Research on arrangements of geometric figures in space

空间几何图形排列研究

• 批准号: 17540127
• 负责人: MAEHARA Hiroshi
• 金额: $ 2.09万
• 依托单位: University of the Ryukyus
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2005
• 资助国家: 日本
• 起止时间: 2005 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540127/
• 关键词: co-unit-distance-graph; lattice points on a sphere; unit-sphere-system; chromatic number of balls; knotted cycle of balls; ball number; 単立球面システム; double six lines; 線球変換; double six spheres; unit-sphere-sytem; escribed sphere; almost halving hy; co-unit-distance-graph; lattice points on a sphere; unit-sphere-system; chromatic number of balls; knotted cycle of balls; ball number; 単立球面システム; double six lines; 線球変換; double six spheres; unit-sphere-sytem; escribed sphere; almost halving hy

(1) To characterize unit distance graphs on the plane, or to determine the sup of their chromatic numbers is very difficult problem, and no big progress has been made so far. In the present study, we enumerate the unit distance graphs in the planes whose complements are also unit distance graphs in the plane. The total number of such graphs is 69, among them, 55 graphs are connected, and 7 graphs are self-complementary. (Ajoint work with S. V. Gervacio and Y. F. Lim of De La Salle University, Manila)(2) Concerning a sphere that passes through a prescribed number of lattice points, we have the following result. For every integer n>d>m>l, there is a (hyper) sphere in the d-dimensional Euclidean space that passes through exactly n lattice points, and these n lattice points span an m dimensional polytope. This is a generalization of a result on a circle in the plane obtained by myself and M. Matsumoto in 1998.(3) Aset of d+2 unit spheres in d-space is called a d-dimensional unit-sphere-system if every d+1 spheres have non-empty intersection, but the intersection of all d+2 spheres is empty. There is no 1-dimensional unit-sphere-system, and there are many 2-dimensional unit-sphere-systems. In the present study, we could prove that for every d>3, there is a d-dimensional unit-sphere-system, and there is no 3-dimensional unit-sphere-system. This settles unit-sphere-systemproblem that has been unsettled since 1989.(4) Concerning families of solid balls in 3-space, we could slightly improve the bounds on their chromatic numbers, and the bounds on the number of balls necessary to make a knotted cycle.

（1）要表征平面上的单位距离图，或确定其色数的SUP是非常困难的问题，到目前为止还没有取得很大进展。在本研究中，我们列举了平面上的单位距离图，其补充也是平面中的单位距离图。此类图的总数为69，其中有55个图形，并且7个图是自相符的。 （Ajoint与Manila De La Salle大学的S. V. Gervacio和Y. F. Lim合作）（2）关于经过规定的晶格点的球体，我们会有以下结果。对于每个整数n> d> m> l，d维欧几里得空间中都有一个（超级）球，可以通过正好的N晶格点，而这些N晶格点横跨M维多型。这是我本人和M. Matsumoto在1998年获得的平面上的圆的概括。（3）d+2个单位球的ASET在D空间中的ASET称为D+1个d+1球，如果每个D+1球都有非空的相交，但是所有D+2个球体的相互作用都是空的。没有一维单元 - 球体系统，并且有许多2维单位球系统。在本研究中，我们可以证明，对于每个d> 3，都有一个D维单元 - 球体系统，并且没有三维单位球系统。这解决了自1989年以来一直未解决的单位 - 球形系统问题。（4）关于3个空间的实心球家族，我们可以稍微提高其色数的界限，以及制作结节循环所需的球数量的界限。