Building with Blocks: Enumerating Polyforms on Tilings
Bert Dobbelaere, Peter Kagey, Drake Thomas, Andr\'es R. Vindas-Mel\'endez

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 -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.
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Taxonomy
TopicsQuasicrystal Structures and Properties · Computational Geometry and Mesh Generation · Advanced Combinatorial Mathematics
