论文标题
在用等离子体三角形的多面体实现上
On Polyhedral Realization with Isosceles Triangles
论文作者
论文摘要
回答约瑟夫·马尔基维奇（Joseph Malkevitch）提出的一个问题，我们证明存在一个多面体图，带有三角形的面孔，使得它的每个实现为凸多面体的图表都至少包含一个面孔，这是鳞状三角形。我们的构建基于kleetopes，并表明存在一个整数$ i $，因此所有凸$ i $ $ $ titerate的kleetopes均具有calceene脸。但是，我们还表明，所有三角形多面体图的kleetopes均具有非凸面的非跨性实现，其中所有面都是同步的。我们通过观察到道森（Dawson（2005）的球形瓷砖导致了第四个无限的凸多面（Convex Polyhedra），我们回答了Malkevitch的另一个问题，其中所有面孔都是一致的等化等质三角形，并将一个均为Malkevitch以前已知的三个家族。我们证明，具有一致性同步脸的凸多面体图的图具有界限，并且具有界限的主导尺寸集。
Answering a question posed by Joseph Malkevitch, we prove that there exists a polyhedral graph, with triangular faces, such that every realization of it as the graph of a convex polyhedron includes at least one face that is a scalene triangle. Our construction is based on Kleetopes, and shows that there exists an integer $i$ such that all convex $i$-iterated Kleetopes have a scalene face. However, we also show that all Kleetopes of triangulated polyhedral graphs have non-convex non-self-crossing realizations in which all faces are isosceles. We answer another question of Malkevitch by observing that a spherical tiling of Dawson (2005) leads to a fourth infinite family of convex polyhedra in which all faces are congruent isosceles triangles, adding one to the three families previously known to Malkevitch. We prove that the graphs of convex polyhedra with congruent isosceles faces have bounded diameter and have dominating sets of bounded size.
