论文标题
圆顶曲线
Domes over curves
论文作者
论文摘要
如果闭合的分段线性曲线由单位间隔组成，则称为积分曲线。肯尼恩的问题询问是否针对$ \ mathbb {r}^3 $中的每个积分曲线$γ$，都有一个超过$γ$的圆顶，即$γ$是否是多面体表面的边界，其面部的面是带有单位边缘的等边三角形。首先，当$γ$是四边形时，我们给出了代数必要的条件，从而为肯尼恩的问题提供了负面的解决方案。然后，我们证明圆顶存在于一组密集的积分曲线上。最后，我们在所有常规$ n $ gons上对圆顶进行了明确的结构。
A closed piecewise linear curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve $γ$ in $\mathbb{R}^3$, there is a dome over $γ$, i.e. whether $γ$ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when $γ$ 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.
