Regular polygonal complexes in space, I

Abstract

Abstract. A polygonal complex in Euclidean 3-space E 3 is a discrete polyhedron-like structure with finite or infinite polygons as faces and finite graphs as vertex-figures, such that a fixed number r 2 of faces surround each edge. It is said to be regular if its symmetry group is transitive on the flags. The present paper and its successor describe a complete classification of regular polygonal complexes in E 3 . In particular, the present paper establishes basic structure results for the symmetry groups, discusses geometric and algebraic aspects of operations on their generators, characterizes the complexes with face mirrors as the 2-skeletons of the regular 4-apeirotopes in E 3 , and fully enumerates the simply flag-transitive complexes with mirror vector (1, 2). The second paper will complete the enumeration.

Regular polygonal complexes in space, II

“…In fact, we proved in [23] that λ 1 , applied to a regular complex with face mirrors, always yields another regular complex with face mirrors. …”

“…As pointed out in [23, p. 6692], there are examples where the new complex K λ 0 is not simply flag-transitive but rather has face-mirrors and possibly a strictly larger symmetry group; the latter depends on whether or not the reflections in the face-mirrors of K λ 0 are also symmetries of K (see Section 4.2). However, by mistake, this possibility was not carried forward to the wording of Lemma 5.1 in [23]. Our new version corrects this error.…”

“…Lemma 2.1 is a slightly revised version of Lemma 5.1 in [23], which was incorrect as stated. As pointed out in [23, p. 6692], there are examples where the new complex K λ 0 is not simply flag-transitive but rather has face-mirrors and possibly a strictly larger symmetry group; the latter depends on whether or not the reflections in the face-mirrors of K λ 0 are also symmetries of K (see Section 4.2).…”

“…Now the faces are helices over squares with their axes parallel to the coordinate axes; in particular, the axis of the base face F 2 is parallel to the y-axis and the projection of [3,3]. Note that the common edge graph of K 4 (1, 1) and K 7 (1, 2) is the famous diamond net modeling the diamond crystal (see [23], as well as [18, p. 241] and [27, pp. 117,118]).…”

“…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.…”