论文标题
完美的正方形包装
Perfect packing of squares
论文作者
论文摘要
众所周知，$ \ sum \ limits_ {i = 1}^\ infty {1/ i^2} = {π^2/6} $。 Meir和Moser问什么是最小的$ε$，以使长度$ 1 $，$ 1/2 $，$ 1/3 $，$ \ ldots $的所有正方形都可以包装到面积$ {π^2/6}+ε$的区域矩形中。在正确区域的矩形中包装称为完美包装。 Chalcraft打包了长度$ 1 $，$ 2^{ - t} $，$ 3^{ - t} $，$ \ ldots $的正方形，他发现以$ 1/2 <t \ le3/5 $的价格找到了完美的包装。我们将根据Chalcraft的算法显示，如果$ 1/2 <t \ le2/3 $，则有完美的包装。此外，我们表明，对于$ \ log_32 \ le t \ le2/3 $，对于所有$ t $的所有$ t $都有完美的包装。
It is known that $\sum\limits_{i =1}^\infty {1/ i^2}={π^2/6}$. Meir and Moser asked what is the smallest $ε$ such that all the squares of sides of length $1$, $1/2$, $1/3$, $\ldots$ can be packed into a rectangle of area ${π^2/6}+ε$. A packing into a rectangle of the right area is called perfect packing. Chalcraft packed the squares of sides of length $1$, $2^{-t}$, $3^{-t}$, $\ldots$ and he found perfect packing for $1/2<t\le3/5$. We will show based on an algorithm by Chalcraft that there are perfect packings if $1/2<t\le2/3$. Moreover we show that there is a perfect packing for all $t$ in the range $\log_32\le t\le2/3$.
