Integral spinors, Apollonian disk packings, and Descartes groups

Abstract

We show that every irreducible integral Apollonian packing can be set in the Euclidean space so that all of its tangency spinors and all reduced coordinates and co-curvatures are integral. As a byproduct, we prove that in any integral Descartes configuration, the sum of the curvatures of two adjacent disks can be written as a sum of two squares. Descartes groups are defined, and an interesting occurrence of the Fibonacci sequence is found.