Regular projective polyhedra with planar faces I

Abstract

Summary. This is the first of two papers in which we classify the regular projective polyhedra in P 3 with planar faces. Here, we develop the basic notions; we introduce a new diophantine trigonometric equation, which plays a key role in the classification theorem, relating the combinatorial and geometric parameters of such polyhedra, and conclude with the case in which the polyhedron is an embedded surface.Mathematics Subject Classification (1991). 52B, 52C, 51M20, 51F15.

Regular polygonal complexes in space, II

“…Following [18, p.166], we write Λ a for the sublattice of aZ 3 generated by a and its images under permutation and changes of sign of coordinates. Then Λ (1,0,0) = Z 3 is the standard cubic lattice, Λ (1,1,0) is the face-centered cubic lattice consisting of all integral vectors with even coordinate sum, and Λ (1,1,1) is the body-centered cubic lattice.…”

“…Moreover, for any four consecutive edges e, f, g, h of a helical face, the two edges shared by C e , C f , C g and C f , C g , C h , respectively, are adjacent edges (of a square face) of C f . Each of the six helical faces of K 1 (1, 1) around an edge e with vertices u, v is now determined by one of the six edges of C e that do not contain u or v. The vertex-figure of K 1 (1,1) at o coincides with the vertex-figure of K 3 (1, 2) at o, and hence is the double-edge graph of the cube with vertices (±a, ±a, ±a). The vertex-figure group is [3,4].…”

“…First note that if L is a simply flag-transitive polygonal complex with mirror vector (1, 1) obtained from a simply flag-transitive complex with mirror vector (1,2) by the operation λ 1 as in Section 3.1, then the axis of the half-turn R 1 for L must lie in the mirror of a plane reflection in G 2 (L), so in particular R 1 must leave this mirror invariant. But then Lemma 4.3 implies that the regular complex obtained from any such complex L by operation λ 0 must actually have face mirrors.…”

“…The full enumeration of the chiral polyhedra in E 3 in [25,26] (see also Pellicer & Weiss [24]), as well as a number of corresponding enumeration results for figures in higherdimensional euclidean spaces by McMullen [14,15,16] (see also Arocha, Bracho & Montejano [1] and [2]), are examples of recent successes of this program; for a survey, see [19]. 1 2 w. Then there is a regular 4-apeirotope in E 3 , denoted apeir Q, with T 0 , T 1 , T 2 , T 3 as the generating reflections of its symmetry group, o as initial vertex, and Q as vertex-figure.…”

“…In our previous paper [23] we already dealt with the complexes with mirror vector (1,2). In this paper, we complete the enumeration of the simply flag-transitive regular complexes and describe the complexes for the remaining mirror vectors.…”