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The use of parallel memories in SIMD machines requires special data mappings, known as ``skewing schemes,'' for storing matrices for the purposes of efficient vector computations. Some schemes have explicitly been implemented in current super-computers. Periodic skewing schemes are of particular interest because they have a regular structure and can be represented by simple formulas. In this paper we show that periodic skewing schemes have an elegant foundation in the mathematical theory of integral lattices and Z-modules, which leads to insightful proofs of a number of general properties of periodic skewing schemes in all dimensions.