Talk:Problem of Apollonius
I've been trying to figure out the algebraic solution... hopefully I'll make progress soon.
It looks to me like you're in need of some signs. You get that simply with a \pm in your equation. (You can also do with \mp).
I'd like to see a picture of that algebraic solution... and maybe clear up what r is as well (I'm assuming it's the radius of the unknown circle). Are you working on finishing up this solution?
What do you think about simply having your whole description in the more basic description? It could go either way, but I just wanted to check and see if you've thought about it.
I also agree with Chris that a gasket construction with three different sized circles would be a great addition to the page.
MaeBeth, this is a most interesting topic (my 9-year-old son thought it was "cool," and your description is very well-written (no surprise, considering your pedigree).
I looked up gasket and didn't find any part of the definition that relates to math but did notice that all the gaskets in the pictures I saw had holes in them. I'd like to have some info in this wiki about why they are called Apollonian gaskets.
I know you're not finished so it may be simply that you haven't gotten to it, but I share with Gene a desire both to have more examples of answers to Apollonian tangencies and more info about Apollonian gaskets. For example, do the filled-in bright white circles in the main math image have a mathematical basis? They don't seem to emerge from the pattern described in your section about the gasket.
Since the circles don't have to have the same radius, it would be interesting to have an example of an Apollonian gasket make of circles of different radii.
Hey, this is shaping up! I'd love to see examples for the situations in the paragraph right before A More Mat Exp,
- Given three points, the problem only has one solution. In the cases of one line and two points; two lines and one point; and one circle and two points, the problem has two solutions. Four solutions exist for the cases of three lines; one circle, one line, and one point; and two circles and one point. There are eight solutions for the cases of two circles and one line; and one circle and two lines, in addition to the three circle problem.
Maybe Drexel pals?
The "Constructing the gasket" paragraph needs to be broken up into pieces. I think you have the basic plan just fine--just break up with illustrations.
I agree with Anna that you don't need a More Math Exp--just title the section Apollonian Gasket.
I don't think you need the sub heading under a more mathematical description. It will flow better without it.
Basic description: Which 3 circles are given? Which is the soln? Maybe make the solution dotted or something?
I'd prefer you just have the first paragraph beside the image and begin the history paragraph after the image.
"The problem usually has ..." Images needed! With mouseovers or whatever, especially if they can be clicked on to be built up to the MathWorld e.g. Some with points and lines, too.
Hope you get back to explaining your Apollonian Gasket image, it's lovely!
You're off to a good start, MaeBeth. Don't feel you have to do everything to fill in the holes. You can ask future readers to contribute if they can!
The download links under "Mathematica Programs" are broken.