An infinite sequence of circles can be constructed, containing rings for each n 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. It begins by sandwiching the unit circle between two parallel halfplanes; in
the geometry of circles, these are considered to be tangent to each other at the
point at infinity. Each successive circle after these first two is tangent to the central unit circle and to the two most recently added circles; see the illustration for the first six circles (including the two halfplanes) constructed in this way. The first n 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. This construction can be perturbed to a ring of n finite circles, without additional tangencies, whose minimum radius is arbitrarily close to this bound. ==History==