Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry

TL;DR

This paper presents algorithms to generate all isohedral tilings with polyominoes or polyiamonds as fundamental domains, focusing on specific rotational symmetries, and provides enumeration data for small tile sizes.

Contribution

The work introduces comprehensive algorithms and enumeration tables for isohedral tilings with polyominoes and polyiamonds under certain rotational symmetries, expanding previous research.

Findings

Complete sets of isohedral tilings generated for small n

No tilings found with certain symmetry groups using polyominoes or polyiamonds

Enumeration tables provided for small tile sizes

Abstract

We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with symmetry groups p3m1, p4m, or p6m that have polyominoes or polyiamonds as fundamental domains. We display the algorithms' output and give enumeration tables for small values of n. This expands on our earlier works (Fukuda et al 2006, 2008).