dansmath > circle page

the circle page 7/01

 Definitions: A CIRCLE is the set of all points in a plane at a distance r from the center C. The CURVATURE k of a circle is the reciprocal of the radius: k = 1 / r.

The original problem for my weekly contest #117: "Kissing Circles"
was taken from the April 21, 2001 issue of Science News:
What are the radii and centers of the circles marked 'a' and 'b'?

(scroll down for more!)

It turns out that the radii are always reciprocals of integers (here a = 1/3 and b = 1/6),
and even more surprising (to me) is that the centers are always at rational coords
with the same denominator (maybe lower when reduced; here (0, 2/3) and (-3/6, 4/6)).

Way back in 1638, Descartes developed the elegant formula for
four mutually tangent "kissing" circles with curvatures k, m, n, p:
k^2 + m^2 + n^2 + p^2 = (1/2)(k + m + n + p)^2 .

I hear the formula for the centers is similar but uses complex
number coordinates a + b i for (a, b). Anyone seen (or can find) it?

I had Mathematica draw a series of pictures and the first
frame looks like this; click the picture for lots more detail!
( Here the whole numbers represent the curvatures. )
(scroll down for more!)

I also have a trippy zooming movie; here are a few selected frames for you! %;-} Dan

For more on the subject, search at www.google.com for 'circle packing' or 'Lagarius' or 'Appollonian tiling'.

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