Holey Hyperbolic Polyforms

TL;DR

This paper investigates polyforms in hyperbolic tessellations, focusing on minimizing tiles for a given number of holes, providing bounds, exact cases, and structural conditions.

Contribution

It introduces the function g_{p,q}(h) for minimal tiles in hyperbolic polyforms with holes and establishes bounds, exact values, and structural criteria.

Findings

Established bounds for g_{p,q}(h)

Computed exact values in small cases

Provided structural conditions for polyforms with holes

Abstract

A polyform is a planar figure formed by gluing congruent regular polygons along entire edges. We study polyforms in hyperbolic ${p,q}$-tessellations and the extremal problem of minimizing the number of tiles needed to realize exactly $h$ holes. Denoting this minimum by $g_{p,q}(h)$, we establish general lower and upper bounds, compute exact values in several small cases, and give a sufficient structural condition for a polyform to have $h$ holes and $g_{p,q}(h)$ tiles.