Geometric point-circle pentagonal geometries from Moore graphs

Klara Stokes, Milagros Izquierdo


We construct isometric point-circle configurations on surfaces from uniform maps. This gives one geometric realisation in terms of points and circles of the Desargues configuration in the real projective plane, and three distinct geometric realisations of the pentagonal geometry with seven points on each line and seven lines through each point on three distinct dianalytic surfaces of genus 57. We also give a geometric realisation of the latter pentagonal geometry in terms of points and hyperspheres in 24 dimensional Euclidean space.
From these, we also obtain geometric realisations in terms of points and circles (or hyperspheres) of pentagonal geometries with k circles (hyperspheres) through each point and k-1 points on each circle (hypersphere).


Uniform map, equivelar map, dessin d'enfants, configuration of points and circles

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ISSN: 1855-3974

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