关于正三角形和正方形吻接数的一个简单证明

The Simple Proof of Kissing Number of Square

设 pn是任意一个正 n边形 ,最大整数 k(pn)称为 pn的吻接数 ,其中 ,在同一平面内有 k(pn)个与 pn全等的正 n边形均与 pn有非空的交集 ,但没有重叠 ,而且 k(pn)个正 n边形两两没有重叠 . Youngs (Amer.Monthly46(1 93 9) 2 0 ) ,Klamkin(Math.Mag. 68(1 995 ) 1 2 8)先后证明了 k(p3) =1 2 ,k(p4 ) =8,作者(Discrete Math.68(1 998) 2 93 )证明了当 n >6时 k(pn) =6.然而 ,Youngs、Klamkin等人关于 k(p3) =1 2 ,k(p4 ) =8的证明非常复杂 .本文将就 k(p3) =1 2 ,k(p4 ) =8给出非常简单的证明 .

Let p n be an arbitrary regular polygon with n sides. What is the maximum number k(p n) of congruent regular polygons (copies of p n) that can be arranged so that each touches p n but no two of them overlap? Youngs (1939), Klamkin (1995) and others established that k(p 3)12 and k(p 4)8. The author (1998) established that k(p n)6 (n>6), k(p 3)12. However, the proofs of Youngs, Klamkin are complicated, In this paper, we will gave a quite simple proof for k(p 3)12 and k(p 4)8.