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The classical definition of regular polytopes is recursive. It says that a polytope is regular if its facets and vertex figures (both smaller-dimensional polytopes) are regular.

The modern definition goes as follows. Let $P\subseteq\mathbb{R}^n$ be a polytope centered at the origin and let $\mathrm{Aut}(P)\leq O(n)$ be its automorphism group. We say that $P$ is regular if its automorphism group acts transitively on maximal flags of faces.

However, there are many equivalent definitions of regularity. Let's say that a polytope is $d$-regular if its automorphism group is transitive on $d$-dimensional faces. The following theorem is stated in several places (for example in McMullen and Schulte's "Abstract Regular Polytopes", pages 9-10):

Theorem: Let $P$ be an $n$-dimensional polytope. If $P$ is $d$-regular for all $0\leq d\leq n-1$ then $P$ is regular.

All statements of this theorem I've seen refer to Peter McMullen's 1968 thesis from the University of Birmingham, which I don't have access to.

So here's my question: Does anyone know where I can find a proof of this theorem or how to gain access to Peter McMullen's thesis?

Bonus Problem: How dependent/independent are the notions of $d$-regularity for different $d$?

Thanks.

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  • $\begingroup$ This seems to need a clarification on automorphisms: clearly the cyclic group of order $n$ acting naturally on the regular $n$-gon is transitive on its vertices and edges, but not on flags. So the statement seems to boil down to a finite Coxeter group in the background... $\endgroup$ Commented Feb 20, 2016 at 23:23
  • $\begingroup$ "How dependent/independent are the notions of $d$-regularity for different $d$": I have looked into this specific question for almost two years now, and it seems that surprisingly little is known (there is even a quote in Grünbaum's "Convex Polytopes" hinting to that lack of knowledge). See e.g. this question of mine, for which the answer is No (at least that is what my results tell me; it is still unpublished). $\endgroup$
    – M. Winter
    Commented May 27, 2020 at 19:57

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This is theorem 4C6 of Peter McMullen's thesis, "On the Combinatorial Structure of Convex Polytopes", on page 73:

4C6. Theorem. A $d$-polytope $P$ is regular if and only if for each $j = 0, \dots, d-1$, its symmetry group $G(P)$ is transitive on the $j$-faces of $P$.

The condition is clearly necessary. For the converse, we deduce that the vertices of $P$ lie on a $(d-1)$-sphere, $S^{d-1}$, and hence that the vertices of any $j$-face $F^j$ $(j=1,\dots,d)$ lie on the $(j-1)$-sphere $S^{d-1} \cap \operatorname{aff} F^j$. If we assume inductively that the facets of $F^j$ are regular and congruent, then by theorem 4C4, we deduce that $F^j$ is itself regular. In particular, for $j = d$, we deduce that $P$ is regular. This completes the proof of theorem 4C6.

Theorem 4C4 says

4C4. Theorem. A $d$-polytope $P$ ($d \geq 3$) is regular if and only if its vertices lie on a sphere, and its facets are regular and combinatorially equivalent.

That the condition is necessary is clear, since the transforms of a point under a finite group of congruent transformations lie on a sphere. We prove that the condition is sufficient by induction on the dimension $d$. Since two adjacent facets share a $(d-2)$-face, using proposition 1C16 we deduce that any two facets of $P$ are congruent. In particular, the edges of $P$ have the same length. This implies that the mid-points of the edges of $P$ through any given vertex lie on a $(d-2)$-sphere. But these points are the vertices of a vertex-figure of $P$ (cf. proposition 1B4); its facets are congruent, and so by the induction hypothesis, it is regular. (Notice that this holds for the case $d=2$ as well.) In 3 dimensions, this condition also implies that the vertex-figures, being regular polygons with the same edge-length inscribed in circles of the same radius, are congruent. Hence, by theorems 4C2 and 4C3, $P$ is regular, which completes the proof of theorem 4C4.

The mentioned results are

  • 1C16: Any two $j$-faces $F^j$ and $G^j$ of a polytope $P$ can be joined by a chain $$F^j = F_0^j, F_1^j, \dots, F_m^j = G^j$$ of $j$-faces of $P$, such that for $k = 1, \dots, m$, $F_{k-1}^j$ and $F_k^j$ are adjacent.

  • 1B4: $H \cap P \approx P / F$, where $F$ is a vertex of a polytope $P$, and $H$ is a hyperplane strictly separating $F$ from the remaining vertices $P \setminus F$ of $P$.

  • 4C2: A $d$-polytope $P$ ($d \geq 3$) is regular if and only if its facets are regular and combinatorially equivalent, and its vertex-figures are combinatorially regular and combinatorially equivalent.

  • 4C3: A $d$-polytope $P$ ($d \geq 4$) is regular if and only if its facets are regular and its vertex-figures are combinatorially regular.

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As far as I can tell, this was published:

MR0221384 (36 #4436) Reviewed McMullen, P. Combinatorially regular polytopes. Mathematika 14 1967 142–150.

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  • $\begingroup$ This seems not to contain the result I'm looking for. Probably it's implicit, but too implicit for me. $\endgroup$ Commented May 21, 2014 at 20:00

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