The Circles of Descartes

[phantom power]

http://blog.wolfram.com/2010/06/16/the-circles-of-descartes/

In 1938, Lester Ford tried putting circles with radii equal to that block’s third row above their position on the number line. These are now called Ford circles. All these radii have a one on top. Sometimes, it’s nice to use fractions, but it’s better to use integers when possible, so the bend (or curvature) of a circle is 1/radius, as seen in the fourth row. The circle in the middle of the sequence above with radius 1/8 has a bend of 8. Here is a picture of the Ford circles along with their bends, along with Mathematica code for making it. Notice how the circles all exactly touch some other circles. Each circle is tangent to the neighboring circles. Touching = tangent = kissing, when it comes to circles.



...

Circles don’t have to be in a line, though. Imagine placing circles on the complex plane, the realm of the Mandelbrot set, where (z=x+i y). For the circle with with curvature c1, let z1 be the center point. On the complex plane, when four circles kiss, Mathematica can make use of this expanded Descartes Circle Theorem:



Here is that theorem in action. All of the given circles are inside the outer circle, which has a negative bend because the others are inside it. If the first few circles have an integer bend, then all the circles, off to infinity, have integers for 1/radius. Here, the circles are colored mod 12, much like the hours on a clock face. In each grouping, only four colors are possible.

[WheelWalker]

1. I gave this an unrec for failure to provide warning of a level 5 psychedelic yield...