Spinors and Descartes' Theorem

Abstract

Descartes' circle theorem relates the curvatures of four mutually externally tangent circles, three "petal" circles around the exterior of a central circle, forming a "$3$-flower" configuration. We generalise this theorem to the case of an "$n$-flower", consisting of $n$ tangent circles around the exterior of a central circle, and give an explicit equation satisfied by their curvatures. The proof uses a spinorial description of horospheres in hyperbolic geometry.