Systematic celestial 4-configurations

english

2014

volume 7, issue 2

configurations   incidence geometry

Univerza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije
(Obvezni izvod spletne publikacije)

Celestial 4-configurations are a class of highly symmetric geometric configurations of points and lines in the plane in which 4 points lie on each line and 4 lines pass through each point (that is, they are ?$(n_4)$? configurations). The set of isometries of the plane that map a configuration to itself (that is, the symmetries of the configuration) partition the points into orbits, called the symmetry classes of points, and likewise the symmetries of the configuration partition the lines into orbits as well, forming the set of symmetry classes of lines. Celestial 4-configurations have the property that two lines from each of two symmetry classes of lines pass through each point, and two points from each of two symmetry classes of points lie on each line; a celestial 4-configuration with ?$k?$ symmetry classes is called ?$k$?-celestial. Celestial configurations may be classified as being trivial, systematic, or sporadic. Previously, three non-trivial classes of 3-celestial 4-configurations were known. This paper presents a number of new systematic families of celestial 4-configurations, including 16 new 3-celestial families, four 4-celestial families, and three classes of ?$h$?-celestial configurations for infinitely many values of ?$h$?, although it does not provide a complete classification.

29.05.2020