Table of Contents
ISRN Geometry
Volume 2012, Article ID 348618, 19 pages
Research Article

On a Diophantine Equation That Generates All Integral Apollonian Gaskets

Department of Mathematics, Southern Illinois University Carbondale, Carbondale, IL 62901, USA

Received 1 December 2011; Accepted 20 December 2011

Academic Editors: L. V. Bogdanov and G. Martin

Copyright © 2012 Jerzy Kocik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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, occurrence of Pythagorean triples in such gaskets is discussed.