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On Polyhedral Realization with Isosceles Triangles
arXiv - CS - Computational Geometry Pub Date : 2020-08-31 , DOI: arxiv-2009.00116 David Eppstein
arXiv - CS - Computational Geometry Pub Date : 2020-08-31 , DOI: arxiv-2009.00116 David Eppstein
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.
中文翻译:
等腰三角形的多面体实现
回答 Joseph Malkevitch 提出的一个问题,我们证明存在一个多面体图,具有三角形面,使得它作为凸多面体图的每个实现都至少包括一个不等边三角形的面。我们的构造基于 Kleetopes,并表明存在一个整数 $i$,使得所有凸 $i$ 迭代的 Kleetopes 都有一个不等边面。然而,我们还表明三角多面体图的所有 Kleetopes 都具有非凸非自交叉实现,其中所有面都是等腰的。我们通过观察 Dawson (2005) 的球形平铺导致第四个无限凸多面体家族来回答 Malkevitch 的另一个问题,其中所有面都是全等等腰三角形,在 Malkevitch 之前已知的三个家族中增加一个。
更新日期:2020-09-02
中文翻译:
等腰三角形的多面体实现
回答 Joseph Malkevitch 提出的一个问题,我们证明存在一个多面体图,具有三角形面,使得它作为凸多面体图的每个实现都至少包括一个不等边三角形的面。我们的构造基于 Kleetopes,并表明存在一个整数 $i$,使得所有凸 $i$ 迭代的 Kleetopes 都有一个不等边面。然而,我们还表明三角多面体图的所有 Kleetopes 都具有非凸非自交叉实现,其中所有面都是等腰的。我们通过观察 Dawson (2005) 的球形平铺导致第四个无限凸多面体家族来回答 Malkevitch 的另一个问题,其中所有面都是全等等腰三角形,在 Malkevitch 之前已知的三个家族中增加一个。