Observations on the hex clusters of the Spectre tilings

TL;DR

This paper studies the structure of hexagonal clusters within Spectre tilings, providing a non-computer-assisted proof of their existence and unique hierarchy.

Contribution

It offers a new, human-readable proof of the existence and hierarchical structure of Spectre tilings with hex clusters, enhancing understanding of their geometric properties.

Findings

Confirmed the existence of Spectre tilings.

Established the unique hierarchical structure of the clusters.

Provided a non-computer-assisted proof.

Abstract

Decorating the Spectre tile with hexagons reveals triangular hexagonal clusters whose structure we study. In the process we reprove that the Spectre tilings exist and are uniquely hierarchical. The proof is not computer-assisted.