Building with Blocks: Enumerating Polyforms on Tilings

TL;DR

This paper presents an algorithm to count and classify polyform structures within periodic tilings, with applications in art, design, and puzzles, extending combinatorial enumeration to complex tiling patterns.

Contribution

It introduces a novel algorithm for enumerating n-celled polyforms on various tilings, accounting for symmetries, and applies it to both 2D and 3D structures.

Findings

Algorithm successfully counts polyforms in specific tilings

Provides numerical data for artistic and structural applications

Suggests new puzzles based on polyform structures

Abstract

In areas as diverse as contemporary art, play structures, climbing equipment, and modular construction toys, we see the presence of building block-like polyhedral complexes, which are generalizations of the pieces in the game Tetris. We give an algorithm for counting the number of $n$-celled structures on polygonal and polyhedral cells of certain periodic two- and three-dimensional tilings; moreover, we count these structures up to translations, rotations, and reflections of the tiling. We describe this algorithm with respect to structures in the snub square tiling, provide numerical data related to existing three-dimensional art and structures, and suggest puzzles based on these constructions.