Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, we introduce and study Heronian friezes, the Euclidean analogues of Coxeter's frieze patterns. We prove that a generic Heronian frieze possesses the glide symmetry (hence is periodic), and establish the appropriate version of the Laurent phenomenon. For a closely related family of Cayley-Menger friezes, we identify an algebraic condition of coherence, which all friezes of geometric origin satisfy. This yields an unambiguous propagation rule for coherent Cayley-Menger friezes, as well as the corresponding periodicity results.

中文翻译：

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苍鹭楣

受欧几里德平面上点配置的计算几何学和 A 类簇代数理论的启发，我们介绍和研究了 Heronian 饰带，Coxeter 饰带图案的欧几里得类似物。我们证明了一个通用的 Heronian 楣具有滑动对称性（因此是周期性的），并建立了 Laurent 现象的适当版本。对于一个密切相关的 Cayley-Menger 带状体家族，我们确定了所有几何起源的带状体都满足的代数相干条件。这产生了相干 Cayley-Menger 带状体的明确传播规则，以及相应的周期性结果。