Study on arrangements of solid balls in 3-space

1.R^3中不重叠的单位球的循环序列，其中每个连续的球相切，称为珍珠链。我们证明，要制作一条打结的珍珠项链，15个单位球就足够了。要制作一条可以镶嵌在两个平行平面之间、相距2+√；2&gt；的打结项链，需要16个单位球，而三叶草是唯一可以由16个单位球组成的结。链是一个有限的球序列，其中每一对连续的球都相切。用球表示顶点，用连接两个顶点球的链条表示边，从而形成一个图。设b_n是构成n个顶点的完全图所需的最小球数。然后我们得到了C1n^3&lt；bn&lt；c2n^3logn的界，当我们把球都放在固定的桌子上时，也得到了类似的界。对于d维空间R^d中的球族F，设λ=λ(F)=(最大。半径)/(最小半径)。证明了对于R^d中的任一族n个球族，存在一个方向，使得该方向的任何直线与至多O(√&lt；(1+logλ)nlogn&gt；)个球相交。另一方面，对于n[大于或等于]d，在R^d中存在一个不重叠的n个球族，使得对于任何方向，都有一条与至少n-d+1个球相交的直线。对于坐在R^3中固定桌子上的一族球，我们还得到了与桌子相交的垂直线穿透的球的平均数的上界。如果R^3中不重叠球族满足logλ=o((n/logn)^lt；1/3)，则存在一个平面，它的两侧都包含n/2-o(N)个完整的球。

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.