Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field $\mathbb{Q}(\sqrt{d})$ where $d<0$. We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non-zero frieze patterns for any given height. We then show that apart from the cases $d\in \{-1,-2,-3,-7,-11\}$ all non-zero frieze patterns over the rings of integers ${\mathcal{O}}_d$ for $d<0$ have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns.

中文翻译：

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代数数上的楣图案

康威和考克塞特已经证明，正有理整数上的饰带图案与多边形的三角剖分是双射的。 Jørgensen 和前两位作者最近发起了一项针对复数其他子集的饰带图案的研究。在本文中，我们首先证明代数数环具有有限多个单元当且仅当它是二次数域中的阶数$\mathbb{问}（\sqrt{d}）$在哪里。我们得出的结论是，这些正是代数数环，对于任何给定的高度，在这些环上都有有限多个非零饰带图案。然后我们证明除了案例之外$d\epsilon \{-1,-2,-3,-7,-11\}$整数环上的所有非零饰带图案${\mathcal{氧}}_d$为了只有完整的条目，因此被称为（扭曲的）康威-考克斯特饰带图案。