Talk:Apollonian gasket

From Wikipedia, the free encyclopedia

WikiProject Systems
Systems science Portal
This article is within the scope of WikiProject Systems, which collaborates on articles related to Systems science.
Systems rating: A-BB+ Class Mid importance  Field: Chaos theory
Please update this rating as the article progresses, or if the rating is inaccurate. Please also add comments to suggest improvements to the article.


[edit] 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)

[edit] Any size

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

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

[edit] 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 − 2Δ
e = − a + b + c + 2Δ
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)