Astral (n4) configurations

A family of n points and n (straight) lines in the Euclidean plane is said to be an (n4) configuration provided each point is on four of the lines and each line contains four of the points. A configuration may have various symmetries, that is, there may exist isometric mappings of the plane onto itself that map the configuration onto itself; all the symmetries of a configuration form its group of symmetries. It is obvious that no more than two points of a line can be in the same transitivity class with respect to the group of symmetries, and no more than two lines passing through one point can be in the same transitivity class unless all lines pass through that point. Hence, under its symmetry group each (n4) configuration must have at least two transitivity classes of points, and at least two equivalence classes of lines. An (n4) configuration is called astral if its points, as well as its lines, form precisely two transitivity classes. While it is not completely trivial that any astral (n4) configurations exist, there is, in fact, a large number of possibilities which will be precisely described below. A few examples of astral (n4) configurations were given in an earlier paper [2], and additional ones are shown in Figure 1. The aim of the present paper is to give this description, and to explain how it was found and established. In order to proceed, we need some notation. It is easily verified that the points of any astral (n4) configuration must lie at the vertices of two concentric regular m-gons, where m = n/2, and that the lines of the configuration must be determined by common diagonals of these m-gons. The size b of a diagonal D of a regular m-gon is the number of sides of P bridged by D; hence the angle subtended by D at the center of P is 2pib/n = pib/m, and we need consider only the range 2 ≤ b ≤ m-1. W