Looking through the
Apollonian Window

Apollonian Seminar, Room 356.
E-mail Jerzy Kocik: "jkocik at math.siu.edu"

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Here you find very basic construction and data

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Art related to Apollonian disk packing

1. Pappus chain --- a 2200 year old geometric construction (interactive) at different sites: "Cut-the-Knot" 1, "Cut-the-Knot" 2, and by Thomas Schoch.

2. Descartes configuration -- an interactive Java applet. Try the same with movable centers and see the construction. (my humble use of Cinderella).

3. Apollonian gasket: Play with Apollonian gasket of the first dozen circles.

4. Template of the Apollonian Window (pdf) for printing. And here is the upper half only.

5. Draw your own Apollonian Window (recipe plus the data for the first seven hundred circles).

6. Poster with the spin structure shown -- version 0.02 (In progress. Please, share your critical remarks).

7. Images. Some of my Povray figures inspired by the Apollonian window

1. Dedekind tessellation --- an interactive page.


See also some pictures.

2. Apollonian Galaxy. See also its center, a big picture, and a third-kind encounter (click on the corresponding image).

3. Chaos game (IFS) done with Java applet. Here is a version made with the graphics program Proceessing (see also this page head, click on the picture to start a new process).

4. Symmetries. Click to see various symmetries: Pappus threads, (a few thousand circles generated iteratively), only the external part (hyperbolic tasselation via triangles!), three-circle symmetries, and finally the galaxies as an extra bonus.

1. Beautifull image of the regular Apollonian gasket by Jos Leys

See more here. The movie Dimensions, which he made together with Etienne Ghys and Aurélien Alvarez, won the "Prix d'Alembert" award from the French Mathematical Society.

My texts related to the Apollonian Window:

A theorem on circle configurations. ( arXiv:0706.0372v2 ). A generalization of the Descartes circle theorem to quite arbitrary configurations. The correspondence between circles in a plane and vectors in Minkowski space is utilized. The "extended Descartes theorem" in "Beyond the Descartes theorem" by Lagarias at al. (see below in Readings) is a special case for circles that are tangent.

Lens sequences. ( arXiv:0710.3226v1). About sequences of curvatures of chains of circles inscribed in lenses, i.e., intersections of two overlaping disks of the same radius. Rather unexpected properties.

Clifford Algebras and Euclid's parameterization of Pythagorean Triples (Advances in Applied Clifford Algebras (Mathematical Structures), 17 2007 pp. 71-93). Why Euclid's parameters of pythagorean triples are "spinors". You may find here also formulas for parameterization of Pythagorean quadruples, hexads and decuples (using quaternions and octonions) as well as a geometric interpretation of Hall matrices in the context of the Apollonian Window.

On a Diophantine equation that generates all integral Apollonian disk packings A simple purely geometric derivation of the Diophantine equation and algorithm to generate all integral Apollonian packings. Includes some remarks on Pythagorean triples occuring in the packings.

Golden window (Mathematics Magazine, 83 Dec 2010, pp. 384-90). A light text where a window built with circlar panels conceals the golden ratio and its various powers. The cover features a figure based on this article.


Jeffrey Lagarias at al.: Beyond the Descartes configuration -- the generalization of Descartes theorem on four circles (2002).

Bo Soderberg: Apollonian tiling, the Lorentz group, and regular trees -- a physicist on the correspondence between the circles and Minkowski space-time, and much more. Rich but very parsimonious in giving credits to others (1992).

Donald Coxeter: The Problem of Apollonius -- (1968)

J.B. Wilker: Four proofs of a generalization of the Descartes circle theorem (1969)

Peter Sarnak: Integral Apollonian Packings Am Math Montly (Apr 2011)

More papers on Apollonian gaskets by Lagarias at al.:

Apollonian Circle packing: Geometry and Group Theory. I. The Apollonian Group (2005)
Apollonian Circle packing: Geometry and Group Theory. II. Super-Apollonian Group and Integral Packing (2005)
Apollonian Circle packing: Geometry and Group Theory. III. Higher Dimensions (2005)
Apollonian Circle Packing: Number theory (2003)
Apollonian Circle Packing: Number theory II. Spherical and Hyperbolic Packing (2005)

More papers of some relevance:

Pfiefer and Van Hook: Circles, Vectors and Linear algebra -- a very simple introduction to the concept of Minkowski space as parameter space of circles in plane (1993)
W.S. Brown: Kiss Precise -- an old paper on tangent spheres (1969)

Kenneth Stephenson: Circle Packing: A Mathematical Tale, Notices of AMS, Vol. 50 (11) Dec 2003, pp. 1376--1388.

M. Borkovec, W. De Paris, and R. Peikert: The fractal dimension of the Apollonian sphere packing/

P.N. Ruane: The curious rectangles of Rollett and Rees from The Mahematical Gazette, July 2001, p. 208.

XKCD: What if?...

Vladimir Arnold: On Teaching Mathematics, an extended text of the address at the discussion on teaching of mathematics in Palais de D'ecouverte in Paris on 7 March 1997. (Some have marked it as "controversial").


The On-Line Encyclopedia of Integer Sequences --- maintained by N. J. A. Sloane.

Related sites:

Tangencies: Three Tangent Circles -- from Geometry Junkyard
Soddy Circles -- from Wolfram MathWorld
Descartes' theorem -- a selection of articles
Apollonian Gasket -- from Wolfram MathWorld
Apollony fractal -- by Paul Bourke
Apollonian Gasket -- from Interactive Mathematics Miscellany and Puzzles
Apollonian Gasket -- Wolfram Demonstrations Project

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