Polynomial generated polygons

Abstract

A turtle geometric construction on the plane, called a polynomial generated polygon (PGP) and represented by 𝒫[sub]f,[sub]pᵐ, is generated from the sequence obtained from evaluating f(𝓍) ∈ ℤ [𝓍] over ℤ modulo pᵐ where p is a prime and m ∈ ℕ. Computational methods are developed to pre-calculate the symmetries exhibited by [sub]f,[sub]pᵐ for a given f and pᵐ.

These include procedures to find whether [sub]f,[sub]pᵐ is bounded or unbounded, the degree of rotational symmetry present, whether lines of reflectional symmetry can be observed, and in the case of 𝒫[sub]f,[sub]pᵐ unbounded, whether the PGP has a glide reflection.

Methods are also sought to find a suitable f and pᵐ to produce a desired 'feasible' shape in a PGP construction, and how the same shape might be generated modulo pᵐ⁺ᵏ if it cannot be produced modulo pᵐ.