Three years before he received the Nobel Prize in Physics, Eugene Wigner published an article entitled "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (1960). He marveled at how often physicists develop concepts to describe the "real" world only to discover that mathematicians--heedless of that real world--have already thought up and explored the concepts. His own experience of the uncanny applicability of mathematical insights to the physical reality of quantum mechanics led Wigner to observe "that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it."
When compressed by a liquid droplet, small groups of colloidal microspheres -- plastic spheres with diameters about one one-hundredth that of a human hair -- pack to form an unusual sequence of structures. At top are packings containing four to eleven spheres, as seen through the scanning electron microscope. At bottom are the polyhedra defined by drawing lines between the centers of touching spheres in each cluster. Some of these polyhedra are familiar structures, such as the tetrahedron (4 spheres) and octahedron (6 spheres), but most of the others -- including the "snub disphenoid" (8 spheres) and the "gyroelongated square dipyramid" (10 spheres) -- are probably unfamiliar, despite their attractive symmetry. Nevertheless, all of these structures obey a single, simple mathematical rule: they all minimize a quantity called the second moment. This is the first observation of this packing motif in nature. [Image credit: V. N. Manoharan]
Doubtless the observation of just such an uncanny correspondence between mathematics and physics prompted the editors of the July 25 issue of "Science" to feature on the cover the colloidal particle clusters that are the subject of research by an engineering professor and his two graduate students at the University of California at Santa Barbara (UCSB). That professor, David Pine, holds a joint appointment in the departments of Chemical Engineering and Materials and chairs the Chemical Engineering Department. The first author of the article, "Dense Packing and Symmetry in Small Clusters of Microspheres," is Vinothan Manoharan; the other author is Mark Elsesser.
Their story begins with the iridescence of opals, which are composed of equal-sized spheres about a micrometer in diameter, or roughly a hundred times smaller than the size of a human hair. The spheres are packed into a structure known as the face-centered cubic (FCC) lattice, which is exactly the same arrangement used by grocers to stack oranges or apples. Because the opals constituent spheres are about the size of the wavelength of light, their orderly arrangement diffracts light and causes iridescence.
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