《自然》杂志介绍填充问题研究中的新进展

数学星空

2009年8月13日《自然》杂志第460卷发表封面文章《Dense packings of the Platonic and Archimedean solids》（柏拉图多面体和阿基米德多面体的最紧密堆积），介绍普林斯顿的师生在填充问题（Packing problem）研究中取得的进展。 基于致密颗粒堆积几何知识的模型，可帮助解释很多体系的结构，包括液体、玻璃、晶体、颗粒和生物体系。这一领域以前的研究工作大部分集中在球状颗粒上，但即使对这种理想化的形状，这个问题仍然极为困难——Kepler关于球体最紧密堆积的猜想直到2005年才得到证明。人们对18个经典几何形状——柏拉图多面体和阿基米德多面体（如本期封面所示）——的最紧密排列知之甚少，虽然它们自古希腊时代以来就为人们所知了。现在，Salvatore Torquato和Yang Jiao报告了5种柏拉图多面体（四面体、立方体、八面体、十二面体和二十面体）和13种阿基米德多面体的已知最紧密排列。这些多面体的对称性在决定它们基本堆积排列中非常关键，具有中心对称性的柏拉图多面体和阿基米德多面体的最紧密堆积被猜想由它们相应的最紧密（布拉菲）晶格堆积给出。 英文报道(from the Princeton University press release) Princeton pair sets world record in packing puzzle Princeton researchers have beaten the present world record for packing the most tetrahedra into a volume. Research into these so-called packing problems have produced deep mathematical ideas and led to practical applications as well. Princeton University/Torquato Lab. "Finding the best way to pack the greatest quantity of a specifically shaped object into a confined space may sound simple, yet it consistently has led to deep mathematical concepts and practical applications, such as improved computer security codes. When mathematicians solved a famed sphere-packing problem in 2005, one that first had been posed by renowned mathematician and astronomer Johannes Kepler in 1611, it made worldwide headlines. Now, two Princeton University researchers [Salvatore Torquato, a professor in the Department of Chemistry and the Princeton Institute for the Science and Technology of Materials, and Yang Jiao, a graduate student in the Department of Mechanical and Aerospace Engineering] have made a major advance in addressing a twist in the packing problem, jamming more tetrahedra--solid figures with four triangular faces--and other polyhedral solid objects than ever before into a space. The work could result in better ways to store data on compact discs as well as a better understanding of matter itself. Henry Cohn, a mathematician with Microsoft Research New England in Cambridge, Mass. says, 'the Princeton researchers, have thrown out a new challenge to the math world. Their results could be considered a 21st Century analogue of Kepler's conjecture about spheres. And, as with that conjecture, I'm sure their work will inspire many future advances.'" 转自:http://159.226.47.206/cms/news/2009/20090817.htm