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Disk Packing

January 29, 2024

One of my uncles was a Marine who fought in World War II. After the war, he worked as a welder and did things very interesting to my youthful self. In the 1950s, glass bottles were the container of choice for soft drinks, and empty bottles has a small bounty for their return. In those days, there was no city-wide recycling, and he would roam the neighborhood with an old Radio Flyer wagon on the night before trash collection, finding discarded bottles to be exchanged for cash.

He would also roll his own cigarettes, and I was fascinated by the mechanical device that he used for that purpose. He also had a huge glass jar of coins that he would sort, looking for some rare coins that might be worth more than their face value. I was reminded of this when I read about the discovery of a huge cache of ancient coins, some dating to 175 BC, at a Japanese construction site about 70 miles outside of Tokyo.[1] What most interested me was their arrangement in neat stacks, quite unlike the jumbled mass you would expect to find.

Financial cost of cigarette smoking (cartoon)

Unlike my uncle, I've never smoked. As this cartoon illustrates, smoking causes a financial burden as well as a health burden.

Governments have a financial interest in smoking cessation, since much of the expense of smoking-related health problems fall on them. One way governments discourage smoking, and also recoup health-related expenses, is through a cigarette tax.

George Harrison (1943-2001) of The Beatles had many health-related issues arising from smoking. These included throat cancer and lung cancer.

(Portion of Wellcome Trust Photo no. L0024904, Library ref., ICV No 51428, by Reginald Mount, via Wikimedia Commons. Click for larger image.)

One reason that cigarettes are packed as twenty items is the efficient stacking of 7, 6, and 7 cylinders. Matchbooks also contain twenty matches for the same reason. Another common packing is the packing of spherical objects. The one obvious candidate for the densest packing of identical spheres is the stacking used by greengrocers when they stack fruit. This close-packing arrangement is shown in the figure.

Close-packed spheres

The well-known close-packing arrangement of spheres (right), along with the illustration that Johannes Kepler (1571-1630) used when making the conjecture that this is the densest possible arrangement.

(Left image by "Jvangiel," and right image from Johannes Kepler's 1611 Strena Seu de Nive Sexangula, both images via Wikimedia Commons. Click for larger image.)

Materials scientists and crystallographers are familiar with this close packing of spheres, since this is the arrangement of atoms in a face-center-cubic crystal lattice, as found for such metals as aluminum and copper. In this arrangement, and in all other close packings, each sphere contacts twelve neighboring spheres. The average density in close packing is 0.74048; that is, nearly three-quarters of space is filled.

In 1611, Johannes Kepler (1571-1630) conjectured that this packing has the maximum possible density for all sphere packings, whether packed in a lattice, or packed in some irregular arrangement. In 1831, Carl Friedrich Gauss (1777-1855) proved the conjecture for lattice packing, but it wasn't until 1998 that a computer-assisted proof by Thomas Hales (b. 1958) showed that an irregular packing couldn't do better.[2] Since this was a computer-assisted proof, verification of the proof was not easy, and it wasn't verified until 2014.[3]

Randomly packed spheres have a density of about 64%, quite a bit less than the 74% for close-packed lattice packing. In 2004, physicist, Paul Chaikin, and chemist, Salvatore Torquato, found that randomly packed oblate spheroids have a higher density. A physical example of an oblate spheroid is an M&M candy, and these will randomly pack at 68%, while their lattice packing is about the same.[4-5]

Monte Carlo simulation of packing of hard ellipses

Monte Carlo simulation of packing of hard ellipses.

(image from the ChaikinLab Condensed Matter Physics Laboratory at New York University.)

Their computer simulations found that a random stacking of ellipsoids, which are closer in shape to a sphere than the lenticular M&Ms, has a larger density than the close-packed lattice packing of spheres. Further work by Torquato and colleagues showed that the densest packings of tetrahedra, icosahedra, dodecahedra, and octahedra were 0.823, 0.836, 0.904, and 0.947, respectively.[6-13] This is an interesting result, since the dodecahedra and icosahedra approximate a sphere to a high degree but pack much better.

A pile of US quarter dollar coins

Because of inflation, the old saying, "a penny for your thoughts," should be updated to offering a quarter, instead.

This pile of US quarter dollar coins illustrates another type of packing, that of disk-shaped objects.

(Photo by the author)

A disk is another type of simple geometric object, and the random stacking behavior of disks has been studies by a researchers from the East China Normal University (Shanghai, China).[14-15] Their experiments with small, coin-sized, plastic disks have revealed the formation of randomly oriented stacks in their packing.[15] Just as the random packing of other shapes has important consequences in the mechanics of materials, disk packing applies to the flow and optical properties of technological materials, such as liquid crystals, and also the mechanical properties of clays.[15] The frictional forces acting between flat flakes in clays stabilize this material against temperature.[15]

Their experiments used hollow disks about 30 millimeters in diameter and 5 millimeters in thickness.[15] The disks were hollow so they could be filled with a gel that was strongly visible in magnetic resonance imaging (MRI) to allow the disk positions to be measured.[15] Twelve hundred disks were poured into a rectangular container and lightly shaken by repeated dropping to a hard surface several times from a height of one centimeter; but, to avoid boundary effects, only about 500 disks at the center were analyzed.[15]

When first loaded, many of the disks lay flat in a random arrangement, but shaking leads to a compacted state with some disks forming short stacks that are randomly aligned with each other.[15] As the density increases, the alignment decreases, and this correlation has been observed in a prior experimental and theoretical study of packings in clays.[15] The researchers speculate that more intense shaking might lead to a globally aligned columnar state.[15] As for potential applications, study coauthor, Chengjie Xia, states that the porosity of concrete could be adjusted by addition of disk-like particles.[15]

Ancient Greek clayware circa 800 BC

Ancient Greek clayware, circa 800 BC, at the National Hellenic Museum.

(Wikimedia Commons image by "GreekDude1." Click for larger image.)


  1. David Pescovitz, "Stash of 100,00 ancient coins found at construction site," BoingBoing, November 15, 2023.
  2. Thomas C. Hales, "An overview of the Kepler conjecture," arXiv, May 20, 2002, https://doi.org/10.48550/arXiv.math/9811071.
  3. Thomas C. Hales, "A proof of the Kepler conjecture, Annals of Mathematics, vol. 162, no. 3 (2005), pp. 1065-1185, https://doi.org/10.4007/annals.2005.162.1065.
  4. Steven Schultz, "Sweet science: Common candies yield physics discovery," Princeton University Press Release, February 12, 2004.
  5. 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.
  6. Aleksandar Donev, Frank H. Stillinger, P. M. Chaikin and Salvatore Torquato, "Superdense Crystal Packings of Ellipsoids," arXiv, March 10, 2004. Also at Physical Review Letters.
  7. S. Torquato, "Reformulation of the Covering and Quantizer Problems as Ground States of Interacting Particles," arXiv, September 8, 2010. Also at Physical Review E.
  8. Salvatore Torquato and Frank H. Stillinger, "Jammed Hard-Particle Packings: From Kepler to Bernal and Beyond," arXiv, August 17, 2010. Also at Reviews of Modern Physics.
  9. S. Torquato and Y. Jiao, "Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra," arXiv, April 30, 2010. Also at Physical Review E.
  10. Y. Jiao, F. H. Stillinger and S. Torquato, "Novel Features Arising in the Maximally Random Jammed Packings of Superballs," arXiv, January 4, 2010. Also at Physical Review E.
  11. S. Torquato and Y. Jiao, "Dense Packings of Polyhedra: Platonic and Archimedean Solids," arXiv, September 9, 2009. Also at Physical Review E.
  12. S. Torquato and Y. Jiao, "Dense Packings of the Platonic and Archimedean Solids," arXiv, August 27, 2009. Also at Nature.
  13. S. Torquato, T. M. Truskett and P. G. Debenedetti, "Is Random Close Packing of Spheres Well Defined?," arXiv, March 25, 2000. Also at Physical Review Letters.
  14. Yunhao Ding, Jing Yang, Chenyang Wang, Zhichao Wang, Jianqi Li, Bingwen Hu, and Chengjie Xia, "Structural Transformation between a Nematic Loose Packing and a Randomly Stacked Close Packing of Granular Disks, " Physical Review Letters, vol. 131, no. 9 (September 1, 2023), Article no. 098202, https://doi.org/10.1103/PhysRevLett.131.098202
  15. Philip Ball, "Packing Disks in a New Way," Physics, vol. 16, no. 150, September 1, 2023.
  16. How to stack coins? ECNU research team reveals the stack structure transformation mechanism of disk-like particles, East China University Press Release, September 9, 2023.

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