In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be denoted as (94 123) or configuration matrix [ 9 4 3 12 ] {\displaystyle \left[{\begin{smallmatrix}9&4\\3&12\\\end{smallmatrix}}\right]} . It is symmetric (point and line transitive) with 432 automorphisms. 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. It was introduced by Colin Maclaurin and studied by Hesse (1844), and is also known as Young's geometry, named after the later work of John Wesley Young on finite geometry.
Description 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. In the language of configurations the Hesse configuration has the notation (94 123), meaning that there are 9 points, 4 lines per point, 12 lines, and 3 points per line. With points indexed 1...9 in a 3x3 grid, can have configuration table. Columns are lines, indexing points.
The Hesse configuration has 18×4! = 432 automorphisms, doubling the symmetries of the related Hessian group. Every pair of points are connected within one line. This is seen in its complete adjacency matrix:
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