Edge lengths determining tetrahedrons
Edge lengths determining tetrahedrons
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边长决定四面体
DOI：
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
A. Dreiding
中科院分区：
文献类型：
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作者：
K. Wirth;A. Dreiding
一个四面体若要真正存在，其四面的边长显然必须服从三角形不等式。这个条件是必要的，但不足以使六个边长组成一个四面体。例如，即使三角形不等式成立，也不存在五条边长为4、一条边长为7的四面体。只要考虑两个边长为4的等边作为四面体的面;剩余的边长必须小于4 <$3（< 6.93），因为这是当四面体退化时达到的极值（见图1）。什么时候六个给定的长度是某个四面体的边长？这个问题在文献中已经被讨论过好几次了（Menger，布卢门塔尔，Dekster and Wilker，Herzog，见下文），大多数甚至是针对d维单形的一般情况。目前的工作，作为一个分支，我们原来的调查有关四面体结构在有机化学中，仅限于三维的情况。此限制
For a tetrahedron to actually exist, the edge lengths of each of its four faces evidently must obey the triangle inequality. This condition is necessary but not sufficent for six edge lengths to make up a tetrahedron. There does, for example, not exist a tetrahedron with five edges of length 4 and one edge of length 7, even though the triangle inequalities are fulfilled. Just consider two equilaterals with edge length 4 as faces of a tetrahedron; the remaining edge length must be smaller than 4 √ 3 (< 6.93), since this is the extreme value reached when the tetrahedron becomes degenerate (see Fig. 1). When are six given lengths the edge lengths of some tetrahedron? This question has been addressed to already several times in the literature (Menger, Blumenthal, Dekster and Wilker, Herzog, see below), mostly even for the general case of d-dimensional simplices. The present work, as an offshoot of our original investigations concerning tetrahedral structures in organic chemistry, is restricted to the 3-dimensional case. This restriction