Princeton Researchers Solve Problem Filling Space — Without Cubes
Whether packing oranges into a crate, fitting molecules into a human cell or getting data onto a compact disc, wasted space is usually not a good thing.
Now, in findings published June 20 in the Proceedings of the National Academy Sciences, Princeton University chemist Salvatore Torquato and colleagues have solved a conundrum that has baffled mathematical minds since ancient times — how to fill three-dimensional space with multi-sided objects other than cubes without having any gaps.
The discovery could lead to scientists finding new materials and could lead to advances in communications systems and computer security.
“You know you can fill space with cubes,” Torquato said, “We were looking for another way.” In the article “New Family of Tilings of Three-Dimensional Euclidean Space by Tetrahedra and Octahedra,” he and his team show they have solved the problem.
Torquato, who is a professor in the Department of Chemistry, Princeton Institute for the Science and Technology of Materials and Princeton Center for Theoretical Science, worked with Yang Jiao, a postdoctoral associate at Princeton, and John Conway, Princeton’s John Von Neumann Professor in Applied and Computational Mathematics, to find that three-dimensional space can be efficiently filled with units consisting of an octahedron and six smaller tetrahedra. Moreover, the resulting patterns can be repeated infinitely.
The researchers’ work involves “tiling,” also known as “tessellation” after the Latin word “tessella,” meaning a small cube or tile. Just as someone tiling a floor repeats a geometric pattern to fill the length and width of the floor space, Torquato and colleagues take that task to a third dimension, adding height and filling the whole room. That process involves fitting more complex three-dimensional “tiles” together, in this case the regular octahedron (a solid figure with eight triangular faces) and the regular tetrahedron (a figure with four triangular faces).
“The cube is one of five Platonic solids,” Torquato explained. “It is known that the other four Platonic solids (tetrahedron, octahedron, the 12-sided dodecahedron and 20-sided icosehedron) individually cannot tile 3D space. However, we’ve discovered new ways to combine tetrahedra and octahedra to tile 3D space.”
The phenomenon of tiling occurs in nature — in structures within certain living cells, in the formation of crystals, and even the honeycombs in a beehive or the repeated densely packed structures in the center of a daisy, Torquato said.
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“Tetrahedra and octahedra are about as simple as it gets — if you ignore cubes, which are the really obvious case — but it's far from obvious how to combine them, and this paper shows that they can fill space in truly unexpected ways.
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