By Peter McMullen, Egon Schulte

Summary general polytopes stand on the finish of greater than millennia of geometrical study, which started with typical polygons and polyhedra. The quick improvement of the topic long ago 20 years has ended in a wealthy new concept that includes an enticing interaction of mathematical components, together with geometry, combinatorics, staff concept and topology. this is often the 1st complete, up to date account of the topic and its ramifications. It meets a severe want for this kind of textual content, simply because no booklet has been released during this zone due to the fact Coxeter's "Regular Polytopes" (1948) and "Regular advanced Polytopes" (1974).

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A) There exists a unique polarity ω of P which ﬁxes the base ﬂag Φ (but reverses the order of its faces). (b) For each j = 0, . . , n − 1, ωρ j ω = ρn−1− j , so that ω induces a group automorphism of Γ (P). In particular, Γ¯ (P) = Γ (P) C2 , a semi-direct product of Γ (P) by C2 . (c) If the group automorphism in (b) is an inner automorphism of Γ (P), given by conjugation with an involution in Γ (P), then Γ¯ (P) = Γ (P) × C2 . Proof. To prove (a), let ψ be any duality of P, and deﬁne Ψ := Φψ. Since P is regular, there exists ϕ ∈ Γ (P) such that Ψ ϕ = Φ.

This group contains Γ (P) as a subgroup of index 2. If P admits a polarity, then Γ¯ (P) = Γ (P) C2 , a semi-direct product of Γ (P) by C2 (see also Proposition 2B17). The semi-direct product G := W Λ means that Λ acts as a group of automorphisms on W on the right; thus, W G, a normal subgroup, with quotient G/W ∼ = Λ. ) According to our deﬁnition, the automorphisms of a polytope P are permutations of the face-set of P. We shall often consider automorphisms as operating on certain subsets of P, or on sets of subsets of P.

Fi−1 , Fi , F j , F j+1 , . . , Fn } is a chain of P, then Γ (P, Ω) is (isomorphic to) a subgroup of Γ (F j /Fi ). (b) If F is a vertex of P, then Γ (P, F) is a subgroup of the group of the vertex-ﬁgure of P at F. Similarly, if F is a facet, then Γ (P, F) is a subgroup of the group of F (or more precisely, of F/F−1 ). Proof. For (a), observe that Γ (P, Ω) clearly acts on the section F j /Fi . To see that this action is faithful, note that Ω is contained in a ﬂag Φ (say) of P. Hence, if ϕ in Γ (P, Ω) acts trivially on F j /Fi , then it must keep Φ ﬁxed, so that ϕ = ε by Proposition 2A4.