mr_e_man wrote:Thanks to Klitzing's list, I found another potential example: The rectified n-simplex has circumradius approaching 1, so its pyramid has height approaching 0 . But I haven't calculated the ditopal angles.
Klitzing wrote:the rectified n-simplex pyramid is nothing but
the vertex figure pyramid of the n+1-demihypercube!
--- rk
quickfur wrote:...
Fascinating! So as the dimension increases these pyramids become shallower and shallower, so they could potentially augment a *lot* of CRFs and still remain convex. This does require the augment to fit on a rectified (n-1)-simplex facet, though. How many polytopes would have such cells? The n-cube truncates would, and the number of those increases exponentially with dimension. Which leads to the question: at what dimension does the rectified n-simplex pyramid become shallow enough that it can augment a birectified (n+1)-cube and still remain convex? That would be an interesting question to answer.
Klitzing wrote:[...]
Thence there the dihedral between the base and those lacings would have the counterintuitive behaviour to increase with the dimension instead.
[...]
So indeed the dihedrals between lacing simplex and base rectified simplex would allow for arbitrary many such components around an ridge, however the neighbouring ridge would prohibite that at the same time!
[...]
mr_e_man wrote:But can we construct a sequence of, say, CRF 6-polytopes, whose angles approach 0° ? If so, that would make it much more difficult to enumerate CRF 7-polytopes, as there'd be no limit on the number of 6-faces around a 4-face.
quickfur wrote:...
Fascinating! So as the dimension increases these pyramids become shallower and shallower, so they could potentially augment a *lot* of CRFs and still remain convex. This does require the augment to fit on a rectified (n-1)-simplex facet, though. How many polytopes would have such cells? The n-cube truncates would, and the number of those increases exponentially with dimension. Which leads to the question: at what dimension does the rectified n-simplex pyramid become shallow enough that it can augment a birectified (n+1)-cube and still remain convex? That would be an interesting question to answer.
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