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There are an infinite number of regular polygons in the plane, five regular polyhedra, six regular polytopes in $\mathbb{R}^4$, and then three regular polytopes in every dimension $d > 4$.

There are eight convex deltahedra (all faces equilateral triangles), five in $\mathbb{R}^4$ (see, "Convex deltahedra in higher dimensions"), and then three deltatopes in every dimension $d > 4$.

Q. Is there some intuitive reason why freedom is removed in higher dimensions?

I ask because, if I didn't know better, I would think the opposite. There is vastly much more "room" in higher dimensions, and one might think forms proliferate, even under constraints. Concerning "much more room," think of the severe contraints on planar graphs vs. the fact that every graph can be realized as embedded in $\mathbb{R}^3$.

I seek a corrective to my faulty intuition. Thanks!

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    $\begingroup$ I like Coxeter's book, Regular Polytopes. He wasn't just a pretty face. $\endgroup$
    – Will Jagy
    Nov 18, 2013 at 1:01
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    $\begingroup$ Meanwhile, our own Jim Humphreys has a book math.umass.edu/~jeh/pub/book.html on Coxeter groups and reflection groups that probably gives a complete answer. $\endgroup$
    – Will Jagy
    Nov 18, 2013 at 1:09
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    $\begingroup$ "Regular" seems like a more important adjective than "convex" in the title. Anyway, my intuition is that while the amount of room might increase, it doesn't increase as quickly as the number of constraints imposed by regularity. Graph embeddings, on the other hand, don't have to have any kind of symmetry properties. $\endgroup$
    – Qiaochu Yuan
    Nov 18, 2013 at 1:14
  • $\begingroup$ Excellent point, @QiaochuYuan! I wonder if that intuition can be made more precise, that constraints outrun freedom...? $\endgroup$
    – Joseph O'Rourke
    Nov 18, 2013 at 1:16
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    $\begingroup$ A rough estimate of the number of constraints is that an $n$-simplex has $n!$ flags, and regularity means that the symmetry group acts transitively on flags. $\endgroup$
    – Qiaochu Yuan
    Nov 18, 2013 at 1:18

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This is a well known "dimension curse" phenomenon. It is easier to explain for spherical polyhedra. Let $P$ be a regular spherical convex polyhedron of dimension $d$. Starting with dimension $d=3$, all spherical convex polyhedra are rigid, so they are determined by their links (intersection of a vertex cones with an $\epsilon$-sphere). The latter are themselves regular (spherical) polyhedra. Therefore, for $d\ge 4$ these links are rigid themselves and regular polyhedra are uniquely determined by their links (which are all congruent, of course, by regularity). Thus, as dimension grows the combinatorial types of regular polyhedra can disappear but new cannot appear.

For Euclidean polyhedra the same type of analysis works, but goes through the links which are spherical polyhedra. Therefore, as regular spherical polyhedra disappear so do Euclidean ones, but this is less intuitive perhaps. I haven't seen "deltatopes" before, but it's the same story there as well, if I understood definitions correctly.

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