14 DAY TRIAL // For centuries, mathematicians have been trying to determine the best way of filling a given space, in such a way that not free spaces remain, and that they do not use cubes. Efficiently filling a volume with multi-faceted objects has proven to be a formidable theoretical challenge, until now.
This conundrum is not only theoretical, as it has a multitude of potential applications. These range from improving transport and fuel efficiency to designing drugs and data-storage mediums such as DVD.
Mathematicians say that the conundrum was formed in unambiguous terms. How do you go about filling a volume with multi-sided objects without leaving any gaps between these objects?
In a paper published in the June 20 issue of the esteemed journal Proceedings of the National Academy Sciences (PNAS), experts propose a possible solution. Based on these data, investigators might even start developing new materials, as well as advanced communications and computer security systems.
“You know you can fill space with cubes. We were looking for another way,” explains Princeton University chemist Salvatore Torquato. He is a professor in the Department of Chemistry at the Princeton Institute for the Science and Technology of Material.
He also holds an appointment at the Princeton Center for Theoretical Science, and is the author of the new paper, which is entitled “New Family of Tilings of Three-Dimensional Euclidean Space by Tetrahedra and Octahedra.”
Together with Princeton postdoctoral associate Yang Jiao and the John Von Neumann Professor in Applied and Computational Mathematics, John Conway, Torquato demonstrated that an octahedron and six smaller tetrahedra can efficiently fill a 3D space.
According to the team, the patterns that appear as the space is filled can be extended indefinitely, which means that spaces of any dimensions could be filled using this approach as well. In other words, their discovery is scalable to larger applications.
“Scientists can devise better and more realistic models for structures of matter; architects can come up with designs that balance function and appearance; mathematicians have new arrangements to analyze and study; and artists can work together with scientists to perhaps make new works of art,” Torquato says.
“This work is elegant in that it reveals packing modes that have been overlooked for years, specifically that the holes in lattices of optimal packings of octahedra are regular tetrahedra of equal size,” comments New York University chemistry professor Michael Ward.
“Octahedra and tetrahedra may be regarded as simple objects themselves, which may suggest that unraveling their optimal packing modes would be straightforward,” he adds.
“But this work illustrates that mathematical approaches […] often can reveal patterns that are difficult, if not impossible, to visualize directly,” the expert concludes.