One of the projections of the $4_{21}$ polytope (https://en.m.wikipedia.org/wiki/4_21_polytope) into four dimensions positions its vertices as those of two concentric 600-cells scaled by the golden ratio (https://vzome.com/home/geometry/gossets-polytopes/). Taking the convex hull of 120 of the $4_{21}$ corresponding to just one of these 600-cells results in a diminishing of the $4_{21}$ with 120 vertices and some number of edges. Are all these edges the same length?
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3$\begingroup$ Dear Daniel. I recommend that you edit your question to include some (a lot?) of background. For example, could you please include a definition of the $4_{21}$ polytope. Also, could you please explain (sketch) why some of the facts that your cited are true? Etc. This will help other users provide useful answers to your question. $\endgroup$– André HenriquesCommented Jul 2, 2021 at 21:02
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$\begingroup$ en.m.wikipedia.org/wiki/4_21_polytope $\endgroup$– Dima PasechnikCommented Jul 2, 2021 at 21:18
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$\begingroup$ Wikipedia gives some relatively simple coordinates for the vertices of the $4_{21}$. Do you have an idea which of them remain after deleting half of them? Or one step further back, can you give the 4-subspace on which you project? $\endgroup$– M. WinterCommented Jul 2, 2021 at 22:13
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