Hesse configuration

In geometry, the Hesse configuration, introduced by Colin Maclaurin and studied by Hesse (1844), is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane.

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). Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board.

Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals. Each line contains three points, so in the language of configurations the Hesse configuration has the notation 9₄12₃.

The automorphism group of the Hesse configuration has order 216 and is known as the Hessian group.

Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8₃8₃, the Möbius–Kantor configuration.

In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9₃9₃, the Pappus configuration.

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