Four mathematicians broke a 75-year-old record by finding a denser way to pack high-dimensional spheres.

四位数学家打破了一项长达 75 年之久的纪录，找到了一种更密集地堆积高维球体的方法。

Mathematicians like to generalize concepts into higher dimensions. Sometimes this is easy.数学家喜欢将概念推广到更高的维度。 有时这很容易。

If you want to efficiently pack squares in two dimensions, you arrange them like a checkerboard. To squeeze together three-dimensional cubes, you stack them like moving boxes. Mathematicians can easily extend these arrangements, packing cubes in higher-dimensional space to perfectly fill it.

如果你想有效地将二维的正方形堆积，你可以像摆棋盘一样摆放它们。 如果想把三维立方体挤在一起，可以像搬箱子一样把它们堆在一起。 数学家们可以很容易地扩展这些排列方式，在更高维度的空间中完美地填满立方体。

Packing spheres is much harder. Mathematicians know how to pack circles or soccer balls together in a way that minimizes the empty space between them. But in four or more dimensions, the most efficient packing scheme is a complete mystery.（With the exception of dimensions 8 and 24, which were solved in 2016.）

堆积球体要难得多。 数学家们知道如何把圆或足球堆积在一起，以尽量减少它们之间的空隙。 但在四维或更多维中，最有效的堆积方案完全是个谜。（2016年解决的8维和24维问题除外）。

“It sounds so simple,” said Julian Sahasrabudhe, a mathematician at the University of Cambridge. “There could be 20 different ways of approaching it. And that seems to be what’s happened — there’s lots of different ideas.”

"剑桥大学数学家朱利安-萨哈斯拉布德说："这听起来很简单。 "可能有 20 种不同的方法。 这似乎就是现在的情况--有很多不同的想法。

The known optimal sphere packings in 2, 3, 8 and 24 dimensions look like lattices, full of patterns and symmetry. But in every other dimension, the best packings might be totally chaotic.

在 2 维、3 维、8 维和 24 维中，已知的最佳球体堆积看起来就像晶格，充满了图案和对称性。 但在其他维度中，最佳堆积可能是完全混乱的。

“That, I think, is a very tantalizing aspect of it. It’s really very open,” said Akshay Venkatesh, a mathematician at the Institute for Advanced Study. “We just do not know.”

"我认为，这是它非常诱人的一面。 它真的非常开放，"高等研究所的数学家阿克谢-文卡特什说。 "我们只是不知道而已。

Last December, Sahasrabudhe, together with his Cambridge colleague Marcelo Campos, Matthew Jenssen of King’s College London and Marcus Michelen of the University of Illinois, Chicago, provided a new recipe for how to densely pack spheres in all arbitrarily high dimensions. It’s the first significant advance on the general sphere-packing problem in 75 years.

去年 12 月，萨哈斯拉布德与他在剑桥大学的同事马塞洛-坎波斯（Marcelo Campos）、伦敦国王学院的马修-延森（Matthew Jenssen）和芝加哥伊利诺伊大学的马库斯-米歇尔恩（Marcus Michelen）一起，为如何在所有任意高维度上密集地堆积球体提供了一个新方法。 这是 75 年来一般球体堆积问题的首次重大进展。

From left：Marcus Michelen, Marcelo Campos, Julian Sahasrabudhe, and Matthew Jenssen in Sahasrabudhe’s Cambridge office on the last day of a month-long visit in which they broke a 75-year-old sphere-packing record.
左起 Marcus Michelen、Marcelo Campos、Julian Sahasrabudhe 和 Matthew Jenssen 在 Sahasrabudhe 剑桥办公室，这是他们为期一个月的访问的最后一天，在此期间，他们打破了有 75 年历史的装球记录。
参考https://www.quantamagazine.org/to-pack-spheres-tightly-mathematicians-throw-them-at-random-20240430/