Study on arrangements of solid balls in 3-space

3维空间中实心球排列的研究

基本信息

批准号：
11640129
负责人：
MAEHARA Hiroshi
金额：
$ 1.54万
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：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640129/
关键词：
arrangement of balls knotted necklace piercing balls almost halving-plane representation of a graph almost halving plane piercing number linked 4-pair

项目摘要

1. A cyclic sequence of nonoverlapping unit balls in R^3 in which each consecutive balls are tangent, is called a necklace of pearls. We show that to make a knotted necklace of pearls, 15 unit balls are sufficient. To make a knotted necklace that can be inscribed between a pair of parallel planes with distance 2+√<2> apart, 16 unit balls are necessary, and the trefoil is the unique knot that can be made by 16 unit balls.2. A chain is a finite sequence of balls in which each consecutive pair of balls are tangent. Make a graph by representing vertices by balls, and edges by chains connecting two vertex-balls. Let b_n be the minimum number of balls necessary to make a complete graph of n vertices. Then we got the bound c_1n^3<b_n<c_2n^3 log n. A similar bound is also obtained when we use balls all sitting on a fixed table.3. For a family F of balls in d-dimensional space R^d, let λ= λ(F)=(the max. radius) / (the min. radius). We proved that for any family of n balls in R^d, there is a direction such that any line with this direction intersects at most O (√<(1+logλ)n log n>) balls. On the otherhand, for n【greater than or equal】d, there is a family of nonoverlapping n balls in R^d such that for any direction, there is a line with this direction that intersects at least n-d+1 balls. For a family of balls sitting on a fixed table in R^3, we also got an upper bound of the average number of balls pierced by a vertical line meeting the table.4. If a family of nonoverlapping balls in R^3 satisfies that logλ=o ((n/log n)^<1/3>), then there is a plane both sides of which contain n/2-o (n) intact balls.

1。r^3中非重叠的单位球的环状序列，其中每个连续的球是切线的，称为珍珠项链。我们表明，要制作一条打结的珍珠项链，15个单位球就足够了。要制作一条打结的项链，可以在一对具有2+√<2>相距的平行平面之间刻有16个单位球，而三叶线是可以由16个单位球制成的独特结。2。链条是一个有限的球序列，其中每对连续的球是切线的。通过用球来表示顶点，并通过连接两个顶点球的链条来制作图形。令b_n为完整的n个顶点所需的最小球数。然后，我们得到了绑定的C_1N^3 <b_n <c_2n^3 log n。当我们使用所有坐在固定桌上的球时，也会获得类似的界限3。对于在d维空间中的小球系列r^d，令λ=λ（f）=（最大半径） /（最小半径）。我们证明，对于在r^d中的任何n个球的家族，都有一个方向使得与此方向的任何线最多相交O（√<（1+logλ）n log n log n>）球。另一方面，对于n【d，r^d中有一个非重叠的n个球的家族，使得在任何方向上，这个方向都有一条线，至少与N-D+1球相交。对于一个坐在r^3的固定桌上的球家族，我们还获得了平均球数的上限，该球的平均球数是通过与桌子相遇的垂直线刺穿的。4。如果r^3中的非重叠球系列满足logλ= o（（（n/log n）^<1/3>），则有一个平面两侧包含n/2-o（n）完整球。