Apollonian circle packings

Some on-line resources

Lecture notes:

• Hee Oh, Apollonian Circle Packings: Dynamics and Number Theory.
• Alexandre Kirillov, A tale of two fractals.

Decartes circle Theorem:

• Frederick Soddy, The Kiss Precise.
• Thorold Gossett, The Kiss Precise.
• Jerzy Kocik, Proof of Descartes circle formula and its generalization clarifiedProof of Descartes circle formula and its generalization clarified .
• Paul Levrie, A straightforward proof of Descartes's circle theorem.
• H. S. M. Coxeter, The Problem of Apollonius.
• Edward Kasner, Fred Supnick, The Apollonian Packing of Circles.
• Jeffrey C. Lagarias, Colin L. Mallows, Allan R. Wilks, Beyond the Descartes Circle Theorem.
• Jerzy Kocik, A theorem on circle configurations.
• Arthur Baragar, Alex Kontorovich, Efficiently constructing tangent circles .
• Sam Northshield, Complex Descartes Circle Theorem.
• Alden Bradford, An even more straightforward proof of Descartes’s circle theorem.
• Richard Evan Schwartz, Serge Tabachnikov, Descartes Circle Theorem, Steiner Porism, and Spherical Designs.
• Daniel V. Mathews, Orion Zymaris, Spinors and Descartes’ Theorem.
• J. B. Wilker, Four Proofs of a Generalization of the Descartes Circle Theorem.
• W. S. Brown, The Kiss Precise.
• Richard E. Pfiefer, Cathleen Van Hook, Circles, Vectors, and Linear Algebra.

Some introductory papers:

• Katherine Stange, The Farey structure of the Gaussian integers .
• Dana Mackenzie, A Tisket, a Tasket, an Apollonian Gasket, in American Scientist.
• David Sheard, On the cover: Apollonian packing, in ChalkDust magazine.
• Jerzy Kocik, Golden window.
• Kenneth Stephenson, Circle Packing: A Mathematical Tale.

Apollonian Packings, Local-Global principle:

• R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, Apollonian Circle Packings: Number Theory.
• R. L. Graham, J. C. Lagarias, C. L. Mallows, A. R. Wilks, C. H. Yan, Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group.
• Elena Fuchs, Katherine Sanden, Some experiments with integral Apollonian circle packings.
• Jean Bourgain, Elena Fuchs, A proof of the positive density conjecture for integer Apollonian circle packings .
• Elena Fuchs, Arithmetic Properties of Apollonian Circle Packings, Ph.D. Thesis (2010), Princeton University.
• Elena Fuchs, Strong approximation in the Apollonian group.
• Jean Bourgain, Integral Apollonian circle packings and prime curvatures.
• Jean Bourgain, Elena Fuchs, On representation of integers by binary quadratic forms.
• Katherine E. Stange, The sensual Apollonian circle packing.
• Alex Kontorovich, From Apollonius To Zaremba: Local-Global Phenomena in Thin Orbits.
• Katherine E. Stange, Visualising the arithmetic of imaginary quadratic fields.
• Jean Bourgain, Alex Kontorovich, On the Local-Global Conjecture for integral Apollonian gaskets.
• Sneha Chaubey, Elena Fuchs, Robert Hines, Katherine E. Stange, The Dynamics of Super-Apollonian Continued Fractions.
• Elena Fuchs, Katherine E. Stange, Xin Zhang, Local-global principles in circle packings.
• Summer Haag, Clyde Kertzer, James Rickards, Katherine E. Stange, The Local-Global Conjecture for Apollonian circle packings is false.

Higher dimensional sphere packings:

• Alex Kontorovich, The Local-Global Principle for Integral Soddy Sphere Packings.
• Arthur Baragar, Higher dimensional Apollonian packings, revisited.
• Arthur Baragar, Apollonian packings in seven and eight dimensions .

Polyhedral packings and Superintegral sphere packings:

• Alex Kontorovich, Kei Nakamura, Geometry and Arithmetic of Crystallographic Sphere Packings.
• Michael Kapovich, Alex Kontorovich, On Superintegral Kleinian Sphere Packings, Bugs, and Arithmetic Groups.
• N. Ahmed, W. Ball, E. Buckminster, E. Rivkin, D. Torrance, J. Viscusi, R. Wang, I. Whitehead, S. Yang, Domains of Convergence for Polyhedral Packings .
• Nikolay Bogachev, Alexander Kolpakov, Alex Kontorovich, Kleinian sphere packings, reflection groups, and arithmeticity.
• Philip Rehwinkel, Ian Whitehead, David Yang, Mengyuan Yang, Circle Packings from Tilings of the Plane .

Other topics:

• Tomislav Došlić, Luka Podrug, Solving the dimer problem on Apollonian gasket.
• Arthur Baragar, The Apollonian circle packing and ample cones for K3 surfaces .
• Ian Whitehead, Apollonian Packings and Kac-Moody Root Systems .
• Jerzy Kocik, Clifford Algebras and Euclid’s Parameterization of Pythagorean Triples.

Sangaku (Japanese Temple Geometry):

• Hidetoshi Fukagawa, Kazunori Horibe, Sangaku--Japanese Mathematics and Art in the 18 th ,19 th and 20 th Centuries.
• Hidetoshi Fukagawa, Symmetry in Traditional Japanese Mathematics.
• Nobuya Nakamura, Formulas in Traditional Geometry English translation of a 1841 text in Japanese.
• Eiichi Ito, Echio Nomura, Hirotaka Kobayashi, Hideaki Tanaka, Isao Kitahara, Kenji Otani, Nobuya Nakamura, Ryutaro Yanagisawa and Tetsuo Sekiguchi Japanese Temple Mathematical Problems in Nagano Pre..
• Hiroshi Kotera collected several hundreds of images of sangaku tables - out of about 820 that survived (click on hieroglyphs to see the pictures!). The English version of the page contains the map of prefectures in Japan.

• Peter Sarnak,Number Theory and the Circle Packings of Apollonius, Mahler Lectures 2011.
• Peter Sarnak,Integral Apollonian Packings, MAA Lecture - January, 2009.
• Peter Sarnak,Letter to J. Lagarias about Integral Apollonian Packings.
• Edna Jones, Apollonian circle packings, integers, and higher-dimensional sphere packings.
• Edna Jones, Local-global I: Apollonian packings.
• Katherine E. Stange, A Farey Tour (from Farey sequences to Apollonian circle packings).

Webpages:

• Jerzy Kocik, Looking through the Apollonian Window.
• Blog: Apolloniana. On Apollonian and other disk arrangements: geometry, number theory, physics.

Some applets:

• Malin Christersson, Apollonian Gasket.

Miscellania:

• Daniel Martin, Katherine Stange, Appuzzle: Apollonian Circle Packing Puzzle.