Abstract: A multidimensional interconnection and routing apparatus for a parallel processing computer connects together processing elements in a three-dimensional structure. The interconnection and routing apparatus includes a plurality of processing element nodes. A communication connects at least one of the processing elements with a host system. An interconnection network connects together the processing element nodes in an X, Y, and Z dimension. The network includes communication paths connecting each of the plurality of processing elements to adjacent processing elements in the plus and minus directions of each of the X, Y, and Z dimensions.