Talk:Apollonian circles

First merger discussion

I would recommend that the pages of Apollonian circles not be merged with the Apollonian gasket, because they seem to be mathematically distinct. The Apollonian circles constitute two families of mutually orthogonal circles, and extend Apollonius' distance-ratio definition of the circle (see blurb in Circle). By contrast, the Apollonian gasket extends Apollonius' problem, which is to identify a circle that is tangent to three fixed circles. The circles in this case are tangent (not orthogonal) and are constructed by a different method to solve a different problem. The gasket looks nothing like the Apollonian circles, which correspond to the orthogonal bipolar coordinate system. The main thing they have in common is their inventor, Apollonius of Perga. WillowW 21:28, 6 May 2006 (UTC)

Second merger discussion

— I'm willing to merge these articles on the Apollonian circles, the Circles of Apollonius and the Apollonian gasket, but I'll confess that the motivation is murky to me. Despite their similar name, these concepts seem as mathematically alien as "linear regression" and "linear dynamical system". Perhaps the motivating idea for merger is that non-professionals might become confused by their similar names? Or that we can make one good-length article from a handful of overly short articles? Please help me understand your reasoning. Willow 06:27, 31 July 2006 (UTC)

Blue circles

Are the blue circles in the currently used image constructed by varying $\lambda$ in $k_{A}=\{X|{\overline {AX}}:{\overline {XB}}=\lambda \}$ (see de:Kreis des Apollonios)? Thanks, --Abdull 15:32, 9 October 2007 (UTC)

Yes, you're exactly right, Abdull! :) I'm sorry that I didn't notice your message earlier. :( Apollonius defined a circle as the locus of points that have the same ratio of distances to two given foci. Interestingly, the red circles are the locus of points that subtend the same angle between the two foci; you might want to read up on the inscribed angle theorem. These two types of circles are perpendicular (they meet at right angles), which is the basis for bipolar coordinates, an important orthogonal coordinate system. Hoping that you enjoy geometry, too, :) Willow (talk) 22:29, 21 March 2008 (UTC)

Third merger discussion

Here we are again at a merger discussion — or at least a merger monologue. ;) As clarified at the disambiguation page, Circles of Apollonius, Apollonius did a lot of work with various types of circles. Although he's not well-known today, he was one of the top three Greek geometers, along with Euclid and Archimedes. Meanwhile, I'm trying to bring one of those types of circles (the solutions to Apollonius' problem) to Featured Article status, and it would not be sensible for any of us to merge that article with other mathematical topics that are unrelated to it. I hope that this clarifies where we're headed with Apollonius and his circles; please consider helping out! :) Willow (talk) 21:40, 21 March 2008 (UTC)

Split of pencil of circles

I suggest to split of the section Apollonian circles#pencils of circles to a new page pencil of circles (now a redirect to this page) I think the subject is important enough to have its own page. WillemienH (talk) 14:32, 1 August 2015 (UTC)

I think this is a good idea. There is also a similar section Cardioid#Cardioid_as_envelope_of_a_pencil_of_circles on the Cardioid article, so it could provide some additional content. Nilesj (talk) 18:44, 19 June 2018 (UTC)

There is already a page for Pencil (mathematics), which seems a good destination for me ....

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IMPRESSIVE! Two Students Debunk a Widely Accepted Math Conjecture - YouTube

Two Students Unravel a Widely Believed Math Conjecture | Quanta Magazine

[2307.02749] The Local-Global Conjecture for Apollonian circle packings is false (arxiv.org) 2603:6011:9600:52C0:F501:1D4E:CF44:5605 (talk) 23:39, 15 August 2023 (UTC)

The appropriate article for this new result/discovery is Apollonian gasket, where it is already included.--Kmhkmh (talk) 07:10, 16 August 2023 (UTC)