Cremona-Richmond configuration
Cremona-Richmond configuration | |
---|---|
Rank | 2 |
Type | Regular generalized polygon |
Elements | |
Edges | 15 triads |
Vertices | 15 |
Vertex figure | Triad |
Related polytopes | |
Dual | Cremona-Richmond configuration |
Abstract & topological properties | |
Flag count | 45 |
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 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]
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An alternative realization using circular arcs to acheive pentagonal symmetry. The name doily comes from the appearance of this realization.
External links[edit | edit source]
- Wikipedia contributors. "Cremona-Richmond configuration".
- Weisstein, Eric W. "Cremona-Richmond Configuration" at MathWorld.
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