Spinors, lattices, and classification of integral Apollonian disk packings

Abstract

A parametrization of integral Descartes configurations (and effectively Apollonian disk packings) by pairs of two-dimensional integral vectors is presented. The vectors, called here tangency spinors defined for pairs of tangent disks, are spinors associated to the Clifford algebra for 3-dimensional Minkowski space. A version with Pauli spinors is given. The construction provides a novel interpretation to the known Diophantine equation parametrizing integral Apollonian packings.

Apollonian depth, spinors, and the super-Dedekind tessellation

“…5. Finally, note that the upper (and the lower) part of the tessellation coincides with the results of [7] where the integral packings were analyzed with the help of the tangency Pauli spinors.…”

“…Recall that the arrows in the figures represent only the order of the disks (not the actual spinor). The remarkable properties of spinors are presented in [12], and recapitulated in [7] (For the first appearance, see [9]) .…”

Integral spinors, Apollonian disk packings, and Descartes groups

“…Since a proper integral tricycle determines a superintegral Apollonian packing, and since the Descartes move travels along tricycles in such a packing, super-integrality is preserved trivially. Since the integrality of two adjacent spinors in any Descartes configuration determines integrality of all its spinors [10], the claim hold trivially.…”

“…In any Apollonian disk packing, if any two adjacent spinors, i.e., spinors spin (A, B) and spin (A, C) for some tricycle in the packing, are integral then all spinors in the packing are integral. For more see [8,7,11,10].…”

“…This paper is companion to [10], where the integral Apollonian packings are derived from spinors only.…”