## Interesting vertex-faceting of the grand antiprism

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

### Interesting vertex-faceting of the grand antiprism

Take a grand antiprism of unit length. You can actually delete vertices and get a uniform runcinated 5-cell of golden ratio length.

This happens because it contains a pentagonal swirl symmetry of the 5-cell (which corresponds to the pentagons of the grand antiprism).
Mercurial, the Spectre
Trionian

Posts: 106
Joined: Mon Jun 19, 2017 9:50 am

### Re: Interesting vertex-faceting of the grand antiprism

Well, we already knew you could inscribe an f-scaled spid (that's f3o3o3f) into ex (the 600-cell). Look at the following tegum sum representation of ex, from Klitzing's site:

Code: Select all
`xffoo3oxoof3fooxo3ooffx&#zx`

if you take the third "layer" (the third letter in each group above) you get f3o3o3f, which is an f-scaled spid. The spid-wise diminishing of ex also exists and has been described on here before.

Never tohught aobut trying to inscribe the same subset into gap before, but I guess it works. (I don't think gap has a pennic subsymmetry, though.) And obviously, you can't diminish any vertices from gap (not in a CRF way at least) because you would get diminished pentagons (with f-scaled edges) as faces.
Trionian

Posts: 128
Joined: Sat Mar 18, 2017 1:42 pm

### Re: Interesting vertex-faceting of the grand antiprism

You even could turn that the other way round: there is a 60-spid-compound, inscribed into ex, using every vertex 10 times: dopix = 10{3,3,5}[60{;3,3,3;}].
--- rk
Klitzing
Pentonian

Posts: 1640
Joined: Sun Aug 19, 2012 11:16 am
Location: Heidenheim, Germany