# Abstract Regular Polytopes (Encyclopedia of Mathematics and by Peter McMullen, Egon Schulte

By Peter McMullen, Egon Schulte

Summary general polytopes stand on the finish of greater than millennia of geometrical examine, which all started with usual polygons and polyhedra. The speedy improvement of the topic some time past 20 years has ended in a wealthy new concept that includes an enticing interaction of mathematical components, together with geometry, combinatorics, workforce conception and topology. this can be the 1st complete, up to date account of the topic and its ramifications. It meets a serious desire for this kind of textual content, simply because no publication has been released during this quarter given that Coxeter's "Regular Polytopes" (1948) and "Regular advanced Polytopes" (1974).

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P4) below. ) The elements of P are called the faces of P. Sometimes we use the term face-set of P to denote the underlying set of P (without reference to the partial order). Two faces F and G of P are said to be incident if F G or F G. A chain of P is a totally ordered subset of P. A chain has length i ( −1) if it contains exactly i + 1 faces. Note that, by deﬁnition, the empty set is a chain (of length −1). The maximal chains are called the ﬂags of P. We denote the set of all ﬂags of P by F(P).

102]. For the next proposition, recall that Γ (P, Ω) denotes the stabilizer of a chain Ω of P. 2B3 Proposition Let P be a regular n-polytope. (a) All sections of P are regular polytopes, and any two sections which are deﬁned by faces of the same ranks are isomorphic. In particular, P has isomorphic facets and isomorphic vertex-ﬁgures. Furthermore, P is equivelar (that is, possesses a Schl¨aﬂi symbol). (b) The group of each section of P is a subgroup of Γ (P). More precisely, if F j is a j-face and Fk a k-face with −1 j < k n and F j < Fk , then Γ (Fk /F j ) is isomorphic to Γ (P, Ω), where Ω is any chain of type {−1, 0, .

266]). By deﬁnition, the 1-sections of an n-polytope P are all of the same kind. Generally this will not remain true for sections of higher rank, indeed not even for the 2-sections determined by faces of the same rank. For n 2 and i = 1, . . , n − 1, if F is an (i − 2)-face and G an (i + 1)-face of P incident with F, then we write pi (F, G) for the number of i-faces (or (i − 1)-faces) of P in the section G/F; then G/F is isomorphic to the 2-polytope with Schl¨aﬂi symbol { pi (F, G)}. If these numbers depend only on i but not on the choice of F and G, then we set pi := pi (F, G) for each i, and call P equivelar of (combinatorial Schl¨aﬂi) type 30 2 Regular Polytopes { p1 , .