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Many problems in scientific computation involve sparse matrices. While dense matrix computations can be parallelized relatively easily, sparse matrices with arbitrary or irregular structure pose a real challenge to the design of highly parallel machines. In this paper we propose a new parallel architecture for sparse matrix computation based on finite projective geometries. Mathematical structure of these geometries play an important role in defining the pattern of interconnection between memories and processors as well as solving several difficult problems arising in parallel systems (such as load balancing, data-routing, memory-access conflicts etc.) in an efficient manner.