GL(3,n) Acting on 3D Points

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The three-dimensional space contains 125 points: , , … , , . The 1,488,000 invertible 3×3 matrices over form the general linear group known as . They act on by matrix multiplication modulo 5, permuting the 125 points.

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More generally, is the set of invertible matrices over the field With shifted to the center, the matrix actions on the points make symmetrical patterns.

The controls let you choose a modulus, which then computes the size of the group. A 3×3 matrix can then be chosen. If the determinant is zero or the determinant and modulus share a factor, the matrix is not invertible and thus not in the group. The chances of being in the group are roughly

The matrix and its inverse (if it exists) are shown, along with their determinants. Finally, a grid of the loops is shown.

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Contributed by: Ed Pegg Jr (December 2017)
Open content licensed under CC BY-NC-SA


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