## Particle ClumpingJuly 9, 2014 Since our earliest days, each of us has been concerned with the common human activities of packing and filling. As children, we needed to pack our toys in our toy chest and our clothes in our dresser drawers and closet. We needed to fill our bowl with breakfast cereal without its overflowing, and the same for cups of milk. Not surprisingly, there's a lot of science behind the optimal packing and filling of regularly- and irregularly-shaped items. As I wrote in a previous article (Packing and Filling, May 17, 2012) the optimal packing of circle in a plane occurs when their centers are on an hexagonal lattice. This packing gives an areal density of (π/2√3) = 0.9069 (see figure). Carl Friedrich Gauss proved that this packing was optimum for a lattice arrangement, and László Fejes Tóth showed that this is the optimal packing, lattice or otherwise.[1]
## References:- Weisstein, Eric W. "Circle Packing." From MathWorld--A Wolfram Web Resource.
- Weisstein, Eric W. "Kepler Conjecture." From MathWorld--A Wolfram Web Resource.
- Steven Schultz, "Sweet science: Common candies yield physics discovery," Princeton University Press Release, February 12, 2004.
- Aleksandar Donev, Ibrahim Cisse, David Sachs, Evan A. Variano, Frank H. Stillinger, Robert Connelly, Salvatore Torquato and P. M. Chaikin, "Improving the Density of Jammed Disordered Packings Using Ellipsoids," Science, vol. 303, no. 5660 (February 13, 2004), pp. 990-993.
- Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato, "Superdense Crystal Packings of Ellipsoids," arXiv Preprint Server, March 10, 2004.
- S. Torquato, "Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles," arXiv Preprint Server, September 8, 2010.
- Salvatore Torquato and Frank H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond," arXiv Preprint Server, August 17, 2010.
- S. Torquato and Y. Jiao, "Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra," arXiv Preprint Server, April 30, 2010.
- Y. Jiao, F. H. Stillinger and S. Torquato, "Novel Features Arising in the Maximally Random Jammed Packings of Superballs," arXiv Preprint Server, January 4, 2010.
- S. Torquato and Y. Jiao, "Dense Packings of Polyhedra: Platonic and Archimedean Solids," arXiv Preprint Server, September 9, 2009.
- S. Torquato and Y. Jiao, "Dense Packings of the Platonic and Archimedean Solids," arXiv Preprint Server, August 27, 2009.
- S. Torquato, T. M. Truskett and P. G. Debenedetti, "Is Random Close Packing of Spheres Well Defined?," arXiv Preprint Server, March 25, 2000.
- Christopher D. Zangmeister, James G. Radney, Lance T. Dockery, Jessica T. Young, Xiaofei Ma, Rian You and Michael R. Zachariah, "Packing density of rigid aggregates is independent of scale," Proc. Natl. Acad. Sci. Early Edition, June 9, 2014, doi:10.1073/pnas.1403768111.
- Supporting information for ref. 13 (PDF File).
- Michael Baum, "Snowballs to Soot: The Clumping Density of Many Things Seems to Be a Standard," NIST Tech Beat, June 10, 2014.
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