Descartes Circle Theorem

The Descartes circle theorem relates the signed curvatures of four mutually tangent circles, with no three tangent at a common point. The relation is

(1)

where for a circle of radius that encloses none of the others, and for a circle enclosing the other three. A line is included as a limiting circle with (Soddy 1936, Eppstein 2001).

For three pairwise externally tangent circles with positive curvatures , , and , the two solutions of Apollonius' problem have curvatures

(2)

The plus branch gives the inner Soddy circle. The minus branch gives the outer Soddy circle, which encloses the other three circles when , is a line when , and is externally tangent when (Eppstein 2001).

This formula was known to Descartes and Viète (Boyer and Merzbach 1991, p. 159), but Soddy extended it to spheres. In -dimensional space, mutually touching -spheres can always be found, and the relationship of their curvatures is

(3)

A generalization of the theorem to the case of an "-flower" consisting of tangent circles around the exterior of a central circle was given by Mathews and Zymaris (2025), who also derived an explicit equation satfied by their curvatures. In particular, let be the curvature of a central circle and be the curvatures for , ..., surrounding mutually tangent circles forming an -flower. Define

			(4)
			(5)

for . Then for odd,

(6)

and for even

(7)

where and (Mathews and Zymaris 2025).