On a Diophantine equation that generates all integral Apollonian Gaskets
Jerzy Kocik
TL;DR
This paper presents a new derivation of a simple Diophantine quadratic equation that generates all integral Apollonian Gaskets, using inversive geometry, and discusses the presence of Pythagorean triples within these gaskets.
Contribution
It introduces a novel derivation of the key Diophantine equation for Apollonian Gaskets based on inversive geometry and explores Pythagorean triples in these packings.
Findings
Derivation of the Diophantine equation using inversive geometry
Identification of Pythagorean triples in Apollonian Gaskets
Comprehensive understanding of the structure of integral Apollonian Gaskets
Abstract
A remarkably simple Diophantine quadratic equation is known to generate all Apollonian integral gaskets (disk packings). A new derivation of this formula is presented here based on inversive geometry. Also occurrences of Pythagorean triples in such gaskets is discussed.
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