I know of a remarkable result from a paper of Matthew Kahle (PDF download), that there are arbitrarily low-density jammed packings of congruent disks in $\mathbb{R}^2$:

In 1964 Böröczky used a subtle disk packing construction to disprove a conjecture of Fejes Tóth, that a locally jammed arrangement of disks in the plane, one where each disk is held in place by its neighbors, must have positive density.

Böröczky, K. "Über stabile Kreis- und Kugelsysteme." Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 7, 79-82, 1964.

Surprisingly (to me), this natural candidate is not the lowest density:
My question is whether or not Böröczky's construction extends to $\mathbb{R}^3$ and higher dimensions. (I do not yet have access to Böröczky's 1964 paper.)

Addendum. (I add this prompted by user j.c.) My question is answered: Yes. First, through the help of Wlodek Kuperberg and other commenters, we learned what exactly is Böröczky's 1964 zero-density packing. Indeed it is "subtle" as Matthew Kahle said. Second, simply layering identical copies of that packing of spheres in $\mathbb{R}^3$ results in a jammed packing of zero density, as confirmed by Benoît Kloeckner. What remains is to correct Wikipedia's misleading entries on the topic, as detailed by Gerry Myerson.

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    $\begingroup$ Of course yours is not lowest density. You can replace each circle with three smaller circles to get a lower density. And so on. $\endgroup$ Oct 20, 2013 at 3:20
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    $\begingroup$ Actually, it is not clear to me that you can iterate and keep the jamming condition without using circles of different sizes. $\endgroup$ Oct 20, 2013 at 3:26
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    $\begingroup$ No picture of the Böröczky packing?! I am appalled! $\endgroup$ Oct 20, 2013 at 5:37
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    $\begingroup$ I wonder now if identical copies of the Böröczky packing in parallel layers are jammed in $\mathbb{R}^3$...? $\endgroup$ Oct 20, 2013 at 11:49
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    $\begingroup$ @Joseph O'Rourke: they certainly should be jammed, as for each sphere the spheres on top and bottom to it prevent it to move otherwise than horizontaly (at first order), which is prevented by its neighbors in Böröczky packing. I can't see what would be wrong with your example. $\endgroup$ Oct 21, 2013 at 7:55

2 Answers 2


Wlodek Kuperberg sent me this 'diagram from the "Lagerungen"' of Böröczky's packing, over which I overlaid several congruent disks:
(Yes, small cracks or overlays are visible between disks that should be tangent. The diagram is not 100% metrically accurate.)
Added: The vertical red lines suggest a slight concavity.

  • $\begingroup$ Are the disks in the boundary of the three tentacles really trapped? $\endgroup$ Oct 20, 2013 at 20:49
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    $\begingroup$ @MarianoSuárez-Alvarez: My guess is that there is a slight concavity to those boundary curves (now indicated by the red lines), achieved by progressive angular scissoring of the central diamond-parallelograms. $\endgroup$ Oct 20, 2013 at 23:09
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    $\begingroup$ Yes, because the boundaries are concave narrowing curves, so each point lies strictly inside the triangle formed by three neighboring points. The picture, however, is of pretty low quality in this respect and the possibility of such an arrangement is far from obvious. $\endgroup$
    – fedja
    Oct 20, 2013 at 23:09
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    $\begingroup$ @j.c.: First, it is an answer to the subsidiary question of what is Böröczky's packing. Second, as my buried comment indicated, layering identical copies of this packing does answer the question in $\mathbb{R}^3$. I guess I should make this clearer through an addendum to the main question. $\endgroup$ Oct 25, 2013 at 21:15
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    $\begingroup$ László Fejes Tóth, Lagerungen in der Ebene auf der Kugel und im Raum, Springer Verlag, Die Grundlehren der mathematischen Wissenschaften (2nd Ed.), Vol. 65 (1972). $\endgroup$ Oct 27, 2013 at 16:22

Wikipedia says, "The locally jammed sphere packing with the lowest density has a density of only 0.55536," and refers the reader to Martin Gardner's New Mathematical Diversions from Scientific American, Chicago: University of Chicago Press, 1983, pp. 82–90, ISBN 0-226-28247-3.

Aleksandar Donev has a presentation, Jamming in Hard-Sphere Packings; on slide 49, it says, "There are locally jammed packings of vanishing density (covering fraction) (K. Baroczky)." It may be that this is intended to apply in two dimensions only, but this is not evident from the context.

I am aware that the two paragraphs above appear to contradict each other.

Another reference: Torquato and Stillinger, Jammed hard-particle packings: from Kepler to Bernal and beyond, write on page 38, "Another important distinction is that it is possible to pack spheres subject only to the weak locally-jammed criterion, so that the resulting packing fraction is arbitrarily close to zero." They cite Boroczky, and also Stillinger, F. H., S. Torquato, and H. Sakai, 2003, Lattice-based random jammed configurations for hard particles,” Phys. Rev. E 67, 031107.

EDIT: I'm looking at the Martin Gardner essay. On page 88, he wrote,

"In his Geometry and the Imagination, first published in Germany in 1932, David Hilbert describes what was then believed to be the loosest [locally-jammed] packing [of spheres]: a structure with a density of .123. In the following year, however, two Dutch mathematicians, Heinrich Heesch and Fritz Laves, published the details of a much looser packing with a density of only .0555 [see Fig. 34]. Whether there are still looser packings is another intriguing question that, like the question of the closest packing, remains undecided."

This suggests that the number given in Wikipedia is a typo.

Gardner doesn't give a reference for the Heesch-Laves result, but I think it's Z. Kristallogr., Mineral. Petrogr. 85, (1933) 443–453 Abt. A.

Probably worth a look is O'Keeffe, M., Dense and rare four-connected nets, Z. Krist. 196 (1991), no. 1-4, 21–37, MR1132437 (92h:82125). The summary says, "A rare stable sphere packing that is rarer than that of Heesch and Laves is described."

MORE EDIT: Lots of pretty pictures related to Heesch-Laves at George Hart's webpage, e.g.:

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    $\begingroup$ Looks like we need to update Wikipedia once this issue becomes clearer... $\endgroup$ Oct 20, 2013 at 23:17
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    $\begingroup$ (I took the liberty of adding George Hart's image of the Heesch-Laves loose-packing.) $\endgroup$ Oct 20, 2013 at 23:26
  • $\begingroup$ By the way, I don't think the Heesch-Laves loose-packing shown above can be locally jammes, because each sphere touches just 3 others. $\endgroup$
    – John Baez
    Apr 13, 2016 at 18:08
  • $\begingroup$ @John, the convention seems to be that if the centers of the sphere and the three it touches are coplanar then the structure is considered to be locally jammed. $\endgroup$ Apr 13, 2016 at 23:24
  • $\begingroup$ Interesting; I wouldn't call that locally jammed, because then you can freely move the central sphere in the direction orthogonal to the plane. I might call it "almost" locally jammed. $\endgroup$
    – John Baez
    Apr 20, 2016 at 16:52

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