Definition
The Reye configuration is a geometric arrangement consisting of a set of points and lines (or, more generally, elements and blocks) with a regular incidence pattern. In its most commonly cited form it comprises twelve points and sixteen lines, each point lying on four lines and each line containing three points.
Overview
The configuration is studied within the fields of combinatorial geometry and incidence geometry. It exemplifies a (12₄, 16₃) configuration, where the subscripts indicate the number of incidences per element: each of the 12 points is incident with 4 lines, and each of the 16 lines is incident with 3 points. Such regular configurations are of interest for their symmetry properties and for their connections to other mathematical structures, including projective planes, cubic surfaces, and certain graph embeddings.
Etymology/Origin
The configuration is named after the mathematician who first described it, though detailed biographical information is limited. The precise origin of the term “Reye” in this context is not well documented in widely available encyclopedic sources.
Characteristics
- Incidence Structure: (12₄, 16₃) – 12 points, 16 lines, with the above incidence counts.
- Symmetry: The configuration possesses a non‑trivial automorphism group; its symmetry can be realized by certain permutations of the points that preserve the line incidences.
- Construction: Various geometric constructions have been proposed, including:
- Deriving the points and lines from the intersection patterns of particular planes and lines in three‑dimensional projective space.
- Obtaining the configuration from special arrangements on a cubic surface that contains 27 lines, by selecting a subset that satisfies the (12₄, 16₃) incidence.
- Applications: It serves as a test case for theories concerning combinatorial designs, incidence geometries, and the classification of finite configurations.
Related Topics
- (pₖ, qₗ) configurations – general notation for regular incidence structures.
- Desargues configuration – a well‑known (10₃, 10₃) configuration.
- Projective geometry – the broader mathematical context in which such configurations are studied.
- Cubic surfaces and the 27 lines – a classical source of related incidence patterns.
- Combinatorial design theory – the study of balanced and symmetric arrangements of elements.
Note: Accurate information about the historical provenance and some specific construction methods of the Reye configuration is not fully confirmed in the publicly available encyclopedic literature.