What is a circle chain?

A circle chain, also known as a Villarceau chain or Pappus chain, is a sequence of circles where each circle is tangent to two others in the sequence and to two fixed circles. These fixed circles can be tangent to each other internally (nested), externally (kissing), or disjoint (not touching).

Key aspects of circle chains include:

• Tangency: Each circle in the chain is tangent to its immediate neighbors in the chain and also to the two fixed, bounding circles. This property is fundamental to the geometry of the configuration. See tangency.

• Fixed Circles: These are the two circles that all the circles in the chain are tangent to. The relative sizes and positions of these fixed circles determine the overall appearance and properties of the chain. See fixed%20circles.

• Villarceau Circles: In the special case where the fixed circles are internally tangent, and the chain is complete (i.e., the last circle in the chain is also tangent to the first circle), it's related to Villarceau circles on a torus. See Villarceau%20circles.

• Number of Circles: The number of circles in the chain that can fit between the fixed circles before repeating is a key property. This number depends on the radii and relative positions of the fixed circles.

• Inversion: Circle chains can be studied using the technique of circle inversion, which preserves tangency and transforms circles into other circles or lines. This can simplify the analysis of complex configurations. See inversion.

• Pappus Chain: A special case where one of the fixed circles is a line. See Pappus%20chain.