Cremona-Richmond configuration

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Cremona-Richmond configuration
Rank2
TypeRegular generalized polygon
Elements
Edges15 triads
Vertices15
Vertex figureTriad
Related polytopes
DualCremona-Richmond configuration
Abstract & topological properties
Flag count45
Configuration symbol(153)

The Cremona-Richmond configuration, or the doily, is a configuration and generalized quadrangle.

Duads and systhemes[edit | edit source]

The Cremona-Richmond configuration with its vertices labeled as duads.

The Cremona-Richmond configuration can be constructed from a system of duads and systhemes from an alphabet of 6 characters, Σ . A duad is a set of two distinct elements from Σ . With an alphabet of size 6 there are 15 duads corresponding to the vertices of the Cremona-Richmond configuration. A systheme is a maximal set of duads such that each pair of duads is disjoint. With a alphabet of size 6 there are 15 systhemes each containing 3 duads. Permutations on Σ  are homomorphisms on the configuration forming its automorphism group. They are trivially transitive on its flags and thus the Cremona-Richmond configuration is regular.

The Cremona-Richmond configuration is the smallest non-trivial configruation which can be constructed this way (the smallest being the triad). The next smallest is a configuration of type (843).

Gallery[edit | edit source]

External links[edit | edit source]