Configuration (geometry) - Higher Dimensions

Higher Dimensions

The concept of a configuration may be generalized to higher dimensions, for instance to points and lines or planes in space. In such cases, the restrictions that no two points belong to more than one line may be relaxed, because it is possible for two points to belong to more than one plane.

Notable three-dimensional configurations are the Möbius configuration, consisting of two mutually inscribed tetrahedra, Reye's configuration, consisting of twelve points and twelve planes, with six points per plane and six planes per point, the Gray configuration consisting of a 3×3×3 grid of 27 points and the 27 orthogonal lines through them, and the Schläfli double six, a configuration with 30 points, 12 lines, two lines per point, and five points per line.

A further generalization is obtained in three dimensions by considering incidences of points, lines and planes, or j-spaces (0 ≤ j < 3), where each j-space is incident with Njk k-spaces (jk). Writing for the number of j-spaces present. a given configuration may be represented by the matrix:

The principle extends generally to n dimensions, where 0 ≤ j < n. Such configurations are related mathematically to regular polytopes.

Read more about this topic:  Configuration (geometry)

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