Counting the Classes of Projectively-Equivalent Pentagons on Finite Projective Planes of Prime Order

In this paper, we examine the number of equivalence classes of pentagons on finite projective planes of prime order under projective transformations. We are interested in those pentagons in general position, meaning that no three vertices are collinear. We consider those planes which can be constructed from finite fields of prime order, and use algebraic techniques to characterize them by their symmetries. We are able to construct a unique representative for each pentagon class with nontrivial symmetries. We can then leverage this fact to count classes of pentagons in general. We discover that there are (1/10)((p+3)(p-3)+4rp) such classes, where p is the order of the plane and rp is the number of roots of x2-x-1 on the finite field of order p, a generalization of the golden ratio, ϕ.