### Packing

November 30, 2010 The arrangement of pennies on a plane that gives an optimal packing (most pennies per area) is so trivial that it's nearly instinctive. You arrange them such that their centers are on an hexagonal lattice. This optimal lattice packing of circles on a plane was proven mathematically by Carl Friedrich Gauss, and the resultant density is (π/2√3) = 0.9069.[1] In 1940, László Fejes Tóth showed that this is the optimal packing, lattice or otherwise.[1] We can still obtain compact packing when the coins are of different size, as shown in the figure. In this case, the coins have diameter 1 and the root of the equation, r^{4}- 10r

^{2}- 8r + 9.[2]

*Compact packing of two sizes of disks of radius 1 and 0.6375559772... Figure 1 of Ref. 2.[2]*

You wouldn't think that stepping up just one dimension from circles in a plane to spheres in a volume would cause much more of a problem. Greengrocers have been stacking fruit in nicely structured dense piles for centuries. However, proving that this arrangement, beloved by Nature and crystallographers as face-centered-cubic (FCC), is indeed optimal, or that there is a non-lattice packing that's more dense, is quite a problem. None other than Johannes Kepler first conjectured that this FCC lattice, also called cubic close-packing, was the optimal packing of spheres. Calculation shows that the packing is (π/√18) = 0.74048, which means that there's quite a lot of empty space between spheres, a fact that allows the creation of the interesting material, inverse opal. Thomas Hales produced a "proof" of this conjecture in 1998. His proof, which is much like the "proof" for the four color conjecture, used computer programs to examine every possible case of sphere packing. Checking such a proof is difficult, and even today mathematicians are not completely certain that the proof is correct.[3] Even then, it would not be a proof in the usual sense of the word. I've commented in several earlier articles that many scientific discoveries have been accidental. One packing discovery happened accidentally at Princeton University when a student was asked to experimentally measure random packing of a geometrical object. This object was M&M candy, which is an oblate spheroid. Surprisingly, a random assemblage of these was found to pack more densely than spheres, 68% as compared to 64% for random sphere packing.[4-6] The result was so surprising that Paul Chaikin, a professor of physics, repeated the experiment himself. He obtained the same result, and that led to a more detailed study. Other experiments showed that there was not much difference between spheres and M&Ms for lattice packing, only random packing. Computer simulations showed that a random stacking of ellipsoids, which are closer in shape to a sphere than the lenticular M&M, yields a greater density than a lattice stacking of spheres.

*Density of the laminate crystal packing of ellipsoids as a function of the aspect ratio, via arXiv, Ref. 6.*

What might be happening in such random packing is the idea that the elongated particles are able to slide past each other to a better equilibrium position, while the spheres jam together. Of course, experiments are tedious, and computer simulations are easier. It was found that there are a greater number of contact points between neighboring particles for the higher packing densities. The number of contact points may depend on the number of directions a particle can move and pivot. Paul Chaikin has moved to New York University, but Salvatore Torquato, a professor of chemistry and co-investigator on the original M&M project, is still at Princeton publishing numerous papers in this area.[7-12] He was an author of a Physical Review Letter on packing in 2000, so he's been active in this area for more than a decade.[13] One interesting result is for the packing of the Platonic solids. A cube, of course, fills all space, but the densest packings of tetrahedra, icosahedra, dodecahedra, and octahedra were found to be 0.823, 0.836, 0.904, and 0.947. This is an interesting result, since the dodecahedra and icosahedra approximate a sphere to a high degree but pack much better.

*Paul Chaikin (left) and Salvatore Torquato (Princeton University)*

During the original research project about a decade ago, Mars Inc., the M&M company, donated 125 pounds of almond M&M's; but men do not live by candy alone. Other funding came from the National Science Foundation, the NASA and the Petroleum Research Fund.

### References:

- Weisstein, Eric W. "Circle Packing." From MathWorld--A Wolfram Web Resource.
- Tom Kennedy, "Compact packings of the plane with two sizes of discs," arXiv Preprint, July 8, 2004.
- Sphere Packing Page on Wikipedia.
- 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, March 10, 2004.
- S. Torquato, "Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles," arXiv Preprint, September 8, 2010.
- Salvatore Torquato and Frank H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond," arXiv Preprint, August 17, 2010.
- S. Torquato and Y. Jiao, "Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra," arXiv Preprint, April 30, 2010.
- Y. Jiao, F. H. Stillinger and S. Torquato, "Novel Features Arising in the Maximally Random Jammed Packings of Superballs," arXiv Preprint, January 4, 2010.
- S. Torquato and Y. Jiao, "Dense Packings of Polyhedra: Platonic and Archimedean Solids," arXiv Preprint, September 9, 2009.
- S. Torquato and Y. Jiao, "Dense Packings of the Platonic and Archimedean Solids," arXiv Preprint, August 27, 2009.
- S. Torquato, T. M. Truskett and P. G. Debenedetti, "Is Random Close Packing of Spheres Well Defined?," Physical Review Letters, vol. 84, p. 2064 (2000).

*Permanent Link to this article*

Linked Keywords: penny; plane; synthetic a priori; hexagonal lattice; Carl Friedrich Gauss; László Fejes Tóth; root of a function; dimension; circle; sphere; volume; greengrocer; Nature; crystallography; crystallographers; face-centered-cubic; Johannes Kepler; Kepler conjecture; opal; Thomas Hales; four color conjecture; mathematician; serendipity; Princeton University; M&M candy; oblate spheroid; Paul Chaikin; physics; Computer simulation; New York University; Salvatore Torquato; Platonic solids; cube; tetrahedron; icosahedron; dodecahedron; octahedron; Mars Inc.; National Science Foundation; NASA; MathWorld; arXiv.

### Google Search

Latest Books by Dev Gualtieri

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

- The Wisdom of Composite Crowds - April 27, 2017

- J. Robert Oppenheimer and Black Holes - April 24, 2017

- Modeling Leaf Mass - April 20, 2017

- Easter, Chicks and Eggs - April 13, 2017

- You, Robot - April 10, 2017

- Collisions - April 6, 2017

- Eugene Garfield (1925-2017) - April 3, 2017

- Old Fossils - March 30, 2017

- Levitation - March 27, 2017

- Soybean Graphene - March 23, 2017

- Income Inequality and Geometrical Frustration - March 20, 2017

- Wireless Power - March 16, 2017

- Trilobite Sex - March 13, 2017

- Freezing, Outside-In - March 9, 2017

- Ammonia Synthesis - March 6, 2017

- High Altitude Radiation - March 2, 2017

- C.N. Yang - February 27, 2017

- VOC Detection with Nanocrystals - February 23, 2017

- Molecular Fountains - February 20, 2017

- Jet Lag - February 16, 2017

- Highly Flexible Conductors - February 13, 2017

- Graphene Friction - February 9, 2017

- Dynamic Range - February 6, 2017

- Robert Boyle's To-Do List for Science - February 2, 2017

- Nanowire Ink - January 30, 2017

- Random Triangles - January 26, 2017

- Torricelli's law - January 23, 2017

- Magnetic Memory - January 19, 2017

- Graphene Putty - January 16, 2017

- Seahorse Genome - January 12, 2017

- Infinite c - January 9, 2017

- 150 Years of Transatlantic Telegraphy - January 5, 2017

- Cold Work on the Nanoscale - January 2, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**