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    \$\begingroup\$ Googling shows there's only 3 families of regular polytope extending beyond 4 dimensions: analogous to cube, octahedron and tetrahedron. It seems it would be simpler to write for these families and hardcode the rest (two 3d polytopes, three 4d polytopes, and the infinite family of 2d polytopes.) As far as I can see that meets the spec but wouldn't be generalizable. Would that be a valid answer? It might be feasible to write a recursive algorithm to generate topological graphs beyond the scope of the spec, but the killer with that approach even within the spec is calculating the coordinates. \$\endgroup\$ Commented Apr 1, 2017 at 8:24
  • \$\begingroup\$ How do we know the actual vertices, by only knowing they are equilateral? \$\endgroup\$ Commented Apr 2, 2017 at 0:12
  • \$\begingroup\$ @SIGSEGV the only requirement specified is that the origin should correspond to either the centre or one of the points. That gives plenty of scope to rotate the shape as you please. en.wikipedia.org/wiki/Simplex gives an algorithm for calculating coordinates of the hypertetrahedrons (which could perhaps be extended to the icosahedron and its 4d analogue but doing that is too much for me, hence my question.) The hypercubes and hyperoctahedrons have nice integer coordinates (and the hypertetrahedrons too actually, but often only in more dimensions than the shape itself, which is untidy.) \$\endgroup\$ Commented Apr 2, 2017 at 0:50
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    \$\begingroup\$ I've cast the closing vote on this question because it is a fastest-guns-in-the-west style challenge, where the first valid answer wins. This is not generally considered to be a valid winning criterion. I don't know how this has been open for so long, it should have been closed. \$\endgroup\$ Commented Feb 5, 2018 at 4:16
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    \$\begingroup\$ @DonHatch I see your point. When I posted this, the first thing to do with the output that came to mind was the draw the resulting figures. openGL has an option: GL_POLYGON for which it is useful to have the faces point to the vertices instead of the edges. It is actually not too hard to use if faces point to edges, but the edges would need to be ordered. However, I can see cases where it would be useful to have faces point to edges, hyperfaces point to faces, etc, and the reverse as well. \$\endgroup\$ Commented Jul 14, 2018 at 22:14