长方体规则打包的最小表面积研究

Studies on the Least Surface Areas of Cuboid Regular Packing

证明了pk(p为素数)个小正方体规则打包的最小表面积方案是p[k3]×p[k3+1]×p[k+32].对于给定的若干个全等的小长方体,当规则打包后的大长方体最短边确定时,周长越小,其表面积越小;当周长确定时,最短边越大,其表面积越小;当最短边确定时,最长边与最短边之差越小,其表面积也越小.上述表明,规则打包后的长方体三边的"集中程度(周长)"和"离散程度(最长边与最短边之差)"可作为衡量一个长方体"越接近"正方体的量化指标.最后给出了寻找一般长方体规则打包后的最小表面积方案的算法和程序.

This paper proved that the scheme of least surface areas of p^k（p are prime numbers） cubes after regular packing was p^[k/3]×p^[k＋1/3]×p^[k＋2/3]. Given several congruent small-sized cuboids, if the length of the shortest side of the large cuboid packed regularly is fixed, as the perimeter decreases, the corresponding surface areas decrease; if the perimeter is fixed, as the length of the shortest side increase, the corresponding surface areas decrease; if the length of the shortest side is fixed, as the difference between the length of the longest side and the length of shortest side decrease, the corresponding surface areas decrease. The results show that, the centralized extent （i.e. the perimeter） and the discrete extent （i.e. the difference between the length of the longest side and the length of shortest side） of cuboid＇s sides should be used as the characteristics of quantization on cuboid approaching cube after regular packing. Finally, the algorithm on seeking the scheme of least surface areas of cuboid regular packing in general terms was put forward.