R70-9 A Graph-Theoretic Model for Periodic Discrete Structures

Wiring schemes which exploit the potential of modern integrated circuit technology are likely to exhibit a high degree of "periodicity," as well as various other attributes. The first contribution of this paper is to formulate a precise definition of periodicity in terms of a graph-theoretic model originated by Yoeli. Roughly speaking, an infinite graph is said to be periodic if it can be generated from a finite template which is moved (periodically) from point to point in the plane. This definition is interesting and useful, and generalizes nicely to infinite graphs in n-space. Now, suppose that in addition to periodicity we require planarity and point symmetry. (Roughly speaking, a graph is point symmetric if it "looks the same" from each of its vertices.) What types of infinite graphs satisfy all three of these requirements? The second contribution of this paper is to reveal that there are exactly eleven such graphs, and they correspond to the three regular and eight semi-regular tesselations of the plane. A planar tesselation is a set of polygons fitting together side by side which cover the plane simply and without gaps. A tesselation is regular if each face is the same regular polygon. The three regular tesselations are composed of triangles, squares, and hexagons, respectively. A semiregular tesselation permits the faces to be different, but the same regular polygons must surround each vertex. An example of a semi-regular tesselation is one in which each vertex is surrounded by a square and two octagons. It is interesting to compare these results with those of Eric Wagner, who has considered various wiring schemes that do not meet all ofTurner's conditions. For example, Wagner proposes an array which would appear