Apollonian Edge Circles

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The curvature (or bend) of a circle of radius is .


The Descartes circle theorem [1] states that if four mutually tangent circles have curvatures , , , , then .

If the centers of the circles are , , , in the complex plane, then


In this Demonstration, the four circles have integer curvatures. The outer circle with center at the origin encloses the other three and has negative curvature, and the other three circles have positive curvatures. At each level, new tangent circles are inscribed into curvilinear triangles tangent to the outer circle; new circles are only added at the edge.

Consider the three largest interior circles for example 42. Their centers have coordinates and their curvatures are . The outer circle has curvature and a piece of data called the global denominator is equal to 5. For each circle, multiplying its center coordinates by the product of its curvature, the outer curvature and the global denominator gives integer values; for the three circles in example 42, the results are .

Any subsequently generated edge circle has the same property: the coordinates of the product of the center, curvature, outer curvature and global denominator are integers. The author does not know if this behavior always occurs.

The global denominator seems to always be from the integer sequence 1, 5, 13, 17, 25, 29, 37, ..., the solutions to with and positive [2]. The author does not know if this behavior always occurs.

Modulus 12, only 4 curvature moduli can occur within an integer curvature circle packing. Circles with a similar modulus get the same color.


Contributed by: Ed Pegg Jr (April 2020)
Open content licensed under CC BY-NC-SA




[1] E. Pegg Jr. "The Circles of Descartes" from the Wolfram Demonstrations Project—A Wolfram Web Resource. demonstrations.wolfram.com/TheCirclesOfDescartes.

[2] OEIS, A004613, Numbers that are divisible only by primes congruent to 1 mod 4.

[3] J. C. Lagarias, C. L. Mallows and A. R. Wilks, "Beyond the Descartes Circle Theorem." arxiv.org/abs/math/0101066.

[4] D. Austin. "When Kissing Involves Trigonometry." American Mathematical Society, March 2006. www.ams.org/featurecolumn/archive/kissing.html.

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