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Geometrydiscrete & Computational © 1997 Springer-Verlag New York Inc Discrete Comput Geom 17:449–478 (1997) GeometryDiscrete & Computational © 1997 Springer-Verlag New York Inc. Regular Polytopes in Ordinary Space P. McMullen1 and E. Schulte2 1Department of Mathematics, University College London, Gower Street, London WC1E 6BT, England [email protected] 2Department of Mathematics, Northeastern University, Boston, MA 02115, USA [email protected] Abstract. The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Gr¨unbaum–Dress polyhedra (that is, the regular apeirohedra, or 3 apeirotopes of rank 3) in ordinary space E in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to 3 describe all the discrete regular apeirotopes of rank 4 in E . The paper gives a complete classification of the discrete regular polytopes in ordinary space. 1. Introduction The theory of regular polytopes has undergone a number of changes since its origins in the classification by the Greeks of the five regular (“Platonic”) solids, but most particularly during this century. At the forefront of this development is Coxeter, whose book Regular Polytopes [3] covers what might be called the classical theory. However, already in Coxeter’s work a more abstract approach begins to be manifested. One generalization of regular polyhedron is that of a regular map (see, for example, [4]). The starting point of the present paper is 1926, when Petrie found the two dual regular apeirohedra (infinite polyhedra) 4, 6 4 and 6, 4 4 in E3, which have planar faces but skew vertex-figures (these technical{ | } terms{ are| defined} in Section 2). Immediately afterwards, Coxeter discovered a third such example: 6, 6 3 (see [2]). Around 1975 Gr¨unbaum (see [8]) restored the symmetry by allowing{ skew| } faces as well (although implicitly these were permitted by Coxeter also in using the Petrie operation); he found The second author was partially supported by NSA Grant MDA904-96-1-0027 and Northeastern Uni- versity’s RDS Fund. 450 P. McMullen and E. Schulte 20 more regular apeirohedra in E3. A final instance was discovered by Dress around 1980 (see [5] and [6]), who also proved the completeness of the enumeration. Our aims here are threefold. First, we describe a far quicker method of arriving at Dress’s characterization result. The key to this method is a “trick” employed in the proofs of Theorems 4.5 and 6.2, the essence of which is to replace a reflexion whose mirror is a line by one whose mirror is a perpendicular plane. This idea also leads to new pairings between the finite regular polyhedra. Second, we give presentations of the symmetry groups of those discrete regular apeirohedra in E3 whose groups are affinely irreducible; an important ingredient here is Theorem 2.5, which says that such a presentation arises solely from the vertex-figure and the edge-circuits. Third, we describe seven new discrete regular apeirotopes of rank 4 in E3, to add to the familiar honeycomb 4, 3, 4 of cubes, and prove that there are no others. Thus the paper contains a complete{ classification} of the discrete regular polytopes in ordinary space. 2. Abstract Regular Polytopes Since we discuss regular polytopes on the abstract as well as the geometric level, we begin with a brief introduction to the underlying general theory (see, for example, [12] and [13]). An (abstract) polytope of rank n, or simply an n-polytope, satisfies the following properties. It is a partially ordered set with a strictly monotone rank function whose range is 1, 0,...,n . The elements ofP rank j are called the j-faces of , and the family { } P of such j-faces is denoted j .For j 0, 1, n 2, or n 1, we also call j-faces vertices, edges, ridges, and facets,P respectively.= The flags (maximal totally ordered subsets) of each contain exactly n 2 faces, including the unique minimal face F 1 and unique P + maximal face Fn of . Further, is strongly flag-connected, meaning that any two flags P P 8 and 9 of can be joined by a sequence of flags 8 80,81,...,8k 9, where P = = 8i 1 and 8i are adjacent (differ by one face), and 8 9 8i for each i. Finally, if F and G are a ( j 1)-face and a ( j 1)-face with F∩< G, then there are exactly two j-faces H such that F < H < G. + When F and G are two faces of a polytope with F G, we call G/F : H F H G asection of . We may usually safelyP identify a face F with the={ section| } P F/F 1.ForafaceFthe section Fn/F is called the co-face of at F,orthevertex-figure at F if F is a vertex. P An n-polytope is regular if its (automorphism) group 0( ) is transitive on its P P flags. Let 8 : F 1,F0,...,Fn 1,Fn beafixedorbase flag of . The group 0( ) ={ } P P of a regular n-polytope is generated by distinguished generators 0,...,n 1 (with P respect to 8), where j is the unique automorphism which keeps all but the j-face of 8 fixed. These generators satisfy relations pij (2.1) (i j ) ε(i,j0,...,n 1), = = with (2.2) pii 1, pij pji 2 (i j), = = 6= and (2.3) pij 2ifij2. = || Regular Polytopes in Ordinary Space 451 The numbers pj : pj 1, j ( j 1,...,n 1) determine the (Schlafli¨ ) type p1,...,pn 1 of . Further, 0(=) has the intersection= property (with respect to the distinguished{ gen- } erators),P namely,P (2.4) i i I i i J i i I J for all I, J 0,...,n 1 . h | ∈ i∩h | ∈ i=h | ∈ ∩ i { } Observe that, in a natural way, the group of the facet of is 0,...,n 2 , while P h i that of the vertex-figure is 1,...,n 1 . By a C-group, we meanh a group which i is generated by involutions such that (2.1), (2.2), and (2.4) hold. If, in addition, (2.3) holds, then the group is called a string C-group. The group of a regular polytope is a string C-group. Conversely, given a string C-group, there is an associated regular polytope of which it is the automorphism group [12]. In verifying that a given group is a C-group, it is usually only the intersection property which causes difficulty. Note that Coxeter groups are examples of C-groups (see [12] and [21]). Given regular n-polytopes 1 and 2 such that the vertex-figures of 1 are isomorphic P P P to the facets of 2, we denote by 1, 2 the class of all regular (n 1)-polytopes P hP P i + P with facets isomorphic to 1 and vertex-figures isomorphic to 2.If 1, 2 , then P P hP P i6=∅ any such is a quotient of a universal member of 1, 2 ; this universal polytope is P hP P i denoted by 1, 2 (see [12], [17], and [20]). We end the{P generalP } discussion of regular polytopes and their groups with a useful remark. Let 0 0,...,n 1 be the group of a regular n-polytope , and suppose that 0. Then=h we can express i in the form P ∈ 0010 k 10k, = with i 00 : 1,...,n 1 , the group of the vertex-figure of at its base vertex ∈ =h i P v : F0, for i 0,...,k. With , we can associate a path in with k edges leading from = = P v to v.Ifk 0, the path consists of v ( v0) alone. For k > 0, let (E10 ,...,Ek0 1) = = be an edge-path associated with 0010 k 1. With is then associated the path (E1,...,Ek), given by E1 : Ek ( E0k ), = = Ei : Ei0 10k for i 2,...,k, = = where E : F1 is the base edge of . Of course, this path will not generally be unique, since it depends= on the particular expressionP for . Conversely, an edge-path (E1,...,Ek)from v corresponds to such an element 0, ∈ in which 0 occurs k times. If k > 0, then there is an k 00 such that E1 Ek . The ∈ = shorter path (E10 ,...,Ek0 1), given by 1 Ei0 : Ei 1k 0 = + for i 1,...,k 1, also starts at v, and we can repeat to obtain as above, with a free = choice of 0. 452 P. McMullen and E. Schulte In the context of group presentations, we deduce: (2.5) Theorem. Let P be a regular polytope. Then the group 0 0( ) of is determined by the group of its vertex-figure, and the relations on the= distinguishedP P generators of 0 induced by the edge-circuits of which contain the initial vertex. P Proof. A relation on 0 can be written in the form 0010 k 10 ε, = with i 00 for i 0,...,k 1, which corresponds to an edge-circuit starting and ∈ = ending at v. Conversely, such an edge-circuit is equivalent under 00 to one beginning with E, and this gives rise to a relation as above (now the element 0 will be determined by the circuit). This is the result. We now come to the geometric aspects of the theory. Following [10] and [11], a realization of a regular polytope is a mapping : 0 E of the vertex-set 0 into some euclidean space E, such thatP each automorphismP → of induces an isometryP of P V : 0; such an isometry extends to one of all of E, uniquely if we make the natural assumption= P that E aff V , the affine hull of V .
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