Does there exist a convex uniform 9-polytope obtained by diminishing the 9-hypercube, removing 480 of its 512 vertices and turning each 8-hypercube facet into an 8-orthoplex?
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$\begingroup$ Can you give some more details on your construction? How do you turn an 8-cube into an 8-orthoplex? What other facets does this polytope have? Do you know more about its combinatorics and symmetry? $\endgroup$– M. WinterCommented Oct 31, 2021 at 22:26
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1$\begingroup$ 1. Explicit coordinates for a 7-simplex inscribed in a 7-hypercube are given at polytope.miraheze.org/wiki/Octaexon. If you turn a pair of opposite 7-hypercube facets of an 8-hypercube into a pair of simplices, you get an orthoplex. $\endgroup$– Daniel SebaldCommented Nov 1, 2021 at 0:22
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$\begingroup$ 2. It probably has 8-simplices as facets, but other than that I can’t tell you much about its elements or symmetries. $\endgroup$– Daniel SebaldCommented Nov 1, 2021 at 0:23
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$\begingroup$ So when I understand you correctly, you say you want to delete a bunch of vertices in such a way so that for each 8-cube facet of the 9-cube, there are two opposite 7-cube facets of this 8-cube from which you left exactly those vertices that form a regular 7-simplex. Do you already know that you can take away the vertices in such a way and you ask whether the result is uniform, or are you also asking whether such a choice is possible at all? In the first case, can you please provide the vertex coordinates of the polytope that you describe? $\endgroup$– M. WinterCommented Nov 1, 2021 at 10:56
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$\begingroup$ I’m asking 1. whether such a choice is possible at all, and 2. if it works, whether it results in a uniform polytope. $\endgroup$– Daniel SebaldCommented Nov 1, 2021 at 14:36
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