Square Matrix Permutations

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Repeated application of a particular permutation of the elements of an matrix based on traversing diagonals results in the original matrix. The number of iterations for to complete the cycle is the same as the length of the period of the periodic sequence that results from the Fibonacci sequence modulus known as the Pisano period.

Contributed by: Noel Patson (March 2011)
Based on work by: Eric W. Weisstein
Open content licensed under CC BY-NC-SA



For a 3×3 matrix the permutation takes into , for example.

The size of the matrix at the top left is given by the "size of first matrix" slider. This matrix and the next 19 matrices are represented at the same time. Holding the mouse over any matrix reveals its size and the corresponding Pisano period.

Use the "permutation" slider to rearrange the elements of each matrix and watch the matrices synchronize and desynchronize with each other. Choose a different "color scheme" to enhance viewing the weaving patterns.

The "cycle length" is the least common multiple of the 20 Pisano periods corresponding to the 20 displayed matrices.

Reference: Pisano Period and Permutations ofn×nMatrices

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