Lattice Computers for Approximating Euclidean Space

In the context of mesh-like, parallel processing computers for (i) Aapproximating continuous space and (ii) analog simulation of the motion of objects and waves in Acontinuous space, the Apresent paper is concerned with which mesh-like interconnection of processors Amight be particularly Asuitable for the task and why. Processor interconnection schemes based on nearest neighbor Aconnections in geometric lattices Aare presented along with motivation. Then two major threads are explored Aregarding which lattices Awould be good: the regular lattices, for their symmetry and other properties Ain common with continuous space, and the well-known mot lattices, for being, in a sense, the Alattices required for Aphysically natural basic algorithms for motion. The main theorem of the present paper implies that the well-known Alattice A(n) is the regular Alattice having the maximum number of nearest neighbors among the n-dimensional Aregular lattices. It is noted that the only n-dimensional lattices that are both regular and Aroot are A(n) and Z^n (Z^n Ais the lattice of n -cubes). The remainder of the paper specifies other Adesirable properties of A(n) Aincluding other ways it is superior to Z^n for our purposes.