Finite geometries associated with dihedral groups of regular polygons.

Abstract

The phrase "finite geometry associated with the dihedral group of a regular polygon" is introduced and defined. Two such geometries are developed in detail: the finite geometries associated with the dihedral groups of the square and the regular hexagon. Models are developed for each of these geometries. A method for developing such a geometry associated with the dihedral group of any even-sided polygon is outlined, as well as a way to develop a model for the geometry. The significance of two subgroups of the dihedral group in the development of both the geometry and the model is emphasized. As stated before, the generalized methods are only applicable to finite geometries associated with dihedral groups of order 4n.