Hesse configuration

[3][4] The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements.

That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points (x, y) satisfying a linear equation ax + by = c (mod 3).

The Hesse configuration has 216 symmetries (its automorphism group has order 216).

[8] The Hessian polyhedron is a representation of the Hesse configuration in the complex plane.

Because the Hesse configuration disobeys the Sylvester–Gallai theorem, it has no Euclidean realization.

The Hesse configuration, with four of its lines (the four broken diagonals of the 3×3 array of points) drawn as curves