Polyiamonds Attaining Extremal Topological Properties, Part II

TL;DR

This paper constructs minimal-tile polyiamonds with any number of holes and proves structural properties, showing multiple solutions exist for three or more holes, advancing understanding of polyiamond topology.

Contribution

It introduces a method to construct minimal polyiamonds with any number of holes and establishes their structural properties, revealing multiple solutions for higher hole counts.

Findings

Constructed polyiamonds with h holes for all h ≥ 1.

Proved structural conditions satisfied by crystallized polyiamonds.

Demonstrated multiple distinct solutions for h ≥ 3.

Abstract

In Part II of this work, we construct crystallized polyiamonds with $h$ holes for every $h\ge1$, that is polyiamonds which use the fewest possible tiles necessary to enclose $h$ holes. Furthermore, we prove that crystallized polyiamonds satisfy a set of structural conditions, and for every $h\ge 3$ there are multiple distinct crystallized polyiamonds with $h$ holes.