Smillie and Weiss proved that the set of the areas of the minimal triangles of Veech surfaces with area 1 can be arranged as a strictly decreasing sequence { a n }. And each a n in the sequence corresponds to finitely many affine equivalent classes of Veech surfaces with area 1. In this article, we give an algorithm for calculating the area of the minimal triangles in a Veech surface and prove that the first element of a n which corresponds to non arithmetic Veech surfaces is (5 - √5)/20, which is uniquely realized by the area of the minimal triangles of the normalized golden L-shaped translation surface up to affine equivalence.

中文翻译：

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关于 Veech 曲面中最小三角形的面积

Smillie 和 Weiss 证明了面积为 1 的 Veech 曲面的最小三角形的面积集合可以排列为严格递减的序列 { an }。并且序列中的每个 an 对应有限多个面积为 1 的 Veech 曲面的仿射等效类。 在本文中，我们给出了计算 Veech 曲面中最小三角形面积的算法，并证明了 an 对应的第一个元素到非算术 Veech 曲面是 (5 - √5)/20，这是由归一化的金色 L 形平移曲面的最小三角形的面积唯一实现的，直到仿射等价。