GL(3,n) Acting on 3D Points
GL(3,n) Acting on 3D Points
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.
5
5
5
(0,0,0)
(0,0,1)
(4,4,3)
(4,4,4)
5
GL(3,5)
5
5
5
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.
GL(n,k)
n×n
k
.
(0,0,0)
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
2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
33% | 57% | 33% | 76% | 19% | 84% | 33% | 57% |
The matrix and its inverse (if it exists) are shown, along with their determinants. Finally, a grid of the loops is shown.