论文标题
Plattenbauten：触摸太空的矩形
Plattenbauten: Touching Rectangles in Space
论文作者
论文摘要
平面两分的图可以表示为$ \ mathbb {r}^2 $中的水平和垂直段的触摸图。我们研究了空间的概括：$ \ mathbb {r}^3 $中的轴对准矩形的触摸图，并证明可以以这种方式表示平面3上色图。结果意味着Eppstein和Mumford先前获得的角落多面体的表征。我们证明的副产品是带有给定骨架的正交表面的分布晶格结构。此外，我们通过$ \ mathbb {r}^3 $在轴线对准的非环形矩形中研究表示形式，使所有区域都是框。我们表明，所得图对应于八面体的八面体。这概括了平面四边形与$ \ Mathbb {r}^2 $的水平和垂直段家族之间的对应关系与所有区域都是矩形的属性。
Planar bipartite graphs can be represented as touching graphs of horizontal and vertical segments in $\mathbb{R}^2$. We study a generalization in space: touching graphs of axis-aligned rectangles in $\mathbb{R}^3$, and prove that planar 3-colorable graphs can be represented this way. The result implies a characterization of corner polytopes previously obtained by Eppstein and Mumford. A by-product of our proof is a distributive lattice structure on the set of orthogonal surfaces with given skeleton. Further, we study representations by axis-aligned non-coplanar rectangles in $\mathbb{R}^3$ such that all regions are boxes. We show that the resulting graphs correspond to octahedrations of an octahedron. This generalizes the correspondence between planar quadrangulations and families of horizontal and vertical segments in $\mathbb{R}^2$ with the property that all regions are rectangles.
