b3xoBoFFfxo&#zx → †
A2B2142C21D42E12F32GHI1:        Bo(-x)AoFof(-x)3(-x)(-x)o(-x)(-x)(-x)ffF3xCxFAoxof *b3xoBoFAfxo&#zx → †
A2B2142C2132DEFGHI:             BooFxfofx3(-x)(-x)(-x)ooofff3xCofFxxof *b3xoCxfFfxo&#zx → †
A2B2142C2132DEF32GHI:           BooFxFofx3(-x)(-x)(-x)oo(-x)fff3xCofFoxof *b3xoCxfAfxo&#zx → †
A2B2142C2132DE12FGHI:           BooFofofx3(-x)(-x)(-x)o(-x)offf3xCofAxxof *b3xoCxFFfxo&#zx → †
A2B2142C2132DE12FGHI1:          BooFofof(-x)3(-x)(-x)(-x)o(-x)offF3xCofAxxof *b3xoCxFFfxo&#zx → †
A2B2142C2132DE12F32GHI:         BooFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3xCofAoxof *b3xoCxFAfxo&#zx → †
A2B2142C2132DE12F32GHI1:        BooFoFof(-x)3(-x)(-x)(-x)o(-x)(-x)ffF3xCofAoxof *b3xoCxFAfxo&#zx → †
A2B2142C2132D42EFGHI:           BooAxfofx3(-x)(-x)(-x)(-x)oofff3xCoFFxxof *b3xoCofFfxo&#zx → †
A2B2142C2132D42EF32GHI:         BooAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3xCoFFoxof *b3xoCofAfxo&#zx → †
A2B2142C2132D42E12FGHI:         BooAofofx3(-x)(-x)(-x)(-x)(-x)offf3xCoFAxxof *b3xoCoFFfxo&#zx → †
A2B2142C2132D42E12FGHI1:        BooAofof(-x)3(-x)(-x)(-x)(-x)(-x)offF3xCoFAxxof *b3xoCoFFfxo&#zx → †
A2B2142C2132D42E12F32GHI:       BooAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3xCoFAoxof *b3xoCoFAfxo&#zx → †
A2B2142C2132D42E12F32GHI1:      BooAoFof(-x)3(-x)(-x)(-x)(-x)(-x)(-x)ffF3xCoFAoxof *b3xoCoFAfxo&#zx → †
A23BC213DE1F3G3HI1:             Bo(-x)F(-x)fof(-x)3oxxxoxFfF3(-x)V(-x)fF(-x)(-x)of *b3xoBxfFfxo&#zx → †
A23B2C2132DEF32G3HI:            BxoFxFofx3o(-x)(-x)oo(-x)Fff3(-x)BofFo(-x)of *b3xxCxfAfxo&#zx → †
A23B2C2132DE12F32G3HI:          BxoFoFofx3o(-x)(-x)o(-x)(-x)Fff3(-x)BofAo(-x)of *b3xxCxFAfxo&#zx → †
A23B2C2132D42EF32G3HI:          BxoAxFofx3o(-x)(-x)(-x)o(-x)Fff3(-x)BoFFo(-x)of *b3xxCofAfxo&#zx → †
A23B2C2132D42E12F32G3HI:        BxoAoFofx3o(-x)(-x)(-x)(-x)(-x)Fff3(-x)BoFAo(-x)of *b3xxCoFAfxo&#zx → †
A23B21C213DE1F3G3HI1:           B(-x)(-x)F(-x)fof(-x)3ooxoxxFfF3(-x)B(-x)fF(-x)(-x)of *b3xxBxfFfxo&#zx → †
A23B21C2132DE12F32G3HI1:        B(-x)oFoFof(-x)3oo(-x)o(-x)(-x)FfF3(-x)BofAo(-x)of *b3xxCxFAfxo&#zx → †
A23B21C2132D42E12F32G3HI1:      B(-x)oAoFof(-x)3oo(-x)(-x)(-x)(-x)FfF3(-x)BoFAo(-x)of *b3xxCoFAfxo&#zx → †
A23B2142C2132DEF32G3HI:         BooFxFofx3o(-x)(-x)oo(-x)Fff3(-x)CofFo(-x)of *b3xoCxfAfxo&#zx → †
A23B2142C2132DE12F32G3HI:       BooFoFofx3o(-x)(-x)o(-x)(-x)Fff3(-x)CofAo(-x)of *b3xoCxFAfxo&#zx → †
A23B2142C2132DE12F32G3HI1:      BooFoFof(-x)3o(-x)(-x)o(-x)(-x)FfF3(-x)CofAo(-x)of *b3xoCxFAfxo&#zx → †
A23B2142C2132D42EF32G3HI:       BooAxFofx3o(-x)(-x)(-x)o(-x)Fff3(-x)CoFFo(-x)of *b3xoCofAfxo&#zx → †
A23B2142C2132D42E12F32G3HI:     BooAoFofx3o(-x)(-x)(-x)(-x)(-x)Fff3(-x)CoFAo(-x)of *b3xoCoFAfxo&#zx → †
A23B2142C2132D42E12F32G3HI1:    BooAoFof(-x)3o(-x)(-x)(-x)(-x)(-x)FfF3(-x)CoFAo(-x)of *b3xoCoFAfxo&#zx → †
A23B2142C23DEF32G3HI:           BoxFxFofx3o(-x)ooo(-x)Fff3(-x)C(-x)fFo(-x)of *b3xoBxfAfxo&#zx → †
A23B2142C23DE12F32G3HI:         BoxFoFofx3o(-x)oo(-x)(-x)Fff3(-x)C(-x)fAo(-x)of *b3xoBxFAfxo&#zx → †
A23B2142C23D42EF32G3HI:         BoxAxFofx3o(-x)o(-x)o(-x)Fff3(-x)C(-x)FFo(-x)of *b3xoBofAfxo&#zx → †
A23B2142C23D42E12F32G3HI:       BoxAoFofx3o(-x)o(-x)(-x)(-x)Fff3(-x)C(-x)FAo(-x)of *b3xoBoFAfxo&#zx → †
A234BC213D4E1F3G3H4I1:          Bo(-x)F(-x)fof(-x)3xxxxxxFFF3(-x)V(-x)fF(-x)(-x)of *b3(-x)oB(-x)fFf(-x)o&#zx → †
A234B214CD4E1F3G3H4I1:          B(-x)oF(-x)fof(-x)3xxxxxxFFF3(-x)BofF(-x)(-x)of *b3(-x)(-x)V(-x)fFf(-x)o&#zx → †
A234B214C213D4E1F3G3H4I1:       B(-x)(-x)F(-x)fof(-x)3xxxxxxFFF3(-x)B(-x)fF(-x)(-x)of *b3(-x)(-x)B(-x)fFf(-x)o&#zx → †
A2342B2C2132DEFGHI:             CxoFxfofx3(-x)(-x)(-x)ooofff3oBofFxxof *b3oxCxfFfxo&#zx → †
A2342B2C2132DEF32GHI:           CxoFxFofx3(-x)(-x)(-x)oo(-x)fff3oBofFoxof *b3oxCxfAfxo&#zx → †
A2342B2C2132DEF32G3HI:          CxoFxFofx3(-x)(-x)(-x)oo(-x)Fff3oBofFo(-x)of *b3oxCxfAfxo&#zx → †
A2342B2C2132DE12FGHI:           CxoFofofx3(-x)(-x)(-x)o(-x)offf3oBofAxxof *b3oxCxFFfxo&#zx → †
A2342B2C2132DE12F32GHI:         CxoFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3oBofAoxof *b3oxCxFAfxo&#zx → †
A2342B2C2132DE12F32G3HI:        CxoFoFofx3(-x)(-x)(-x)o(-x)(-x)Fff3oBofAo(-x)of *b3oxCxFAfxo&#zx → †
A2342B2C2132D42EFGHI:           CxoAxfofx3(-x)(-x)(-x)(-x)oofff3oBoFFxxof *b3oxCofFfxo&#zx → †
A2342B2C2132D42EF32GHI:         CxoAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3oBoFFoxof *b3oxCofAfxo&#zx → †
A2342B2C2132D42EF32G3HI:        CxoAxFofx3(-x)(-x)(-x)(-x)o(-x)Fff3oBoFFo(-x)of *b3oxCofAfxo&#zx → †
A2342B2C2132D42E12FGHI:         CxoAofofx3(-x)(-x)(-x)(-x)(-x)offf3oBoFAxxof *b3oxCoFFfxo&#zx → †
A2342B2C2132D42E12F32GHI:       CxoAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3oBoFAoxof *b3oxCoFAfxo&#zx → †
A2342B2C2132D42E12F32G3HI:      CxoAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)Fff3oBoFAo(-x)of *b3oxCoFAfxo&#zx → †
A2342B21C21DE12FGHI1:           C(-x)(-x)Fofof(-x)3(-x)ooo(-x)offF3oBxfAxxof *b3oxBxFFfxo&#zx → †
A2342B21C21DE12F32GHI1:         C(-x)(-x)FoFof(-x)3(-x)ooo(-x)(-x)ffF3oBxfAoxof *b3oxBxFAfxo&#zx → †
A2342B21C21D42E12FGHI1:         C(-x)(-x)Aofof(-x)3(-x)oo(-x)(-x)offF3oBxFAxxof *b3oxBoFFfxo&#zx → †
A2342B21C21D42E12F32GHI1:       C(-x)(-x)AoFof(-x)3(-x)oo(-x)(-x)(-x)ffF3oBxFAoxof *b3oxBoFAfxo&#zx → †
A2342B21C2132DE12FGHI1:         C(-x)oFofof(-x)3(-x)o(-x)o(-x)offF3oBofAxxof *b3oxCxFFfxo&#zx → †
A2342B21C2132DE12F32GHI1:       C(-x)oFoFof(-x)3(-x)o(-x)o(-x)(-x)ffF3oBofAoxof *b3oxCxFAfxo&#zx → †
A2342B21C2132DE12F32G3HI1:      C(-x)oFoFof(-x)3(-x)o(-x)o(-x)(-x)FfF3oBofAo(-x)of *b3oxCxFAfxo&#zx → †
A2342B21C2132D42E12FGHI1:       C(-x)oAofof(-x)3(-x)o(-x)(-x)(-x)offF3oBoFAxxof *b3oxCoFFfxo&#zx → †
A2342B21C2132D42E12F32GHI1:     C(-x)oAoFof(-x)3(-x)o(-x)(-x)(-x)(-x)ffF3oBoFAoxof *b3oxCoFAfxo&#zx → †
A2342B21C2132D42E12F32G3HI1:    C(-x)oAoFof(-x)3(-x)o(-x)(-x)(-x)(-x)FfF3oBoFAo(-x)of *b3oxCoFAfxo&#zx → †
A2342B2142C2DEFGHI:             CoxFxfofx3(-x)(-x)(-x)ooofff3oCxfFxxof *b3ooBxfFfxo&#zx → †
A2342B2142C2DEF32GHI:           CoxFxFofx3(-x)(-x)(-x)oo(-x)fff3oCxfFoxof *b3ooBxfAfxo&#zx → †
A2342B2142C2DE12FGHI:           CoxFofofx3(-x)(-x)(-x)o(-x)offf3oCxfAxxof *b3ooBxFFfxo&#zx → †
A2342B2142C2DE12F32GHI:         CoxFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3oCxfAoxof *b3ooBxFAfxo&#zx → †
A2342B2142C2D42EFGHI:           CoxAxfofx3(-x)(-x)(-x)(-x)oofff3oCxFFxxof *b3ooBofFfxo&#zx → †
A2342B2142C2D42EFGH4I:          CoxAxfofx3(-x)(-x)(-x)(-x)oofFf3oCxFFxxof *b3ooBofFf(-x)o&#zx → †
A2342B2142C2D42EF32GHI:         CoxAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3oCxFFoxof *b3ooBofAfxo&#zx → †
A2342B2142C2D42EF32GH4I:        CoxAxFofx3(-x)(-x)(-x)(-x)o(-x)fFf3oCxFFoxof *b3ooBofAf(-x)o&#zx → †
A2342B2142C2D42E12FGHI:         CoxAofofx3(-x)(-x)(-x)(-x)(-x)offf3oCxFAxxof *b3ooBoFFfxo&#zx → †
A2342B2142C2D42E12FGH4I:        CoxAofofx3(-x)(-x)(-x)(-x)(-x)ofFf3oCxFAxxof *b3ooBoFFf(-x)o&#zx → †
A2342B2142C2D42E12F32GHI:       CoxAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3oCxFAoxof *b3ooBoFAfxo&#zx → †
A2342B2142C2D42E12F32GH4I:      CoxAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fFf3oCxFAoxof *b3ooBoFAf(-x)o&#zx → †
A2342B2142C21DE12FGHI1:         Co(-x)Fofof(-x)3(-x)(-x)oo(-x)offF3oCxfAxxof *b3ooBxFFfxo&#zx → †
A2342B2142C21DE12F32GHI1:       Co(-x)FoFof(-x)3(-x)(-x)oo(-x)(-x)ffF3oCxfAoxof *b3ooBxFAfxo&#zx → †
A2342B2142C21D42E12FGHI1:       Co(-x)Aofof(-x)3(-x)(-x)o(-x)(-x)offF3oCxFAxxof *b3ooBoFFfxo&#zx → †
A2342B2142C21D42E12FGH4I1:      Co(-x)Aofof(-x)3(-x)(-x)o(-x)(-x)ofFF3oCxFAxxof *b3ooBoFFf(-x)o&#zx → †
A2342B2142C21D42E12F32GHI1:     Co(-x)AoFof(-x)3(-x)(-x)o(-x)(-x)(-x)ffF3oCxFAoxof *b3ooBoFAfxo&#zx → †
A2342B2142C21D42E12F32GH4I1:    Co(-x)AoFof(-x)3(-x)(-x)o(-x)(-x)(-x)fFF3oCxFAoxof *b3ooBoFAf(-x)o&#zx → †
A2342B2142C2132DEFGHI:          CooFxfofx3(-x)(-x)(-x)ooofff3oCofFxxof *b3ooCxfFfxo&#zx → †
A2342B2142C2132DEF32GHI:        CooFxFofx3(-x)(-x)(-x)oo(-x)fff3oCofFoxof *b3ooCxfAfxo&#zx → †
A2342B2142C2132DEF32G3HI:       CooFxFofx3(-x)(-x)(-x)oo(-x)Fff3oCofFo(-x)of *b3ooCxfAfxo&#zx → †
A2342B2142C2132DE12FGHI:        CooFofofx3(-x)(-x)(-x)o(-x)offf3oCofAxxof *b3ooCxFFfxo&#zx → †
A2342B2142C2132DE12FGHI1:       CooFofof(-x)3(-x)(-x)(-x)o(-x)offF3oCofAxxof *b3ooCxFFfxo&#zx → †
A2342B2142C2132DE12F32GHI:      CooFoFofx3(-x)(-x)(-x)o(-x)(-x)fff3oCofAoxof *b3ooCxFAfxo&#zx → †
A2342B2142C2132DE12F32GHI1:     CooFoFof(-x)3(-x)(-x)(-x)o(-x)(-x)ffF3oCofAoxof *b3ooCxFAfxo&#zx → †
A2342B2142C2132DE12F32G3HI:     CooFoFofx3(-x)(-x)(-x)o(-x)(-x)Fff3oCofAo(-x)of *b3ooCxFAfxo&#zx → †
A2342B2142C2132DE12F32G3HI1:    CooFoFof(-x)3(-x)(-x)(-x)o(-x)(-x)FfF3oCofAo(-x)of *b3ooCxFAfxo&#zx → †
A2342B2142C2132D42EFGHI:        CooAxfofx3(-x)(-x)(-x)(-x)oofff3oCoFFxxof *b3ooCofFfxo&#zx → †
A2342B2142C2132D42EFGH4I:       CooAxfofx3(-x)(-x)(-x)(-x)oofFf3oCoFFxxof *b3ooCofFf(-x)o&#zx → †
A2342B2142C2132D42EF32GHI:      CooAxFofx3(-x)(-x)(-x)(-x)o(-x)fff3oCoFFoxof *b3ooCofAfxo&#zx → †
A2342B2142C2132D42EF32GH4I:     CooAxFofx3(-x)(-x)(-x)(-x)o(-x)fFf3oCoFFoxof *b3ooCofAf(-x)o&#zx → †
A2342B2142C2132D42EF32G3HI:     CooAxFofx3(-x)(-x)(-x)(-x)o(-x)Fff3oCoFFo(-x)of *b3ooCofAfxo&#zx → †
A2342B2142C2132D42EF32G3H4I:    CooAxFofx3(-x)(-x)(-x)(-x)o(-x)FFf3oCoFFo(-x)of *b3ooCofAf(-x)o&#zx → †
A2342B2142C2132D42E12FGHI:      CooAofofx3(-x)(-x)(-x)(-x)(-x)offf3oCoFAxxof *b3ooCoFFfxo&#zx → †
A2342B2142C2132D42E12FGHI1:     CooAofof(-x)3(-x)(-x)(-x)(-x)(-x)offF3oCoFAxxof *b3ooCoFFfxo&#zx → †
A2342B2142C2132D42E12FGH4I:     CooAofofx3(-x)(-x)(-x)(-x)(-x)ofFf3oCoFAxxof *b3ooCoFFf(-x)o&#zx → †
A2342B2142C2132D42E12FGH4I1:    CooAofof(-x)3(-x)(-x)(-x)(-x)(-x)ofFF3oCoFAxxof *b3ooCoFFf(-x)o&#zx → †
A2342B2142C2132D42E12F32GHI:    CooAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fff3oCoFAoxof *b3ooCoFAfxo&#zx → †
A2342B2142C2132D42E12F32GHI1:   CooAoFof(-x)3(-x)(-x)(-x)(-x)(-x)(-x)ffF3oCoFAoxof *b3ooCoFAfxo&#zx → †
A2342B2142C2132D42E12F32GH4I:   CooAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)fFf3oCoFAoxof *b3ooCoFAf(-x)o&#zx → †
A2342B2142C2132D42E12F32GH4I1:  CooAoFof(-x)3(-x)(-x)(-x)(-x)(-x)(-x)fFF3oCoFAoxof *b3ooCoFAf(-x)o&#zx → †
A2342B2142C2132D42E12F32G3HI:   CooAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)Fff3oCoFAo(-x)of *b3ooCoFAfxo&#zx → †
A2342B2142C2132D42E12F32G3HI1:  CooAoFof(-x)3(-x)(-x)(-x)(-x)(-x)(-x)FfF3oCoFAo(-x)of *b3ooCoFAfxo&#zx → †
A2342B2142C2132D42E12F32G3H4I:  CooAoFofx3(-x)(-x)(-x)(-x)(-x)(-x)FFf3oCoFAo(-x)of *b3ooCoFAf(-x)o&#zx → †
A2342B2142C2132D42E12F32G3H4I1: CooAoFof(-x)3(-x)(-x)(-x)(-x)(-x)(-x)FFF3oCoFAo(-x)of *b3ooCoFAf(-x)o&#zx → †
A2342B2142C23DEF32G3HI:         CoxFxFofx3(-x)(-x)ooo(-x)Fff3oC(-x)fFo(-x)of *b3ooBxfAfxo&#zx → †
A2342B2142C23DE12F32G3HI:       CoxFoFofx3(-x)(-x)oo(-x)(-x)Fff3oC(-x)fAo(-x)of *b3ooBxFAfxo&#zx → †
A2342B2142C23D42EF32G3HI:       CoxAxFofx3(-x)(-x)o(-x)o(-x)Fff3oC(-x)FFo(-x)of *b3ooBofAfxo&#zx → †
A2342B2142C23D42EF32G3H4I:      CoxAxFofx3(-x)(-x)o(-x)o(-x)FFf3oC(-x)FFo(-x)of *b3ooBofAf(-x)o&#zx → †
A2342B2142C23D42E12F32G3HI:     CoxAoFofx3(-x)(-x)o(-x)(-x)(-x)Fff3oC(-x)FAo(-x)of *b3ooBoFAfxo&#zx → †
A2342B2142C23D42E12F32G3H4I:    CoxAoFofx3(-x)(-x)o(-x)(-x)(-x)FFf3oC(-x)FAo(-x)of *b3ooBoFAf(-x)o&#zx → †
A2342B24C2132D42EFGH4I:         CxoAxfofx3(-x)o(-x)(-x)oofFf3oBoFFxxof *b3o(-x)CofFf(-x)o&#zx → †
A2342B24C2132D42EF32GH4I:       CxoAxFofx3(-x)o(-x)(-x)o(-x)fFf3oBoFFoxof *b3o(-x)CofAf(-x)o&#zx → †
A2342B24C2132D42EF32G3H4I:      CxoAxFofx3(-x)o(-x)(-x)o(-x)FFf3oBoFFo(-x)of *b3o(-x)CofAf(-x)o&#zx → †
A2342B24C2132D42E12FGH4I:       CxoAofofx3(-x)o(-x)(-x)(-x)ofFf3oBoFAxxof *b3o(-x)CoFFf(-x)o&#zx → †
A2342B24C2132D42E12F32GH4I:     CxoAoFofx3(-x)o(-x)(-x)(-x)(-x)fFf3oBoFAoxof *b3o(-x)CoFAf(-x)o&#zx → †
A2342B24C2132D42E12F32G3H4I:    CxoAoFofx3(-x)o(-x)(-x)(-x)(-x)FFf3oBoFAo(-x)of *b3o(-x)CoFAf(-x)o&#zx → †
A24B214CD4E1FGH4I1:             B(-x)oF(-x)fof(-x)3oxxxxofFF3xBofFxxof *b3(-x)(-x)V(-x)fFf(-x)o&#zx → †
A24B214C21D4E1FGH4I1:           B(-x)(-x)F(-x)fof(-x)3oxoxxofFF3xBxfFxxof *b3(-x)(-x)B(-x)fFf(-x)o&#zx → †
A24B2142C2D42EFGH4I:            BoxAxfofx3o(-x)(-x)(-x)oofFf3xCxFFxxof *b3(-x)oBofFf(-x)o&#zx → †
A24B2142C2D42EF32GH4I:          BoxAxFofx3o(-x)(-x)(-x)o(-x)fFf3xCxFFoxof *b3(-x)oBofAf(-x)o&#zx → †
A24B2142C2D42E12FGH4I:          BoxAofofx3o(-x)(-x)(-x)(-x)ofFf3xCxFAxxof *b3(-x)oBoFFf(-x)o&#zx → †
A24B2142C2D42E12F32GH4I:        BoxAoFofx3o(-x)(-x)(-x)(-x)(-x)fFf3xCxFAoxof *b3(-x)oBoFAf(-x)o&#zx → †
A24B2142C21D42E12FGH4I1:        Bo(-x)Aofof(-x)3o(-x)o(-x)(-x)ofFF3xCxFAxxof *b3(-x)oBoFFf(-x)o&#zx → †
A24B2142C21D42E12F32GH4I1:      Bo(-x)AoFof(-x)3o(-x)o(-x)(-x)(-x)fFF3xCxFAoxof *b3(-x)oBoFAf(-x)o&#zx → †
A24B2142C2132D42EFGH4I:         BooAxfofx3o(-x)(-x)(-x)oofFf3xCoFFxxof *b3(-x)oCofFf(-x)o&#zx → †
A24B2142C2132D42EF32GH4I:       BooAxFofx3o(-x)(-x)(-x)o(-x)fFf3xCoFFoxof *b3(-x)oCofAf(-x)o&#zx → †
A24B2142C2132D42E12FGH4I:       BooAofofx3o(-x)(-x)(-x)(-x)ofFf3xCoFAxxof *b3(-x)oCoFFf(-x)o&#zx → †
A24B2142C2132D42E12FGH4I1:      BooAofof(-x)3o(-x)(-x)(-x)(-x)ofFF3xCoFAxxof *b3(-x)oCoFFf(-x)o&#zx → †
A24B2142C2132D42E12F32GH4I:     BooAoFofx3o(-x)(-x)(-x)(-x)(-x)fFf3xCoFAoxof *b3(-x)oCoFAf(-x)o&#zx → †
A24B2142C2132D42E12F32GH4I1:    BooAoFof(-x)3o(-x)(-x)(-x)(-x)(-x)fFF3xCoFAoxof *b3(-x)oCoFAf(-x)o&#zx → †
```
Stott expansion:
(derived CRFs)
```2:A2B2C2DEFGHI: Bxx|Fxf|ofx 3 ooo|xxx|FFF 3 xBx|fFx|xof *b3 xxB|xfF|fxo &#zx
→ CRF with cell list:
24  coes
24  ikes
192 oct
144 pip
288 squippies (J1)
96  thawroes (J92)
96  tricues (J3)
96  trip
```
using here node marks / (pseudo) edge lengths: F=f+x=ff, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x=fff, C=2F=2f+2x, R=A+x=f+3x, S=C+x=2f+3x

in o5o2o5o subsymmetry   (up)
Representation:
```Aooo|Fxox|Vofo|Fofx|xf 5 oAoo|xFxo|oVof|oFxf|xf 2 ooAo|xoFx|ofVo|xfFo|fx 5 oooA|oxxF|fooV|fxoF|fx &#zx  (rox)

with cyclical layer symmetries: A(1234) → B(2143) → C(4312) → D(3421) → A(1234)
E(1234) → F(2143) → G(4312) → H(3421) → E(1234)
I(1234) → J(2143) → K(4312) → L(3421) → I(1234)
M(1234) → N(2143) → O(4312) → P(3421) → M(1234)
Q(1234) → Q(2143) → R(4312) → R(3421) → Q(1234)
```
All layers &
kaleido-facetings per layer:
```A: A5o o5o
B: o5A o5o
C: o5o A5o
D: o5o o5A

E: F5x x5o  → E2: B5(-x) x5o  → E23: B5(-x) (-x)5f
↳ E3: F5x (-x)5f
F: x5F o5x  → F1: (-x)5B o5x  → F14: (-x)5B f5(-x)
↳ F4: x5F f5(-x)
G: o5x F5x  → G2: f5(-x) F5x  → G24: f5(-x) B5(-x)
↳ G4: o5x B5(-x)
H: x5o x5F  → H1: (-x)5f x5F  → H13: (-x)5f (-x)5B
↳ H3: x5o (-x)5B
```
```I: V5o o5f
J: o5V f5o
K: f5o V5o
L: o5f o5V

M: F5o x5f  → M3: F5o (-x)5V
N: o5F f5x  → N4: o5F V5(-x)
O: f5x F5o  → O2: V5(-x) F5o
P: x5f o5F  → P1: (-x)5V o5F

Q: x5x f5f  → Q1: (-x)5F f5f
↳ Q2: F5(-x) f5f
R: f5f x5x  → R3: f5f (-x)5F
↳ R4: f5f F5(-x)
```
A priori invalid combinations:
```E   + G2,G24,H3,H13,M3,O2,Q2,R3
E2  + G,G4,H3,H13,M3,O,Q,R3
E3  + G2,G24,H,H1,M,O2,Q2,R
E23 + G,G4,H,H1,M,O,Q,R
F   + G4,G24,H1,H13,N4,P1,Q1,R4
F1  + G4,G24,H,H3,N4,P,Q,R4
F4  + G,G2,H1,H13,N,P1,Q1,R
F14 + G,G2,H,H3,N,P,Q,R
G   + N4,O2,Q2,R4
G2  + N4,O,Q,R4
G4  + N,O2,Q2,R
G24 + N,O,Q,R
```
```H   + M3,P1,Q1,R3
H1  + M3,P,Q,R3
H3  + M,P1,Q1,R
H13 + M,P,Q,R
M   + R3
M3  + R
N   + R4
N4  + R
O   + Q2
O2  + Q
P   + Q1
P1  + Q
```
Other layer-combinations:
```E2F1G2H1MNO2P1Q1R:        AoooB(-x)f(-x)VofoFoV(-x)(-x)f 5 oAoo(-x)B(-x)foVofoF(-x)VFf 2 ooAoxoFxofVoxfFofx 5 oooAoxxFfooVfxoFfx &#zx
E23F14G24H13M3N4O2P1Q1R3: AoooB(-x)f(-x)VofoFoV(-x)(-x)f 5 oAoo(-x)B(-x)foVofoF(-x)VFf 2 ooAo(-x)fB(-x)ofVo(-x)VFof(-x) 5 oooAf(-x)(-x)BfooVV(-x)oFfF &#zx
```
Stott expansion:
(derived potential CRFs)
```12:E2F1G2H1MNO2P1Q1R: RxxxCoFoBxFxAxBooF 5 xRxxoCoFxBxFxAoBAF 2 ooAoxoFxofVoxfFofx 5 oooAoxxFfooVfxoFfx &#zx → ‡ (F in G)
```
```1234:E23F14G24H13M3N4O2P1Q1R3: RxxxCoFoBxFxAxBooF 5 xRxxoCoFxBxFxAoBAF 2 xxRxoFCoxFBxoBAxFo 5 xxxRFooCFxxBBoxAFA &#zx → ‡ (F in G)
```
using here node marks / (pseudo) edge lengths: F=f+x=ff, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x=fff, C=2F=2f+2x, R=A+x=f+3x

#### EKF of the small prismated icositetrachoron (x3o4o3x)

in o2o3o4o subsymmetry   (up)
Representation:
```oxqwQ 2 qowxx 3 xxooo 4 oxoxo &#zx  (spic)
```
All layers &
kaleido-facetings per layer:
```A: o2q3x4o  → A3: o2w3(-x)4q
B: x2o3x4x  → B1: (-x)2o3x4x  → B13: (-x)2x3(-x)4w  → B132: (-x)2(-x)3o4w
↳ B14: (-x)2o3w4(-x)
↳ B3: x2x3(-x)4w  → (B31 = B13)
↳ B32: x2(-x)3o4w     → (B321 = B132)
↳ B4: x2o3w4(-x)  → (B41 = B14)
C: q2w3o4o
D: w2x3o4x  → D2: w2(-x)3x4x  → D23: w2o3(-x)4w
↳ D24: w2(-x)3w4(-x)
↳ D4: w2x3q4(-x)  → (D42 = D24)
E: Q2x3o4o  → E2: Q2(-x)3x4o  → E23: Q2o3(-x)4q
```
A priori invalid combinations:
```A    + B3,B13,D23,E23
A3   + B,B1,D2,E2
B    + D4,D23,D24,E23
B1   + D4,D23,D24,E23
B3   + D2,D24,E2
B4   + D,D2
B13  + D2,D24,E2
B14  + D,D2
B32  + D,D4,E
B132 + D,D4,E
D    + E2
D2   + E,E23
D4   + E2
D23  + E2
D24  + E   (all giving rise to u edges)

E23   (giving rise to q or w edges within extremal layer)
```
Other layer-combinations:
```ABCD2E2        oxqwQ 2 qow(-x)(-x) 3 xxoxx 4 oxoxo &#zx  → ‡ (dead end at C)
AB1CDE         o(-x)qwQ 2 qowxx 3 xxooo 4 oxoxo &#zx  → ‡ (q in A)
AB1CD2E2       o(-x)qwQ 2 qow(-x)(-x) 3 xxoxx 4 oxoxo &#zx  → †
AB4CD4E        oxqwQ 2 qowxx 3 xwoqo 4 o(-x)o(-x)o &#zx  → ‡ (q in A)
AB4CD24E2      oxqwQ 2 qow(-x)(-x) 3 xwowx 4 o(-x)o(-x)o &#zx  → ‡ (dead end at C)
AB14CD4E       o(-x)qwQ 2 qowxx 3 xwoqo 4 o(-x)o(-x)o &#zx  → ‡ (w in B)
AB14CD24E2     o(-x)qwQ 2 qow(-x)(-x) 3 xwowx 4 o(-x)o(-x)o &#zx  → †
AB32CD2E2      oxqwQ 2 q(-x)w(-x)(-x) 3 xooxx 4 owoxo &#zx  → †
AB32CD24E2     oxqwQ 2 q(-x)w(-x)(-x) 3 xoowx 4 owo(-x)o &#zx  → †
AB132CD24E2    o(-x)qwQ 2 q(-x)w(-x)(-x) 3 xoowx 4 owo(-x)o &#zx  → †
A3B3CDE        oxqwQ 2 wxwxx 3 (-x)(-x)ooo 4 qwoxo &#zx
A3B3CD4E       oxqwQ 2 wxwxx 3 (-x)(-x)oqo 4 qwo(-x)o &#zx  → ‡ (dead end at B)
A3B3CD23E      oxqwQ 2 wxwox 3 (-x)(-x)o(-x)o 4 qwowo &#zx  → †
A3B4CD4E       oxqwQ 2 wowxx 3 (-x)woqo 4 q(-x)o(-x)o &#zx  → ‡ (w in C)
A3B4CD23E      oxqwQ 2 wowox 3 (-x)wo(-x)o 4 q(-x)owo &#zx  → †
A3B13CDE       o(-x)qwQ 2 wxwxx 3 (-x)(-x)ooo 4 qwoxo &#zx  → ‡ (w in C)
A3B13CD4E      o(-x)qwQ 2 wxwxx 3 (-x)(-x)oqo 4 qwo(-x)o &#zx  → ‡ (w in D)
A3B13CD23E     o(-x)qwQ 2 wxwox 3 (-x)(-x)o(-x)o 4 qwowo &#zx  → †
A3B14CD4E      o(-x)qwQ 2 wowxx 3 (-x)woqo 4 q(-x)o(-x)o &#zx  → †
A3B14CD23E     o(-x)qwQ 2 wowox 3 (-x)wo(-x)o 4 q(-x)owo &#zx  → †
```
Stott expansion:
(derived CRFs)
```3:A3B3CDE: oxqwQ 2 wxwxx 3 ooxxx 4 qwoxo &#zx
→ CRF with cell list:
6  coes
16 hips
12 sircoes
12 squobcues (J28)
2  toes
16 tricues (J3)
80 trips

related: oxqw. 2 wxwx. 3 ooxx. 4 qwox. &#zx
→ CRF with cell list:
6  coes
2  gircoes
12 sircoes
12 squacues (J4)
16 tricues (J3)
56 trips
```
using here node marks / (pseudo) edge lengths: Q=w+x=q+u, W=Q+x=w+u
in . o3o4o subsymmetry   (up)
not prismatically symmetric
combinations of formers:
```none
```

in o3o3o *b3o subsymmetry   (up)
Representation:
```qoo 3 xxx 3 oqo *b3ooq &#zx  (spic)

with layer cycle:  A(134) → B(341) → C(413) → A(134)
```
All layers &
kaleido-facetings per layer:
```A: q3x3o *b3o  →  A2: w3(-x)3x *b3x  →  A23: w3o3(-x) *b3x  →  A234: w3x3(-x) *b3(-x)  →  A2342: Q3(-x)3o *b3o
↳  A24: w3o3x *b3(-x)  →  (A243 = A234)           →  (A2432 = A2342)
B: o3x3q *b3o  →  B2: x3(-x)3w *b3x  →  B21: (-x)3o3w *b3x  →  B214: (-x)3x3w *b3(-x)  →  B2142: o3(-x)3Q *b3o
↳  B24: x3o3w *b3(-x)  →  (B241 = B214)           →  (B2412 = B2142)
C: o3x3o *b3q  →  C2: x3(-x)3x *b3w  →  C21: (-x)3o3x *b3w  →  C213: (-x)3x3(-x) *b3w  →  C2132: o3(-x)3o *b3Q
↳  C23: x3o3(-x) *b3w  →  (C231 = C213)           →  (C2312 = C2132)
```
A priori invalid combinations:
```A     + B2,B2142,C2,C2132
A2    + B,B24,B214,C,C23,C213
A23   + B24,B214,C2,C21
A24   + B21,C2,C23,C213
A234  + B2,B21,B2142,C2,C21,C2132
A2342 + B,B214,C,C213
```
```B     + C2,C213
B2    + C,C21,C213
B21   + C2,C23
B24   + C21,C213
B214  + C2,C23,C2132
B2142 + C,C213
```
Other layer-combinations:
```ABC21           qo(-x) 3 xxo 3 oqx *b3 oow &#zx  → †
ABC23           qox 3 xxo 3 oq(-x) *b3 oow &#zx  → †
AB21C21         q(-x)(-x) 3 xoo 3 owx *b3 oxw &#zx  → †
AB21C213        q(-x)(-x) 3 xox 3 ow(-x) *b3 oxw &#zx  → †
AB24C23         qxx 3 xoo 3 ow(-x) *b3 o(-x)w &#zx  → †
AB214C21        q(-x)(-x) 3 xxo 3 owx *b3 o(-x)w &#zx  → †
AB214C213       q(-x)(-x) 3 xxx 3 ow(-x) *b3 o(-x)w &#zx  → †
A2B2C2          wxx 3 (-x)(-x)(-x) 3 xwx *b3 xxw &#zx
A2B2C2132       wxo 3 (-x)(-x)(-x) 3 xwo *b3 xxQ &#zx  → †
A2B21C21        w(-x)(-x) 3 (-x)oo 3 xwx *b3 xxw &#zx  → †
A2B21C2132      w(-x)o 3 (-x)o(-x) 3 xwo *b3 xxQ &#zx  → †
A2B2142C21      wo(-x) 3 (-x)(-x)o 3 xQx *b3 xow &#zx  → †
A2B2142C2132    woo 3 ( -x)(-x)(-x) 3 xQo *b3 xoQ &#zx  → †
A23B21C213      w(-x)(-x) 3 oox 3 (-x)w(-x) *b3 xxw &#zx  → †
A23B21C2132     w(-x)o 3 oo(-x) 3 (-x)wo *b3 xxQ &#zx  → †
A23B2142C23     wox 3 o(-x)o 3 (-x)Q(-x) *b3 xow &#zx  → †
A23B2142C2132   woo 3 o(-x)(-x) 3 (-x)Qo *b3 xoQ &#zx  → †
A24B2142C2132   woo 3 o(-x)(-x) 3 xQo *b3 (-x)oQ &#zx  → †
A234B214C213    w(-x)(-x) 3 xxx 3 (-x)w(-x) *b3 (-x)(-x)w &#zx  → †
A2342B2142C2132 Qoo 3 (-x)(-x)(-x) 3 oQo *b3 ooQ &#zx  → †
```
Stott expansion:
(derived CRFs)
```1:ABC wxx 3 xxx 3 oqo *b3 ooq &#zx (owauprit)
→ CRF with cell list:
24  esquidpies (J15)
32  hips
8   octs
160 trips
16  tuts

related: .xx 3 .xx 3 .qo *b3 .oq &#zx
= Wythoffian x3x3o4x (prit) with cell list:
24 cubes
32 hips
8  sircoes
16 tuts
```
```(-2):ABC qoo 3 ooo 3 oqo *b3 ooq &#zx
= Wythoffian x3o4o3o (ico) with cell list:
24 octs

related: .oo 3 .oo 3 .qo *b3 .oq &#zx
= Wythoffian o3o3o4x (tes) with cell list:
8 cubes
```
```1(-2):ABC wxx 3 ooo 3 oqo *b3 ooq &#zx (poxic)
→ CRF with cell list:
24 esquidpies (J15)
16 tets
32 trips

related: .xx 3 .oo 3 .qo *b3 .oq &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```
```13:ABC wxx 3 xxx 3 xwx *b3 ooq &#zx (poc prico)
→ CRF with cell list:
64  hips
24  squobcues (J28)
8   toes
128 trips
16  tuts

related: wx. 3 xx. 3 xw. *b3 oo. &#zx
= Wythoffian o3x3x4x (grit) with cell list:
8  gircoes
32 trips
16 tuts
```
```1(-2)3:ABC wxx 3 ooo 3 xwx *b3 ooq &#zx (pocsric)
→ CRF with cell list:
8  coes
24 squobcues (J28)
16 tets
64 trips

related: wx. 3 oo. 3 xw. *b3 oo. &#zx
= Wythoffian x3o3x4x (tat) with cell list:
16 tets
8  tics
```
```134:ABC wxx 3 xxx 3 xwx *b3 xxw &#zx
= Wythoffian x3x4o3x (prico) with cell list:
96 hips
24 sircoes
24 toes
96 trips

related: .xx 3 .xx 3 .wx *b3 .xw &#zx
= Wythoffian x3x3x4x (gidpith) with cell list:
8  gircoes
32 hips
24 ops
16 toes
```
```1(-2)34:ABC wxx 3 ooo 3 xwx *b3 xxw &#zx
= Wythoffian x3o4x3o (srico) with cell list:
24 coes
24 sircoes
6  trips

related: wx. 3 oo. 3 xw. *b3 xx. &#zx
= Wythoffian x3o3x4x (proh) with cell list:
16 coes
24 ops
8  tics
32 trips
```
```2:A2B2C2 wxx 3 ooo 3 xwx *b3 xxw &#zx
= Wythoffian x3o4x3o (srico) with cell list:
24 coes
24 sircoes
6  trips

related: wx. 3 oo. 3 xw. *b3 xx. &#zx
= Wythoffian x3o3x4x (proh) with cell list:
16 coes
24 ops
8  tics
32 trips
```
```(-1)2:A2B2C2 qoo 3 ooo 3 xwx *b3 xxw &#zx (pocsric)
→ CRF with cell list:
8  coes
24 squobcues (J28)
16 tets
64 trips

related: .oo 3 .oo 3 .wx *b3 .xw &#zx
= Wythoffian x3o3x4x (tat) with cell list:
16 tets
8  tics
```
```(-1)2(-3):A2B2C2 qoo 3 ooo 3 oqo *b3 xxw &#zx (poxic)
→ CRF with cell list:
24 esquidpies (J15)
16 tets
32 trips

related: qo. 3 oo. 3 oq. *b3 xx. &#zx
= Wythoffian x3o3o4x (sidpith) with cell list:
32 cubes
16 tets
32 trips
```
```(-1)2(-3)(-4):A2B2C2 qoo 3 ooo 3 oqo *b3 ooq &#zx
= Wythoffian x3o4o3o (ico) with cell list:
24 octs

related: .oo 3 .oo 3 .qo *b3 .oq &#zx
= Wythoffian o3o3o4x (tes) with cell list:
8 cubes
```

using here node marks / (pseudo) edge lengths: Q=w+x=q+u, W=Q+x=w+u

#### EKF of the snub (dis)icositetrachoron (s3s4o3o)

in o3o3o *b3o subsymmetry   (up)
Representation:
```fox 3 ooo 3 xfo *b3 oxf &#zx  (sadi)

with layer cycle:  A(1234) → B(3241) → C(4213) → A(1234)
```
All layers &
kaleido-facetings per layer:
```A: f3o3x *b3o  →  A3: f3x3(-x) *b3o  →  A32: F3(-x)3o *b3x  →  A324: F3o3o *b3(-x)
B: o3o3f *b3x  →  B4: o3x3f *b3(-x)  →  B42: x3(-x)3F *b3o  →  B421: (-x)3o3F *b3o
C: x3o3o *b3f  →  C1: (-x)3x3o *b3f  →  C12: o3(-x)3x *b3F  →  C123: o3o3(-x) *b3F
```
A priori invalid combinations:
```A    + C123
A3   + B42,C12
A32  + B4,C1
A324 + B
B4   + C12
B42  + C1
B421 + C   (all giving rise to u edges)
```
Other layer-combinations:
```ABC1         fo(-x)3oox3xfo *b3oxf&#zx
ABC12        foo3oo(-x)3xfx *b3oxF&#zx → ‡ (f in A)
AB4C1        fo(-x)3oxx3xfo *b3o(-x)f&#zx → †
AB42C12      fxo3o(-x)(-x)3xFx *b3ooF&#zx → †
AB421C1      f(-x)(-x)3oox3xFo *b3oof&#zx → †
AB421C12     f(-x)o3oo(-x)3xFx *b3ooF&#zx → †
A3B4C1       fo(-x)3xxx3(-x)fo *b3o(-x)f&#zx → †
A3B4C123     foo3xxo3(-x)f(-x) *b3o(-x)F&#zx → †
A3B421C123   f(-x)o3xoo3(-x)F(-x) *b3ooF&#zx → †
A32B42C12    Fxo3(-x)(-x)(-x)3oFx *b3xoF&#zx → †
A32B42C123   Fxo3(-x)(-x)o3oF(-x) *b3xoF&#zx → †
A32B421C123  F(-x)o3(-x)oo3oF(-x) *b3xoF&#zx → †
A324B421C123 F(-x)o3ooo3oF(-x) *b3(-x)oF&#zx → †
```
Stott expansion:
(derived CRFs)
```1:ABC1: Fxo 3 oox 3 xfo *b3 oxf &#zx (pretasto)
→ CRF with cell list:
24 bilbiroes (J91)
8  coes
40 octs
32 teddies (J63)
40 tets
```
```2:ABC: fox 3 xxx 3 xfo *b3 oxf &#zx (prissi)
→ CRF with cell list:
24 ikes
96 tricues (J3)
96 trips
24 tuts
```
using here node marks / (pseudo) edge lengths: F=f+x

in o2o2o2o subsymmetry   (up)
Representation:
```ooo|xxx|fff|FFF 2 Fxf|oFf|xFo|fxo 2 xfF|Ffo|Fox|xof 2 fFx|foF|oxF|ofx &#zx  (sadi)

with layer cycles:
(ABCDEFGHIJKL)(1234) → (KFHAJIELBGDC)(3124) → (ILDGBKCFJAHE)(2431) → (BCAEFDHIGKLJ)(1423) → (JEGCLHDKAIFB)(4132) → (GJEHCLADKBIF)(4321) → (ABCDEFGHIJKL)(4321)
```
All layers &
kaleido-facetings per layer:
```A: o2F2x2f  →  A3: o2F2(-x)2f
B: o2x2f2F  →  B2: o2(-x)2f2F
C: o2f2F2x  →  C4: o2f2F2(-x)

D: x2o2F2f  →  D1: (-x)2o2F2f
E: x2F2f2o  →  E1: (-x)2F2f2o
F: x2f2o2F  →  F1: (-x)2f2o2F

G: f2x2F2o  →  G2: f2(-x)2F2o
H: f2F2o2x  →  H4: f2F2o2(-x)
I: f2o2x2F  →  I3: f2o2(-x)2F

J: F2f2x2o  →  J3: F2f2(-x)2o
K: F2x2o2f  →  K2: F2(-x)2o2f
L: F2o2f2x  →  L4: F2o2f2(-x)
```
A priori invalid combinations:
```A  + I3,J3
A3 + I,J
B  + G2,K2
B2 + G,K
C  + H4,L4
C4 + H,L
D  + E1,F1
D1 + E,F
```
```E  + F1
E1 + F
G  + K2
G2 + K
H  + L4
H4 + L
I  + J3
I3 + J

(all giving rise to u edges)
```
Other layer-combinations:
```ABCD1E1F1GHIJKL          ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
ABC4D1E1F1GH4IJKL4       ooo(-x)(-x)(-x)fffFFF 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx → ‡ (extremal f-edge DG)
AB2C4D1E1F1G2H4IJK2L4    ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 xfFFfoFoxxof 2 fF(-x)foFo(-x)Fof(-x) &#zx → †
A3B2C4D1E1F1G2H4I3J3K2L4 ooo(-x)(-x)(-x)fffFFF 2 F(-x)foFf(-x)Fof(-x)o 2 (-x)fFFfoFo(-x)(-x)of 2 fF(-x)foFo(-x)Fof(-x) &#zx → †
```
Stott expansion:
(derived CRFs)
```1:ABCD1E1F1GHIJKL: xxxoooFFFAAA 2 FxfoFfxFofxo 2 xfFFfoFoxxof 2 fFxfoFoxFofx &#zx
→ CRF with cell list:
6  bilbiroes (J91)
2  ikes
36 squippies (J1)
40 teddies (J63)
36 tets
8  trips
```

using here node marks / (pseudo) edge lengths: F=f+x, A=f+u=f+2x

#### EKF of the rectified tesseract (o3o3x4o)

in o2o3o4o subsymmetry   (up)
Representation:
```qo 2 oo 3 xo 4 oq &#zx  (rit)
```
All layers &
kaleido-facetings per layer:
```A: q2o3x4o  →  A3: q2x3(-x)4q
B: o2o3o4q
```
A priori invalid combinations:
```neither can A+A3 be applied on either side (providing extremal u-edges)
(which thus excludes any mere . o3o4o symmetry)

nor can A3 be applied without an additional applied partial q-contraction
in the last node (else providing extremal q-edges)

even when using A only, then node 1 and node 4 still has to correspond,
in order not to produce non-regular hexagons (i.e. diagonally elongated
squares) there
```
Stott expansion:
(derived potential CRFs)
```2:AB qo 2 xx 3 xo 4 oq &#zx (pabdirico)
→ CRF with cell list:
6  coes
12 cubes
2  toes
16 tricues (J3)
```
```14:AB wx 2 oo 3 xo 4 xw &#zx
= Wythoffian o3o3x4x (tat) with cell list:
16 tets
8  tics
```
```124:AB wx 2 xx 3 xo 4 xw &#zx (pabdiproh)
→ CRF with cell list:
2  gircoes
12 ops
6  tics
16 tricues (J3)
8  trips
```
```3(-q4):A3B qo 2 xo 3 ox 4 oo &#zx
= Wythoffian x3o4o3o (ico) with cell list:
24 octs
```
```13(-q4):A3B wx 2 xo 3 ox 4 oo &#zx (pexic)
→ CRF with cell list:
6  esquidpies (J15)
18 octs
8  trips
```
```34(-q4):A3B qo 2 xo 3 ox 4 xx &#zx (pacsrit)
→ CRF with cell list:
16 octs
2  sircoes
6  squobcues (J28)
24 trips
```
```134(-q4):A3B wx 2 xo 3 ox 4 xx &#zx
= Wythoffian o3x3o4x (srit) with cell list:
16 octs
8  sircoes
32 trips
```

in . o3o3o subsymmetry   (up)
Representation:
```ooxx 3 oxxo 3 xxoo &#xt  (rit)

with inversive top-down-symmetry
```
All layers &
kaleido-facetings per layer:
```A: o3o3x  →  A3: o3x3(-x)  →  A32: x3(-x)3o  →  A321 (-x)3o3o
B: o3x3x  →  B2: x3(-x)3u  →  B21: (-x)3o3u
↳  B3: o3u3(-x)
C: x3x3o  →  C1: (-x)3u3o
↳  C2: u3(-x)3x  →  C23: u3o3(-x)
D: x3o3o  →  D1: (-x)3x3o  →  D12: o3(-x)3x  →  D123: o3o3(-x)
```
A priori invalid combinations:
```A    + B3,C23,D123
A3   + B,B2,B21,C2,D12
A32  + B,B21,B3,C,C1,D1
A321 + B2,C,C2,C23,D
B    + C2,C23,D12,D123
B2   + C,C1,C23,D1,D123
B21  + C,C2,C23,D,D123
B3   + C2,D12
C    + D1,D12
C1   + D,D12
C2   + D1,D123
C23  + D1,D12,D123
```
Other layer-combinations:
```ABC1D1        oo(-x)(-x) 3 oxux 3 xxoo &#xt
AB2C2D        oxux 3 o(-x)(-x)o 3 xuxo &#xt
AB2C2D12      oxuo 3 o(-x)(-x)(-x) 3 xuxx &#xt → †
AB21C1D1      o(-x)(-x)(-x) 3 ooux 3 xuoo &#xt → †
A3B3CD123     ooxo 3 xuxo 3 (-x)(-x)o(-x) &#xt → †
A3B3C1D1      oo(-x)(-x) 3 xuux 3 (-x)(-x)oo &#xt
A3B3C1D123    oo(-x)o 3 xuuo 3 (-x)(-x)o(-x) &#xt → †
A32B2C2D12    xxuo 3 (-x)(-x)(-x)(-x) 3 ouxx &#xt → †
A321B3C1D123  (-x)o(-x)o 3 ouuo 3 o(-x)o(-x) &#xt → †
```
Stott expansion:
(derived potential CRFs)
```1:ABC1D1 xxoo 3 oxux 3 xxoo &#xt
→ CRF with cell list:
1 co
6 cubes
1 oct
6 squippies (J1)
8 tricues (J3)
8 tuts

related: .xoo 3 .xux 3 .xoo &#xt (octum)
→ CRF with cell list:
1 oct
6 squippies (J1)
1 toe
8 tuts
```
```2:AB2C2D oxux 3 xoox 3 xuxo &#xt
= Wythoffian x3x3o4o (thex) with cell list:
8  octs
16 tuts
```
```1(-2)3:A3B3C1D1 xxoo 3 oxxo 3 ooxx &#xt
= Wythoffian o3o3x4o (rit) with cell list:
8  coes
16 tets

related: .xo. 3 .xx. 3 .ox. &#x (tuta)
→ CRF (segmentochoron) with cell list:
6 tets
8 tricues (J3)
2 tuts
```