Spread polynomials, rotations and the butterfly effect

TL;DR

This paper explores the properties of spread polynomials in rational trigonometry, revealing periodicity patterns and connections to finite fields and non-Euclidean geometries, offering new insights into geometric transformations.

Contribution

It uncovers a novel periodicity in spread polynomials related to powers of rational spread rotations and links rational trigonometry with finite field and non-Euclidean geometries.

Findings

Discovered periodicity in prime power decomposition of spread polynomial values.

Linked spread polynomials to Chebyshev polynomials in rational trigonometry.

Connected rational trigonometry concepts with finite fields and non-Euclidean geometries.

Abstract

The spread between two lines in rational trigonometry replaces the concept of angle, allowing the complete specification of many geometrical and dynamical situations which have traditionally been viewed approximately. This paper investigates the case of powers of a rational spread rotation, and in particular, a curious periodicity in the prime power decomposition of the associated values of the spread polynomials, which are the analogs in rational trigonometry of the Chebyshev polynomials of the first kind. Rational trigonometry over finite fields plays a role, together with non-Euclidean geometries.