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:

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`A: q2x3o4o > A2: q2(-x)3x4o > A23: q2o3(-x)4q`

B: 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:

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`+ooo -: wx2xo3ox4oo&#zx = pexic`

o+oo A2: qo2ox3xx4oo&#zx

A2B32: qo2oo3xo4oq&#zx = rit

oo+o B3: qo2xx3xo4oq&#zx

ooo+ -: qo2xo3ox4xx&#zx = pacsrit

++oo A2: wx2ox3xx4oo&#zx

A2B32: wx2oo3xo4oq&#zx

+o+o B3: wx2xx3xo4oq&#zx

+oo+ -: wx2xo3ox4xx&#zx = srit

o++o

o+o+ A2: qo2ox3xx4xx&#zx

A2B32: qo2oo3xo4xw&#zx

oo++ B3: qo2xx3xo4xw&#zx

+++o

++o+ A2: wx2ox3xx4xx&#zx

A2B32: wx2oo3xo4xw&#zx = tat

+o++ B3: wx2xx3xo4xw&#zx

o+++

++++

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