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Lots of them are listed on my CRF page, in my CRF list (downloadable Excel), or some on Quickfur's page.

In the past there were just some sporadic finds, say

--- rk

In the past there were just some sporadic finds, say

- via snubbing (alternated facetings),
- via lace prisms (or stacks thereof, aka lace towers, resp. even the closely related segmentotopes) or
- via (partial) Stott expansions.

--- rk

- Klitzing
- Pentonian
**Posts:**1527**Joined:**Sun Aug 19, 2012 11:16 am**Location:**Heidenheim, Germany

Just yesterday "found" a further CRF, so far not being listed within my spreadsheet.

So I'll describe it here for you. Thereby you might get a glimps on what we have done all these years before.

I reconsidered ico = 24-cell = x3o4o3o (full F4 sym.) = o3x3o4o (C4 subsym.) = o3x3o *b3o (D4 subsym.).

Esp. its A2 x A2 subsymmetric representation. Here you get as according lace city representation:

where x=1 (unit edge) and q=sqrt(2).

Now I was interested into its cyclo-tri-diminishing, i.e. in the structure with the lace city

which, as it turns out, happens to be bounded by 18 square pyramids, 6 octahedra, and 3 cubes.

Just remind yourself that the cells of the 24-cell (ico) are nothing but 24 octahedra. So remaining octahedra and itself diminished octahedra (the square pyramids) shouldn't be surprising here. If you further remember that the vertex figure of ico is nothing but the cube, you'll recognize these as well in the above total as the bases of the 3 withdrawn vertex pyramids.

Finally you should consider the cube in ist axial orientation. Then that one has stacked vertex layers, which are describable as o3o || q3o || o3q || o3o. And just These sequences you could recognize at the outline of the above representation. I.e. those cubes not only are the bases underneath the removed alternate vertices of the outermost hexagon of this projection, they moreover connect vertex-wise at the other, remaining vertices of this very hexagon.

--- rk

PS: this tiny little cutie might be a nice candidate for Quickfur's next "polychoron of the month" rendering (after long)?

PPS: despite the occurances of all these "q" in the above lace cities, all truely being used edges of that "tridiminished icositetrachoron" indeed are all of size x=1. Those q-sized pseudo edges are nothing but the diagonals of squares or of octahedra. Thus this figure is indeed CRF, i.e. convex and regular faced.

So I'll describe it here for you. Thereby you might get a glimps on what we have done all these years before.

I reconsidered ico = 24-cell = x3o4o3o (full F4 sym.) = o3x3o4o (C4 subsym.) = o3x3o *b3o (D4 subsym.).

Esp. its A2 x A2 subsymmetric representation. Here you get as according lace city representation:

- Code: Select all
`o3o -- o3o4o (point)`

o3o q3o o3q o3o -- o3o4x (cube)

o3q q3o -- q3o4o (q-oct)

o3o q3o o3q o3o -- o3o4x (cube)

o3o -- o3o4o (point)

where x=1 (unit edge) and q=sqrt(2).

Now I was interested into its cyclo-tri-diminishing, i.e. in the structure with the lace city

- Code: Select all
`o3o`

q3o o3q

o3q q3o

o3o q3o o3q o3o

which, as it turns out, happens to be bounded by 18 square pyramids, 6 octahedra, and 3 cubes.

Just remind yourself that the cells of the 24-cell (ico) are nothing but 24 octahedra. So remaining octahedra and itself diminished octahedra (the square pyramids) shouldn't be surprising here. If you further remember that the vertex figure of ico is nothing but the cube, you'll recognize these as well in the above total as the bases of the 3 withdrawn vertex pyramids.

Finally you should consider the cube in ist axial orientation. Then that one has stacked vertex layers, which are describable as o3o || q3o || o3q || o3o. And just These sequences you could recognize at the outline of the above representation. I.e. those cubes not only are the bases underneath the removed alternate vertices of the outermost hexagon of this projection, they moreover connect vertex-wise at the other, remaining vertices of this very hexagon.

--- rk

PS: this tiny little cutie might be a nice candidate for Quickfur's next "polychoron of the month" rendering (after long)?

PPS: despite the occurances of all these "q" in the above lace cities, all truely being used edges of that "tridiminished icositetrachoron" indeed are all of size x=1. Those q-sized pseudo edges are nothing but the diagonals of squares or of octahedra. Thus this figure is indeed CRF, i.e. convex and regular faced.

- Klitzing
- Pentonian
**Posts:**1527**Joined:**Sun Aug 19, 2012 11:16 am**Location:**Heidenheim, Germany

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