《自然》刊文：数学家揭开三维空间气泡的变化规律

（转自__ 科学网__，任霄鹏/编译）

图片说明：科学家找到了三维空间气泡膨胀收缩的变化规律。（图片来源：剑桥大学）

最近，两位美国数学家解开了一个困扰科学界长达50年的“简单”问题：啤酒泡和肥皂泡在膨胀、收缩及合并时的数学规律。该研究成果将对工程学的泡沫材料设计、生物学的组织结构研究以及物理学的晶体颗粒排列探测产生深远的影响，相关论文发表在__ 《自然》杂志__上。

金属、泡沫以及细胞组织都是一个个类似马赛克的空间区域相互作用形成的，这些小区域不断变化——胀大、收缩或者合并，背后的驱动力都是表面张力。1952年，著名的数学家冯•诺依曼（John von Neumann）揭开了二维气泡的一些规律，即气泡的变化取决于表面总曲率。此外，他还将复杂的曲率计算简化为考虑气泡的侧面数量。半个世纪以来，科学家都努力地将冯•诺依曼的结论推广到三维空间。

现在，美国普林斯顿高等研究中心（Institute of Advanced Study in Princeton）的数学家__ Robert
D. MacPherson__和犹太大学（Yeshiva University）的材料学家

在欧拉特征数的基础上，MacPherson和Srolovitz创造了一个抽象概念——“平均宽度”（mean width），利用这一概念，研究人员可以对任何物体进行计算而不用考虑它的具体形状。他们在论文中指出，在三维空间中，气泡不同表面之间交界边缘的总长度如果超过平均宽度的6倍，那么气泡将会膨胀；反之，气泡则会收缩。

研究人员证实，他们的结论简化到二维空间时就是冯•诺依曼提出的规律，并且已经将这一结论推广到4维甚至更多维的假想气泡。Srolovitz说，“新发现的这一规律非常普遍，它将改变我们对几何物体的认识方式。”Srolovitz认为，新的发现将可以帮助科学家研制出更持久更有效的材料，并将它们应用于机翼、微处理器乃至核反应堆。

美国西北大学的应用数学家Sascha Higlenfeldt表示，之前许多研究已经发现了气泡变化经验性的规律和关系，一般都是考虑气泡的面数。而最新研究得出的精确结论为这些规律找到了坚实的理论基础。

不过，科学家还需要进行更为艰苦的工作，那就是要精确描述泡沫整体结构随着气泡消失与合并的变化规律。

相关报道

**Solution to Bubble Puzzle Pops Out**

By Adrian Cho

__Science____NOW Daily News__

25 April 2007

With a key mathematical insight, a pair of theorists has solved a
5-decades-old puzzle as easily as you might burst a soap bubble with a pin. The
new result lets researchers predict whether a bubble in foam will grow or
shrink. More than a mere curiosity, the mathematical relation could aid
engineers designing foamy materials, biologists studying the architecture of
tissues, and physicists probing how crystalline grains are arranged within a
solid.

Foam looks simple, but researchers can't explain how it evolves as bubbles
grow, shrink, and merge--a process known as coarsening. In 1952, famed
mathematician John von Neumann deciphered one aspect of 2-dimensional foams,
such as soap bubbles squeezed between glass plates. Whether a bubble grows or
shrinks depends on the sum total of the curvature of its faces. But Von Neumann
reduced the messy problem of adding up curvature to the much simpler task of
counting a bubble's sides. He proved that, regardless of their sizes or shapes,
2-D bubbles with five or fewer sides shrink, those with seven or more grow, and
those with six remain the same. For half a century, researchers have struggled
to extend von Neumann's result to 3 dimensions.

Now, mathematician Robert MacPherson of the

Using the Euler characteristic, MacPherson and Srolovitz also invented an
abstract "mean width" that they could calculate for any object
regardless of its shape. In 3 dimensions, a bubble's faces meet at distinct
edges, and the researchers found that a bubble will grow if the sum of the
lengths of its edges is greater than 6 times its mean width. If the sum of all
the edge lengths is smaller, the bubble will shrink, as the team reports
tomorrow in Nature. The researchers have shown that in 2 dimensions their
result reduces to von Neumann's rule and have extended the relation to
hypothetical bubbles in 4 or more dimensions.

Other researchers had already developed empirical relations that, on
average, tied the growth of a bubble to the number of faces, and the new, exact
result might be useful for to putting those rules of thumb on a firmer
theoretical foundation, says Sascha Higlenfeldt, an applied mathematician at

原文摘要

__ Robert
D. MacPherson__ &