dansmath > **circle page****the circle page****7/01**This page is about the "Appollonian Circle Packing" as seen in Science News, April 21, 2001 **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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