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Ever since I first read Coxeter’s definition of a regular compound (which seems to be the most commonly used), I didn’t like it on account of it being completely different than for properly connected polytopes. This made me begin to wonder what the set of regular compounds would be if the same definition (flag-transitivity) was used. After searching the internet and discovering a few of my own, my list includes two infinite families plus 8 others. However, I have no idea if it’s complete. Could there be others?

  • 2 n-simplices (Aut(An) symmetry)
  • n p/q-gons (Inp symmetry)
  • 3 tesseracts (F4 symmetry)
  • 3 16-cells (F4 symmetry)
  • 2 24-cells (Aut(F4) symmetry)
  • 6 tesseracts (Aut(F4) symmmetry)
  • 6 16-cells (Aut(F4) symmetry)
  • 120 5-cells (H4 symmetry)
  • 72 7-simplices (E7 symmetry)
  • 1920 8-simplices (E8 symmetry)
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  • $\begingroup$ I had never even heard the term "compound of polytopes" before your post but Wikipedia seems to have a list here: en.wikipedia.org/wiki/…. I also note the quote "Coxeter lists 32 regular compounds of regular 4-polytopes in his book Regular Polytopes. McMullen adds six in his paper New Regular Compounds of 4-Polytopes." which might be especially relevant to your question. $\endgroup$
    – Sam Hopkins
    Commented Jan 23, 2021 at 2:19
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    $\begingroup$ Most of those are not flag-transitive, though. Wikipedia seems to prefer Coxeter’s definition. $\endgroup$
    – Daniel Sebald
    Commented Jan 23, 2021 at 3:41
  • $\begingroup$ To find such compounds you would need to take a symmetry group of a regular polytope, and embed it as subgroup of a larger point group. I suspect that your list is complete. Though I wonder whether there is an easy way to see that $\mathrm{Aut}(A_n)$ cannot be enlarged in dimensions $n\ge 9$. For sufficiently large dimensions, the answers to this question suggests that $B_n$ can never be enlarged because it is the largest point group there is. $\endgroup$
    – M. Winter
    Commented Jan 23, 2021 at 10:41

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