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

Authors

Maxwell HoslerFollow

Abstract

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)+4r_p) such classes, where p is the order of the plane and r_p is the number of roots of x²-x-1 on the finite field of order p, a generalization of the golden ratio, ϕ.

Author Bio

While studying at the College of Wooster, Maxwell Hosler completed an independent research project as part of the College of Wooster’s senior capstone program. This paper is a condensation of the ideas developed in that paper. He is currently applying to graduate programs in mathematics, and hopes to pursue teaching and mathematical research as a career. Outside of mathematics, he finds creative outlets in things like digital drawing, 3D modeling, and small programming projects.

Faculty Sponsor

Dr. Robert Kelvey

Recommended Citation

Hosler, Maxwell (2024) "Counting the Classes of Projectively-Equivalent Pentagons on Finite Projective Planes of Prime Order," Rose-Hulman Undergraduate Mathematics Journal: Vol. 25: Iss. 2, Article 3.
Available at: https://scholar.rose-hulman.edu/rhumj/vol25/iss2/3