Algebraic Formulation and Program Generation of Three-Dimensional Hilbert Space-Filling Curves

We use a tensor product based multi-linear algebra theory to formulate three-dimensional Hilbert space-filling curves. A 3-D Hilbert space-filling curve is specified as a permutation which rearranges threedimensional n n n data elements stored in the row major order as in C language or the column major order as in FORTRAN language to the order of traversing a 3-D Hilbert space-filling curve. The tensor product formulation of 3-D Hilbert space-filling curves uses stride permutation, reverse permutation, and Gray permutation. We present both recursive and iterative tensor product formulas of 3-D Hilbert space-filling curves. In addition, we derive a tensor product formula of inverse 3-D Hilbert space-filling curve permutation. The tensor product formulas are directly translated into computer programs which can be used in various applications. The process of program generation is explained in the paper.