How to find a CRF

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

How to find a CRF

Postby ubersketch » Sat Jul 13, 2019 9:12 pm

TItle says it all.
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Re: How to find a CRF

Postby Klitzing » Sun Jul 14, 2019 8:44 am

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
Based on that student91 once tried to derive a more systematical approach, called the "expanded kaleido facetings", cf. the outline here.

--- rk
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Re: How to find a CRF

Postby Klitzing » Fri Dec 06, 2019 6:58 pm

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:
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vertex figure of ico is nothing but the cube, you'll recognize these as well in the above total as the bases of the 3 withdrawn vertex pyramids.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.
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