<p><b>Abstract</b>—The honeycomb mesh, based on hexagonal plane tessellation, is considered as a multiprocessor interconnection network. A honeycomb mesh network with n nodes has degree 3 and <tmath>${\rm diameter}\approx 1.63\sqrt {\rm n}-1,$</tmath> which is 25 percent smaller degree and 18.5 percent smaller diameter than the mesh-connected computer with approximately the same number of nodes. Vertex and edge symmetric honeycomb torus network is obtained by adding wraparound edges to the honeycomb mesh. The network cost, defined as the product of degree and diameter, is better for honeycomb networks than for the two other families based on square (mesh-connected computers and tori) and triangular (hexagonal meshes and tori) tessellations. A convenient addressing scheme for nodes is introduced which provides simple computation of shortest paths and the diameter. Simple and optimal (in the number of required communication steps) routing, broadcasting, and semigroup computation algorithms are developed. The average distance in honeycomb torus with n nodes is proved to be approximately <tmath>$0.54\sqrt {\rm n}.$</tmath> In addition to honeycomb meshes bounded by a regular hexagon, we consider also honeycomb networks with rhombus and rectangle as the bounding polygons.</p>