Unilateral and Equitransitive Tilings by Squares of Sizes 1, 2, 3, and 4
Abstract
In this article we examine a problem stated in Grübaum and Shephard's classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes 1, 2, 3, …, n where n ∈ ℕ. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard's question for the case n = 4, as well as developing methods and computer algorithms that may be extended to the cases where n ≥ 5.
Related
- Automata, groups, limit spaces and tilings
- Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles
- Two examples in amplituhedron tilings
- Regular hyperbolic tilings have no ℓ2 eigenfunctions
- The 1/3-phenomenon of placement probabilities of tilings in the semiregular hexagon
BibTeX
@misc{arrabi2026unilateral,
title = {{Unilateral and Equitransitive Tilings by Squares of Sizes 1, 2, 3, and 4}},
author = {Layan Arrabi and Casey Mann and Fatima Muniz and Harmony Vargas and Rachel Villafranca},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10566},
eprint = {2610.10566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10566},
}
RIS
TY - UNPB
TI - Unilateral and Equitransitive Tilings by Squares of Sizes 1, 2, 3, and 4
AU - Layan Arrabi
AU - Casey Mann
AU - Fatima Muniz
AU - Harmony Vargas
AU - Rachel Villafranca
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.10566
UR - https://arxiv.org/abs/2610.10566
AB - In this article we examine a problem stated in Grübaum and Shephard's classic book on tiling theory, Tilings and Patterns, which asks the reader to find all unilateral and equitransitive tilings by squares of sizes 1, 2, 3, …, n where n ∈ ℕ. The unilaterality condition requires that two squares of the same size in such a tiling may not intersect along an entire side, and the equitransitivity requires that two tiles of the same size in such a tiling must be in the same transitivity class with respect to the symmetry group of the tiling. These two conditions on tilings by squares allows for surprisingly interesting and complex tilings by squares. In this work we provide the answer to Grübaum and Shephard's question for the case n = 4, as well as developing methods and computer algorithms that may be extended to the cases where n ≥ 5.
ER -