A closed piecewise linear curve is called integral if it is composed of unit intervals. Kenyon’s problem asks whether for every integral curve $\gamma $ in ${\mathbb{R}}^3$, there is a dome over $\gamma $, that is, whether $\gamma $ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $\gamma $ is a quadrilateral, thus giving a negative solution to Kenyon’s problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular $n$-gons.

中文翻译：

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曲线上的圆顶

如果一条闭合的分段线性曲线由单位区间组成，则称为积分曲线。肯扬问题问对于${\mathbb{R}}^3$中的每条积分曲线$\gamma$，是否在$\gamma$上都有一个圆顶，即$\gamma$是否是多面体曲面的边界其面是具有单位边长的等边三角形。首先，当$\gamma $ 是一个四边形时，我们给出一个代数必要条件，从而全面地给出了肯扬问题的负解。然后我们证明圆顶存在于一组密集的积分曲线上。最后，我们在所有常规 $n$-gons 上给出了一个明确的圆顶构造。