Talk:Apollonian gasket

Image

any topic like this just cries out for illustrations so us folks that aren't so hot at math can understand what's being discussed. Suppafly

yeah, this page def needs a pic, or at least a link to one.

The image was added 22:06, 8 Oct 2004 [1], before the second comment (Matt me's) above (06:02, 9 Oct 2004, [2]). Hyacinth 18:11, 9 Oct 2004 (UTC)

I think the page needs the construction image https://commons.m.wikimedia.org/wiki/File:Apollonian_gasket_construction.svg in order to make clear exactly which are the 4th & 5th circles. As a clueless non-editor, I don't dare attempt such a potentially space-disrupting change. Snowsim (talk) 04:24, 13 August 2015 (UTC)

Any size

Article should explain from the start that C1, C2, C3 can be any size

Done. Gandalf61 09:17, Oct 11, 2004 (UTC)

The problem remains that there is currently no diagram accompanying the text that identifies C1, C2, C3, etc. rowley (talk) 23:36, 6 July 2010 (UTC)

Descartes circle theorem?

Isn't there a "Descartes circle theorem" or some such, relating circle sizes using integers? A very short google showed articles at mathworld, but no mention of integers ... maybe I'm imagining the bit about integers?? linas 02:16, 11 November 2005 (UTC)

Do you mean [3]? Black Carrot (talk) 22:30, 21 February 2008 (UTC)

I added a huge section on Integral Apollonian Circle Packing, and included the reference for what you are thinking about. Feedback on the section is appreciated. NefariousPhD (talk) 01:31, 1 March 2008 (UTC)

The new section on Integral Apollonian Circle Packing is interesting - well done. A few corrections:

• Each gasket is completely described by the curvatures of its first three circles. not four. If these curvatures are -a, b and c (where -a is the curvature of the bounding circle) then the curvatures of the next two circles d and e are give by

$d=-a+b+c-2\Delta$

$e=-a+b+c+2\Delta$

where

$\Delta ={\sqrt {bc-ab-ac}}$

(see Descartes' theorem). If a, b, c and Δ are integers then d and e are also integers, and it then follows that all the circles in the gasket will have integer curvature.

• (-1,2,2,3) is definitely the only integer gasket with D₂ symmetry. To have D₂ symmetry we must have b=c and d=e, which means that Δ=0 and so b=2a. Therefore, up to a scaling factor, the gasket must be (-1,2,2,3).
• There are definitely no integer gaskets with D₃ symmetry. To have D₃ symmetry we must have b=c=d, which means that

${\frac {a}{b}}=2{\sqrt {3}}-3$

so a and b cannot both be integers.

It is possible that someone might suggest that this new section is original research. To avoid this, it would help if you gave more references that discuss integer gaskets - the Lagarias, Mallows and Wilks paper that you reference is about extensions of Descartes' circle theorem, and only mentions integer gaskets in passing. Gandalf61 (talk) 16:07, 1 March 2008 (UTC)

I have now made changes in the article to correct these errors. Gandalf61 (talk) 14:17, 9 March 2008 (UTC)

Questions ???

To obtain an integer packing one needs four integer curvatures that satisfy DCE, not just three. The first three in the tables happen to produce the fourth, but how were these first 3 obtained to begin with? Are all possible packings accounted for? Also, DCE solutions [2,3,6,23] yield the same packing as [-1,2,2,3]. How does one move from the one characterization to the other? This could be explained a little better. Otherwise, very nice article. —Preceding unsigned comment added by 67.253.170.147 (talk) 15:17, 1 July 2010 (UTC)

If we know the curvatures of any three mutually tangent circles in an Apollonian gasket then we can use repeated applications of Descartes' theorem to construct the whole gasket. And if any four mutually tangent circles in the gasket have integer curvatures, then all circles in the gasket will have integer curvatures. We can characterise each gasket by giving the curvature of the outermost circle (this will be negative) and the curvatures of the two largest circles inside the gasket - so we have a triplet [-a, b, c] where 0 < a < b ≤ c. Given a, we can place an upper limit on b, which is

$b\leq a\left(1+{\frac {2}{\sqrt {3}}}\right)$

and, given a and b, there is also an upper limit on c. So if we require a, b and c to be integers, then there are a finite number of possibile triplets [-a, b, c] for each value of a. Furthermore, if we require the next circle (and hence the whole gasket) to have an integer curvature also, then bc − ab − ac must be a square number. So for each value of a we can test the finite number of possibilities for b and c to find those for which bc − ab − ac is a square number. This gives us a way of systematically generating all integer gaskets. I think the list in the article is exhaustive up to a = 15.

To answer your second question, [2,3,6,23] and [-1,2,2,3] generate the same gasket because, applying Descartes' theorem to [2,3,6], we have

$k_{4}=k_{1}+k_{2}+k_{3}\pm 2{\sqrt {k_{1}k_{2}+k_{2}k_{3}+k_{3}k_{1}}}=2+3+6\pm 2{\sqrt {36}}=-1{\text{ or }}23$

Gandalf61 (talk) 13:11, 2 July 2010 (UTC)

Thanks for that, Gandalf61. My guess is that all possible packings could also be accounted for by proceeding in some fashion from the simplest packing [1,1,0,0]and reflecting each curvature in turn. Perhaps a tree structure is the result? On the Hausdorff dimension calculation can you say what values for N,S one should use in the this equation? $D={\frac {\ln N}{\ln S}}\approx 1.3057...$. —Preceding unsigned comment added by 67.253.170.147 (talk) 05:19, 8 July 2010 (UTC)

types

Hi. Can apollonian gaskets be divided in types like :

• integral
• super integral
• nested
• non-integral

? --Adam majewski (talk) 18:37, 17 July 2010 (UTC)

Interactive JavaScript demo

I created an interactive JavaScript demo at http://www.jasondavies.com/apollonian-gasket/ but I couldn't add the link to the External Links section due to potential conflict of interest. Can someone review it and add if appropriate? Thanks! Jason Davies (talk) 13:42, 16 August 2010 (UTC)

|very nice effort! 124.170.34.249 (talk) 02:14, 6 August 2014 (UTC)

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Special case (0,0,1,1) and reference to Ford Circles

I'm surprised not to see a picture of the special case (0,0,1,1), and it's relationship to Ford Circles. Is there any objection to adding one, for example, as in figure 2 of this article http://www.math.ucsd.edu/~ronspubs/03_02_appolonian.pdf ? The figure shown in the article is similar to the one shown on the Ford Circle page, but has more circles and is, I think, more clear. If there are no objections, I will (eventually) create the picture and add a description somewhere in the section on integral packings. Nilesj (talk) 19:53, 22 January 2018 (UTC)

I have done this now. Nilesj (talk) 16:47, 28 June 2018 (UTC)

General construction

"(in the general construction, these three circles have to be different sizes, and they must have a common tangent)" I'm not an expert, but shouldn't that be "three circles may be different sizes" ? Chris2crawford (talk) 13:41, 3 December 2020 (UTC)

Add: The Local-Global Conjecture for Apollonian circle packings is false

I'd love to see this article worked in with a nicely-explained example: The Local-Global Conjecture for Apollonian circle packings is false

https://www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/ ★NealMcB★ (talk) 20:53, 12 August 2023 (UTC)