Notable projective configurations include the following:
- (11), the simplest possible configuration, consisting of a point incident to a line.
- (32), the triangle. Each of its three sides meets two of its three vertices, and vice versa. More generally any polygon of n sides forms a configuration of type (n2)
- (43 62) and (62 43), the complete quadrangle and complete quadrilateral respectively.
- (73), the Fano plane. This configuration exists as an abstract incidence geometry, but cannot be constructed in the Euclidean plane.
- (83), the Möbius–Kantor configuration. This configuration describes two quadrilaterals that are simultaneously inscribed and circumscribed in each other. It cannot be constructed in Euclidean plane geometry but the equations defining it have nontrivial solutions in complex numbers.
- (93), the Pappus configuration.
- (94 123), the Hesse configuration of nine inflection points of a cubic curve in the complex projective plane and the twelve lines determined by pairs of these points. This configuration shares with the Fano plane the property that it contains every line through its points; configurations with this property are known as Sylvester–Gallai configurations due to the Sylvester–Gallai theorem that shows that they cannot be given real-number coordinates (Kelly 1986).
- (103), the Desargues configuration.
- (125302), the Schläfli double six, formed by 12 of the 27 lines on a cubic surface
- (153), the Cremona–Richmond configuration, formed by the 15 lines complementary to a double six and their 15 tangent planes
- (124 163), the Reye configuration.
- (166), the Kummer configuration.
- (6015), the Klein configuration.
Read more about this topic: Configuration (geometry)
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