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arXiv · 2307.00440

Friezes over $\mathbb Z[\sqrt{2}]$

Esther Banaian·Libby Farrell·Amy Tao·Kayla Wright·Joy Zhichun Zhang
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Abstract

A frieze on a polygon is a map from the diagonals of the polygon to an integral domain which respects the Ptolemy relation. Conway and Coxeter previously studied positive friezes over $\mathbb{Z}$ and showed that they are in bijection with triangulations of a polygon. We extend their work by studying friezes over $\mathbb Z[\sqrt{2}]$ and their relationships to dissections of polygons. We largely focus on the characterization of unitary friezes that arise from dissecting a polygon into triangles and quadrilaterals. We identify a family of dissections that give rise to unitary friezes and conjecture that this gives a complete classification of dissections which admit a unitary frieze.

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Esther Banaian, Libby Farrell, Amy Tao, Kayla Wright, Joy Zhichun Zhang. 2024-07-25. Friezes over $\mathbb Z[\sqrt{2}]$. https://doi.org/10.2140/involve.2025.18.683

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