TL;DR
This paper introduces a novel perspective on integral Apollonian circle packings by linking their curvatures to the bases of Z[i]^2 and exploring their Diophantine properties through Conway's sensual quadratic form.
Contribution
It provides a new description of Apollonian packings based on the study of bases of Z[i]^2, connecting geometric packings with algebraic number theory.
Findings
New description of Apollonian circle packings
Connection to bases of Z[i]^2
Relation to Conway's sensual quadratic form
Abstract
The curvatures of the circles in integral Apollonian circle packings, named for Apollonius of Perga (262-190 BC), form an infinite collection of integers whose Diophantine properties have recently seen a surge in interest. Here, we give a new description of Apollonian circle packings built upon the study of the collection of bases of Z[i]^2, inspired by, and intimately related to, the `sensual quadratic form' of Conway.
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