Integral spinors, Apollonian disk packings, and Descartes groups

TL;DR

This paper demonstrates that all tangency spinors and reduced coordinates in irreducible integral Apollonian packings are integral, and explores properties of Descartes configurations, including sums of curvatures as sums of squares and connections to Fibonacci numbers.

Contribution

It establishes integrality of spinors and coordinates in Apollonian packings and introduces Descartes groups with novel properties linking to Fibonacci sequences.

Findings

All tangency spinors and reduced coordinates are integral in irreducible integral Apollonian packings.

Sum of curvatures of adjacent disks can be expressed as a sum of two squares.

Discovery of Fibonacci sequence occurrence in 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.