Study on arrangements of solid balls in 3-space

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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.