Ring lemma

Then the minimum radius of any circle in the ring is at least the unit fraction

that exactly meet the bound of the ring lemma, showing that it is tight.

The construction allows halfplanes to be considered as degenerate circles with infinite radius, and includes additional tangencies between the circles beyond those required in the statement of the lemma.

circles of this construction form a ring, whose minimum radius can be calculated by Descartes' theorem to be the same as the radius specified in the ring lemma.

finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound.

[4] A version of the ring lemma with a weaker bound was first proven by Burton Rodin and Dennis Sullivan as part of their proof of William Thurston's conjecture that circle packings can be used to approximate conformal maps.

[5] Lowell Hansen gave a recurrence relation for the tightest possible lower bound,[6] and Dov Aharonov found a closed-form expression for the same bound.

[2] Beyond its original application to conformal mapping,[5] the circle packing theorem and the ring lemma play key roles in a proof by Keszegh, Pach, and Pálvölgyi that planar graphs of bounded degree can be drawn with bounded slope number.