## Expanded Kaleido-Facetings based on the Icositetrachoron

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Expanded Kaleido-Facetings based on the Icositetrachoron

Still being in the run of the research for those based on the hexacosachoron (cf. this thread) but meanwhile thought about applying these ideas, which already where discussed to be published (cf. this thread), onto some different starting figure as well.

My choice fell onto the icositetrachoron x3o4o3o.
More just for a trial I chose its subsymmetry o2o3o4o. Then its according decomposition is
x3o4o3o = q2x3o4o + o2o3x4o = qo2xo3ox4oo&#zx

Considering this subsymmetry, resp. that representation one derives the following layer-wise kaleido facetings:
Code: Select all
`A: q2x3o4o  >  A2: q2(-x)3x4o  >  A23: q2o3(-x)4qB: o2o3x4o  >  B3: o2x3(-x)4q  >  B32: o2(-x)3o4q`

When intending to use these for a CRF research by means of an according partial Stott expansion of such kaleido facetings of the whole starting figure, we get the following a priori restrictions:
• A23 generally produces non-unit edges at an extremal layer, so these will contradict to CRF.
• B32, when using neither A2 nor A23, will produce inner-layer edges of size u = 2x.
• The combination A2 + B3 will do that likewise.

The so far mere combinatorical output here then would run like this:
Code: Select all
`+ooo  -:      wx2xo3ox4oo&#zx = pexico+oo  A2:     qo2ox3xx4oo&#zx      A2B32:  qo2oo3xo4oq&#zx = ritoo+o  B3:     qo2xx3xo4oq&#zxooo+  -:      qo2xo3ox4xx&#zx = pacsrit++oo  A2:     wx2ox3xx4oo&#zx      A2B32:  wx2oo3xo4oq&#zx+o+o  B3:     wx2xx3xo4oq&#zx+oo+  -:      wx2xo3ox4xx&#zx = srito++oo+o+  A2:     qo2ox3xx4xx&#zx      A2B32:  qo2oo3xo4xw&#zxoo++  B3:     qo2xx3xo4xw&#zx+++o++o+  A2:     wx2ox3xx4xx&#zx      A2B32:  wx2oo3xo4xw&#zx = tat+o++  B3:     wx2xx3xo4xw&#zxo+++++++`

Finally we have to consider all these possibilities individually.

wx2xo3ox4oo&#zx = oct || co || co || oct
is a true multistratic polychoron, but formerly known as pexic
Cells being: 2+16 octs, 6 esquidpies (J15), 8 trips.

qo2ox3xx4oo&#zx = co || toe || co
is a mere stack of segmentochora, thus close to be known too.
Cells being: 2 coes, 16 tricues (J3), 12 cubes.

qo2oo3xo4oq&#zx = co || q-cube || co
is nothing but rit = o3o3x4o.
Cells being: 2+6 coes, 16 tets.

qo2xx3xo4oq&#zx = toe || x,q-sirco || toe
is an axis-orthogonal partial Stott expanded rit.
Cells being: 2 toes, 12 cubes, 6 coes, 16 tricues (J3).

qo2xo3ox4xx&#zx = sirco || tic || sirco
is a true multistratic polychoron, but formerly known as pacsrit
Cells being: 2 sircoes, 6 squobcues (J28), 16 octs, 24 trips.

wx2ox3xx4oo&#zx = co || toe || toe || co
is a mere stack of segmentochora, thus close to be known too.
Cells being: 2 coes, 16 tricues (J3), 12+6 cubes, 8 hips.

wx2oo3xo4oq&#zx = co || q-cube || q-cube || co
asks for non-regular hexagons: wx .. .. oq&#zx (resp. "4-fold axially elongated co"), i.e. not a CRF!

wx2xx3xo4oq&#zx = toe || x,q-sirco || x,q-sirco || toe
asks for non-regular hexagons: wx .. .. oq&#zx (resp. "4-fold axially elongated co"), i.e. not a CRF!

wx2xo3ox4xx&#zx = sirco || tic || tic || sirco
is nothing but srit = o3x3o4x.
Cells being: 2+6 sircoes, 16 octs, 24+8 trips.

qo2ox3xx4xx&#zx = tic || girco || tic
is a mere stack of segmentochora, thus close to be known too.
Cells being: 2 tics, 16 tricues (J3), 24 trips, 12 ops.

qo2oo3xo4xw&#zx = tic || w-cube || tic
asks for non-regular hexagons: qo .. .. xw&#zx (resp. "4-fold axially contracted tic"), i.e. not a CRF!

qo2xx3xo4xw&#zx = girco || x,w-sirco || girco
asks for non-regular hexagons: qo .. .. xw&#zx (resp. "4-fold axially contracted tic"), i.e. not a CRF!

wx2ox3xx4xx&#zx = tic || girco || girco || tic
is a mere stack of segmentochora, thus close to be known too.
Cells being: 2 tics, 16 tricues (J3), 24 trips, 12 ops, 8 hips, 12 cubes.

wx2oo3xo4xw&#zx = tic || w-cube || w-cube || tic
is nothing but tat = o3o3x4x.
Cells being: 2+6 tics, 16 tets.

wx2xx3xo4xw&#zx = girco || x,w-sirco || x,w-sirco || girco
is a true multistratic polychoron.
Cells being: 2 gircoes, 12 ops, 6 tics, 16 tricues (J3), 8 trips.

Thus, even so several turn out to be known polychora, this post at least shows that these techniques are applicable here too, and indeed do produce non-empty output. - And in the sequel here further subsymmetries of the icositerachoron might get considered resp. elaborated similarily ...

--- rk
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Klitzing wrote:My choice fell onto the icositetrachoron x3o4o3o.
More just for a trial I chose its subsymmetry o2o3o4o.

...

And in the sequel here further subsymmetries of the icositerachoron might get considered resp. elaborated similarily ...

Today I'll present o3o3o4o subsymmetry.
Then its according decomposition is
x3o4o3o = q3o3o4o + o3o3o4x = qo3oo3oo4ox&#zx

Considering this subsymmetry, resp. that representation one derives the following layer-wise kaleido facetings:
Code: Select all
`A: q3o3o4o B: o3o3o4x  >  B4: o3o3q4(-x)`

(Sure, not too many possibilities here, I'd agree.)

The combinatorical output here then would run like this:
Code: Select all
`+ooo  -:      wx 3 oo 3 oo 4 ox &#zxo+oo  -:      qo 3 xx 3 oo 4 ox &#zxoo+o  -:      qo 3 oo 3 xx 4 ox &#zxooo+  B4:     qo 3 oo 3 oq 4 xo &#zx++oo  -:      wx 3 xx 3 oo 4 ox &#zx+o+o  -:      wx 3 oo 3 xx 4 ox &#zx+oo+  B4:     wx 3 oo 3 oq 4 xo &#zxo++o  -:      qo 3 xx 3 xx 4 ox &#zxo+o+  B4:     qo 3 xx 3 oq 4 xo &#zxoo++  B4:     qo 3 oo 3 xw 4 xo &#zx+++o  -:      wx 3 xx 3 xx 4 ox &#zx++o+  B4:     wx 3 xx 3 oq 4 xo &#zx+o++  B4:     wx 3 oo 3 xw 4 xo &#zxo+++  B4:     qo 3 xx 3 xw 4 xo &#zx++++  B4:     wx 3 xx 3 xw 4 xo &#zx`

Finally we have to consider all these possibilities individually.

wx 3 oo 3 oo 4 ox &#zx is formerly known as poxic.
Cells being: 24 esquidpies (J15), 16 tets, 32 trips.

qo 3 xx 3 oo 4 ox &#zx = Wythoffian x3o4o3x (spic).
Cells being: 48 octs, 192 trips.

qo 3 oo 3 xx 4 ox &#zx is formerly known as pocsric.
Cells being: 8 coes, 24 squobcues (J28), 16 tets, 64 trips.

qo 3 oo 3 oq 4 xo &#zx = Wythoffian o3x4o3o (rico).
Cells being: 24 coes, 24 cubes.

wx 3 xx 3 oo 4 ox &#zx is formerly known as owau prit.
Cells being: 24 esquidpies (J15), 32 hips, 8 octs, 160 trips, 16 tuts.

wx 3 oo 3 xx 4 ox &#zx = Wythoffian x3o4x3o (srico).
Cells being: 24 coes, 24 sircoes, 96 trips.

wx 3 oo 3 oq 4 xo &#zx
asks for non-regular hexagons: wx .. oq ..&#zx (resp. "4-fold axially elongated co"), i.e. not a CRF!

qo 3 xx 3 xx 4 ox &#zx is formerly known as poc prico.
Cells being: 64 hips, 24 squobcues (J28), 8 toes, 128 trips, 16 tuts.

qo 3 xx 3 oq 4 xo &#zx = Wythoffian x3o4x3o (srico),
again (cf. above).

qo 3 oo 3 xw 4 xo &#zx
asks for non-regular hexagons: qo .. xw ..&#zx (resp. "4-fold axially contracted tic"), i.e. not a CRF!

wx 3 xx 3 xx 4 ox &#zx = Wythoffian x3o4x3o (prico).
Cells being: 96 hips, 24 sircoes, 24 toes, 96 trips.

wx 3 xx 3 oq 4 xo &#zx
asks for non-regular hexagons: wx .. oq ..&#zx (resp. "4-fold axially elongated co"), i.e. not a CRF!

wx 3 oo 3 xw 4 xo &#zx = Wythoffian o3x4x3o (cont).
Cells being: 48 tics.

qo 3 xx 3 xw 4 xo &#zx
asks for non-regular hexagons: qo .. xw ..&#zx (resp. "4-fold axially contracted tic"), i.e. not a CRF!

wx 3 xx 3 xw 4 xo &#zx = Wythoffian x3x4x3o (grico).
Cells being: 24 gircoes, 24 tics, 96 trips.

Sadly, even so being applicable here as well, we do not get new CRFs in this subsymmetry.

--- rk
Klitzing
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Thought about these Expanded Kaleido-Facetings a bit.

When applied to lace towers then the main restriction wrt. the lacing edges, i.e. restricting them to remain unit sized - and for sure, by construction, keeping the hyperplane distances of the vertex layers - can be deduced from the lacing 2D faces!
• For lacing triangles we have no restrictions wrt. to the quirks (edge reversals).
• For lacing x||x squares we restrict wrt. quirks (edge reversals) to like ones in both parallel sides only.
• For any other lacing polygon no quirks (edge reversals) can be applied, as these necessarily would result in some cross-sectioning lines becoming the new lacing edges, which then surely are different from the asked for unit size.
• And wrt. to the then following partial Stott expansions we most often have to disallow any lacing polygon to be elongated in its face plane, as it then either will get somewhere some doubled up side lengths, or at least would no longer be a regular polygon any more. (E.g. qo2oq&#zx = oqo&#xt becomes qo2xw&#zx = xwx&#xt, which still is unit sided, but then no longer would be a regular polygon (CRF)! (It then obviously would become a 90°-135°-135°-90°-135°-135°-hexagon.) Otoh. its double expansion into wx2xw&#zx = xwwx&#xt becomes a regular octagon again.)

This is why we know of applications of Expanded Kaleido-Facetings wrt. the icosahedron, to the hexacosachoron, and from this thread also to the icositetrachoron. But that outside of those it often has no success.

But, having this insight now, we well could apply EKFs e.g. to the hexadecachoron, to the pentachoron, and even to several other Wythoffians which are bounded by triangles and squares only. - The above observation not even totally disallows its application beyond! It just asks that those other lacing polygons (up to some exceptions) have to be maintained as such in the whole process (but e.g. well can be displaced).

And, for sure, we just have considered lace tower (= ...&#xt) subsymmetric displays of the starting figures in the above. Nothing was said so far about tegmatic sum (= ...&#zx) subsymmetric displays here. But as long as we are searching for CRFs as resulting outcomes, similar restrictions should be found there too.

--- rk
Klitzing
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Klitzing wrote:But, having this insight now, we well could apply EKFs e.g. to the hexadecachoron, to the pentachoron, and even to several other Wythoffians which are bounded by triangles and squares only.

Tried its application onto x3o3x3o (srip = small rhombated pentachoron), as a first non-regular starting figure.
Considered . o3o3o subsymmetry.
In this subsymmetry it can be rewritten as oxx3xxo3oox&#xt = o3x3o || x3x3o || x3o3x.

Then we have the following quirks:
Code: Select all
`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`

But all of the following combinations can be disallowed a priori:
Code: Select all
`A     + B2,B23A     + C12,C32A2    + B,B1,B23A2    + C1,C3,C13A21   + B,B1,B2,B23 → generallyA23   + B1,B2A23   + C,C1,C12A213  + B,B2,B23A213  + C,C1,C3,C12,C32A2132 + B,B1,B2,B23 → generallyB     + C1,C12,C13,C32B1    + C,C3,C12,C32B2    + C1,C3,C12,C32B23   + C,C1,C12,C13C12   generally (u in extremal layer)C13   generally (u in extremal layer)C32   generally (u in extremal layer)`

So we are left with combinatorical cases:
Code: Select all
`B1C1:     o(-x)(-x) 3 xux 3 oox &#xtA2B2:     xux 3 (-x)(-x)o 3 xxx &#xtC3:       oxx 3 xxx 3 oo(-x) &#xtA23C3:    xxx 3 oxx 3 (-x)o(-x) &#xt  → †)A23B23C3: xux 3 oox 3 (-x)(-x)(-x) &#xt`

where the dagger points out, that this case cannot be connected by unit lacing, and thus will have to be rejected as well.

The outcome then is not too surprising so:
• 1:B1C1: xoo 3 xux 3 oox &#xt is nothing but deca (o3x3x3o, decachoron).
• 2:A2B2: xux 3 oox 3 xxx &#xt then is a true CRF with cells: 1 co + 6 hips + 1 toe + 4 tricues + 4 trips + 4 tuts.
• 3:C3: oxx 3 xxx 3 xxo &#xt also is a CRF with cells: 8 hips + 8 tricues + 12 trips + 2 tuts, but that one comes out as a bistratic segmentochoral stack.
• 3:A23B23C3: xux 3 oox 3 ooo &#xt is nothing but tip (x3x3o3o, truncated pentachoron).

Even so it shows its applicability!

--- rk
Klitzing
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Recently I was considering a further non-regular starting figure:
The I had chosen spic = x3o4o3x.
For a first subsymmetry to look at I then took o2o3o4o.

Under this subsymmetry spic then reads
o2q3x4o + x2o3x4x + q2w3o4o + w2x3o4x + Q2x3o4o = oxqwQ2qowxx3xxooo4oxoxo&#zx
(using here and in what follows: Q=w+x=q+u=q+2x, i.e representing an edge of size 2+sqrt(2)).

Next I considered the possible quirks:
Code: Select all
`A: o2q3x4o  -> A3: o2w3(-x)4qB: 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: q2w3o4oD: 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`

That is we have in total 2.8.1.5.3 = 240 combinations.

Next we would try to cancel out impossible ones. As we further-on will apply Stott expansions in order to get rid of negative nodes (for sure, then being equally applied to all 5 layers the same), there cannot be mixed symbols with "x" and "(-x)" at the same Position, else These would provide non-CRF edges of size u (=2x). Thus a priori forbidden combis are here:
Code: Select all
`A    + B3,B13,D23,E23A3   + B,B1,D2,E2B    + D4,D23,D24,E23B1   + D4,D23,D24,E23B3   + D2,D24,E2B4   + D,D2B13  + D2,D24,E2B14  + D,D2B32  + D,D4,EB132 + D,D4,ED    + E2D2   + E,E23D4   + E2D23  + E2D24  + E`

Looking now for the number of remaining combinations this reduces to manageable 36, others than spic itself.

Edit: furthermore would be E23 forbidden as well, as that one produces either q- or w-sized remaining edges at the limitting layer! This then reduces the previous count down to 21 others.

Okay, so far I have not run through all of them. - In fact I just started with the first, ABCD2E2 = oxqwQ2qow(-x)(-x)3xxoxx4oxoxo&#zx, which then (after Stott expansion at the second node position) would result in oxqwQ2wxQoo3xxoxx4oxoxo&#zx. - Halas, this one turns out not to be a CRF. This can be seen as follows:
..... ..... xx.xx4ox.xo&#xt here obviously is a sirco (= x3o4x) facet. But the there used equatorial squares ..... ..... x....4o.... rather are a pseudo faces, as those are diametral to o.q.. ..... x.o..4o.o..&#zx (oct = x3o4o). The distances of those tips (C vertices) to the other layers is unity only wrt. layer A vertices. Thus those sircoes are just facets underneath those C vertices, and there would be outer lacing edges (connecting C to B,D, resp. E) which are larger than unity. Thus the figure cannot be CRF any more.

But then it occured to me that just those vertices of type C are problematic. So I tried to chop off those, and to consider the remainder instead. I.e. now being looking at oxwQ2wxoo3xxxx4oxxo&#zx = o3x4o || o3x4x || x3x4x || w3x4o || x3x4x || o3x4x || o3x4o (with height(1,2) = height(3,4) = height(4,5) = height(6,7) = 0.5 and height(2,3) = height(5,6) = 0.7071).

That one then turned out to be CRF indeed! Its total cell consist is
Code: Select all
` 2 coes (o3x4o)12 esquidpies (J15) 8 hips (x6o x)12 sircoes (x3o4x)16 tricues (J3)64 trips (x3o x)`

Before going on with the above task, I further investigated the Stott contraction of the former wrt. .2.3x4., i.e. considered oxwQ2wxoo3oooo4oxxo&#zx as well. That one then has a total cell consist of
Code: Select all
`24 esquidpies (J15)16 tets (x3o3o)32 trips (x3o x)`
and turns out to be an otherwise well-known fellow: poxic = wx3oo3oo4ox&#zx. (Poxic in fact is closely related to hex (= x3o3o4o). Just consider blowing up its skeleton. Esquidpies occur instead of the edges of hex, trips instead of its triangles, and tets remain as is for cells.) Thus the former one even would have been accessible already before, if poxic would have been investigated with that special subsymmetry. But as this wasn't the case, that expanded one had to wait for its discovery up to my here outlined EKF research.

What shall be emphasized here for conclusion: it is not only that positive results of EKF (= expanded kaleido-facetings) provide new CRFs (= convex regular faced polytopes), but sometimes the negative ones can too!

--- rk
Last edited by Klitzing on Tue Dec 30, 2014 1:31 pm, edited 2 times in total.
Klitzing
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Klitzing wrote:Recently I was considering a further non-regular starting figure:
The I had chosen spic = x3o4o3x.
For a first subsymmetry to look at I then took o2o3o4o.

Under this subsymmetry spic then reads
o2q3x4o + x2o3x4x + q2w3o4o + w2x3o4x + Q2x3o4o = oxqwQ2qowxx3xxooo4oxoxo&#zx
(using here and in what follows: Q=w+x=q+u=q+2x, i.e representing an edge of size 2+sqrt(2)).

...

What shall be emphasized here for conclusion: it is not only that positive results of EKF (= expanded kaleido-facetings) provide new CRFs (= convex regular faced polytopes), but sometimes the negative ones can too!

--- rk

Okay, non-EKF - even so still CRF - results are fine. Better suited here are true EKFs - under current preconditions, if possible.
And indeed I luckily found one!

The one to be considered here was derived from the layer quirks A3B3CDE = oxqwQ2wxwxx3(-x)(-x)ooo4qwoxo&#zx. Stott expansion wrt. the 3rd node then gives oxqwQ2wxwxx3ooxxx4qwoxo&#zx = hull( o2w3o4q + x2x3o4w + q2w3x4o + w2x3x4x + Q2x3x4o ). For sure, Q = w+x = q+u again.

And this one then indeed solves as true CRF. Its total cell count reads:
6 coes (o3x4o)
16 hips (x6o x)
12 sircoes (x3o4x)
12 squobcues (J28)
2 toes (x3x4o)
16 tricues (J3)
80 trips (x3o x)

The geometry here in fact is that of a tower: x3x4o || x3x4x || w3x4o || x3o4w || w3o4q || x3o4w || w3x4o || x3x4x || x3x4o. Accordingly most of the cells align in this symmetry too. E.g. for #3#4. we have a stack of toe-hexagon || hip || tricu || trip || tricu || hip || toe-hexagon for that column. Or for #3.4# column we get hexagon-hexagon-edge of toe || digonal cupola (trip) || sirco (standing on its rhombical square) || digonal cupola || toe-edge. And for column .3#4# we get the stack toe-square || squobcu || co || squobcu || toe-square. The only here missing cells then are the 48 diametral trips, which occur herein as .....2.....3ooxx.4.....&#xr, where the cycle (&#xr-part) runs as (ABDC).

--- rk
Klitzing
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### Re: Expanded Kaleido-Facetings based on the Icositetrachoron

Klitzing wrote:Recently I was considering a further non-regular starting figure:
The I had chosen spic = x3o4o3x.
For a first subsymmetry to look at I then took o2o3o4o.

Today I further considered for that spic = x3o4o3x also its subsymmetry o3o3o *b3o.
Under that subsymmetry it can be written as
Code: Select all
`qoo-3-xxx-3-oqo *b3-ooq-&#zx`
.

For possible flips we get
Code: Select all
`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)`

Rejecting all those which would provide (after corresponding partial Stott expansion) some u-edges, and seeving out multiples (by internal symmetry of the starting diagram), we can drop these 6*6*6=216 combinations down to just 20 additional ones, which then have to be checkt individually. But again all but one drop out because of one layer (at least) that cannot be laced to the others using unit edges only.

The resulting possibilities then come out to be all priviously known CRFs:
A) direct partial Stott transforms of ABC = spic itself:
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`1:ABC       = wxx 3 xxx 3 oqo *b3 ooq &#zx  = owau prit(-2):ABC    = qoo 3 ooo 3 oqo *b3 ooq &#zx  = ico1(-2):ABC   = wxx 3 ooo 3 oqo *b3 ooq &#zx  = poxic13:ABC      = wxx 3 xxx 3 xwx *b3 ooq &#zx  = poc prico1(-2)3:ABC  = wxx 3 ooo 3 xwx *b3 ooq &#zx  = pocsric134:ABC     = wxx 3 xxx 3 xwx *b3 xxw &#zx  = prico1(-2)34:ABC = wxx 3 ooo 3 xwx *b3 xxw &#zx  = srico`

B) true EKFs, all being based on the single possible KF A2B2C2:
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`2:A2B2C2             = wxx 3 ooo 3 xwx *b3 xxw &#zx  = srico(-1)2:A2B2C2         = qoo 3 ooo 3 xwx *b3 xxw &#zx  = pocsric(-1)2(-3):A2B2C2     = qoo 3 ooo 3 oqo *b3 xxw &#zx  = poxic(-1)2(-3)(-4):A2B2C2 = qoo 3 ooo 3 oqo *b3 ooq &#zx  = ico`

--- rk
Klitzing
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### Expanded Kaleido-Facetings not based on the 600-cell

(just corrected the subject line finally )

Well, I did not find new EKF-CRFs recently, but at least I stumbled across some feature we should take into account more generally in the EKF context, and esp. when considering to write some article...

Yesterday eve I was looking for the possible EKFs when taking the rectified tesseract as starting figure and using axial cubic subsymmetry.
The corresponding representation of rit then is qo2oo3xo4oq&#zx = ooo3xox4oqo&#xt.
The possible layers here are
A: q2o3x4o - with possible quirk A3: q2x3(-x)4q, and
B: o2o3o4q

In fact, the mere axial subsymmetry ". o3o4o" happens to be impossible here: it would ask for A': o3x4o at the one end and for A": x3(-x)4q at the opposite end. But then the needed for partial Stott expansion wrt. the (here) 2nd node would result in some u-edges at A' (at least), and as this is an extremal layer, these u-edges would remain in the whole resulting (hypothetical) EKF, which then shows that it cannot happen to be a CRF, right for that reason. - This is why we can restrict to full "o2o3o4o" subsymmetry here.

When using A (and B) we further have the restriction that the 1st and 4th node cannot be partially Stott expanded independently, we have to exclude the potentially therefrom defined non-regular polygons: it might happen to become a non-regular hexagon in shape of a diagonally elongated square. - therefore the here only possible cases then are qo2xx3xo4oq&#zx (a.k.a. "pabdirico" = parabidimin. rectified ico) with cells being 6 coes + 12 cubes + 2 toes + 16 tricues, wx2oo3xo4xw&#zx (a.k.a. "tat" = truncated tesseract, i.e. already uniform), and wx2xx3xo4xw&#zx (a.k.a. "pabdiproh" = parabidimin. prismatorhombated hexadecachoron) with cells being 2 gircoes + 12 ops + 6 tics + 16 tricues + 8 trips.

When using A3 (and B) we seem to have a q-edge (4th node) within the extremal layer, which would survive into the full EKF and therefore should result in a non-CRF. - But, despite of erasing that case completely we should be more careful here! The layer B as well has a q-edge at this node. And so we could apply a conform partial Stott q-contraction wrt. to that node (plus further ones as desired). Accordingly we still get some possible cases here too: qo2xo3ox4oo&#zx (a.k.a. "ico" = icositetrachoron, i.e. already uniform), wx2xo3ox4oo&#zx (a.k.a. "pexic" = partially expanded ico) with cell list: 6 esquidpies + 18 octs + 8 trips, qo2xo3ox4xx&#zx (a.k.a. "pacsrit" = partially contracted small rhombated tesseract) with cell list: 16 octs + 2 sircoes + 6 squobcues + 24 trips, and wx2xo3ox4xx&#zx (a.k.a. "srit" = small rhombated tesseract, i.e. already uniform).

Thus, in total, some valide EKF-CRFs - but sadly none of those is truely a new-found shape.

What I wanted to point out in this post nonetheless, was that we should have in mind partial Stott expansions or contractions not only of unit size, but additionally of any other size as well! (Herein being used: sizes q and (q-x).) Esp. non-unit edges within extremal layers - as such - are not an erasement criterium alone! (As shown above, those might happen to be compensated somehow...)

--- rk
Klitzing
Pentonian

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