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The Apollonian Circle Problem and Apollonian Gaskets

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The Apollonian Circle Problem and Apollonian Gaskets

Jen Kokoska

Math 335

- Apollonius 'The Great Geometer' of Perga
- Student of Euclid (p. 7 in our book)
- Book Conics introduces terms parabola, ellipse, and hyperbola we use today
- Made large contributions to inverse geometry (p.313)

- Book Tangencies introduces Apollonius' circle problem:
- Given any three points, lines, or circles in a plane, construct a circle which contains the points and is tangent to the lines and circles

- We can see there are ten distinct combinations of cases to find solutions for...

- Book IV of Euclid's Elements shows us how to construct a circle tangent to three sides of a given triangle, and a circle containing three noncollinear points

- Case of three circles becomes the most difficult
- up to 8 solution circles can exist

- Special case of Apollonian circles- three circles all tangent to one another.
- Two solutions exist

- Decartes became the first to discover this case (1643), Beecroft rediscovers the solutions (1842)
- Fredrick Soddy writes a poem about them in 1936 titled "The Kiss Precise"
- nobel prize winner in chemistry
- circles become known as Soddy Circles
- Theorem also extended to analogous formula in 3 dimensional space

- Form a triangle connecting the three circle centers (black), and drop a perpendicular line from each center to the opposite triangle edge (blue). This line cuts its circle at two points;
- Draw a line from each cut point to the point of tangency of the other two circles (green). These green lines cut their circles in two more points, which are the points of tangency of the Apollonian circles.

- Reason behind why this construction works deals with inversion of circles

- After constructing an inner soddy circle, we have three sets of tangent circles
- reiteration constructs a proportional figure known as an Apollonian Gasket

Works cited:

http://www.uni.edu/ajur/v3n1/Gisch%20and%20Ribando.pdf

And http://www.ics.uci.edu/~eppstein/junkyard/tangencies/apollonian.html