A circle packing game

Date started: Spring 2017, revisited Fall 2019
Leads: Arthur Baragar, and Daniel Lautzenheiser


Given two circles of radius one a distance 2d apart, and two parallel lines tangent to both circles, find a way to pack circles into the space so that the circles never overlap, they fill the space, and they all have integer curvature.

Like any game, there are more rules (see the references for details).  The game can be won if d^2=n is an integer.  We learn the rules of the game when n is small.  The case n =1 gives us the Apollonian packing; the case n=2 was discovered by Boyd (1974), and rediscovered by Guettler and Mallows (2010); and the case n=3 is a cross section of the Soddy sphere packing.


These illustrations were produced using McMullen’s Kleinian group program (see the references).  It is eerily(?) similar to a game played by Bianchi in 1892.

The Apollonian packing together with its symmetries:

The The Boyd-Guettler-Mallows packing:

A cross section of the Soddy sphere packing (two different perspectives):

Levels n = 5, 6, 7, 9, and 10:

All good games have Boss levels.  In Level 11, the group of symmetries is not representable by a finite number of reflections.  Inverting in the cluster point makes it obvious how to generate the group using reflections and translations (or rotations by pi).

Level 21 (inverted in a point):

High-quality image:


  • Baragar, Arthur, “Higher dimensional Apollonian packings, revisited.” Geom. Dedicata 195 (2018), 137 — 161.  DOI: 10.1007/s10711-017-0280-7
  • >Baragar, Arthur, “The Apollonian circle packing and ample cones for K3 surfaces.”   https://arxiv.org/pdf/1708.06061.pdf
  • Bianchi, Luigi. “Sui gruppi di sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginarî.” (Italian) Math. Ann. 40 (1892), no. 3, 332-412.  DOI: 10.1007/BF01443558
  • Boyd, David W. “A new class of infinite sphere packings.” Pacific J. Math. 50 (1974), 383 — 398.  DOI: 10.2140/pjm.1974.50.383
  • Guettler, Gerhard; Mallows, Colin. “A generalization of Apollonian packing of circles.” J. Comb. 1 (2010), no. 1, [ISSN 1097-959X on cover], 1 — 27.  DOI: 10.4310/JOC.2010.v1.n1.a1
  • Curt McMullen, Kleinian groups program (lim.tar):  http://www.math.harvard.edu/~ctm/programs/index.html