1. The Apollonian Circle Problem and Apollonian Gaskets
Jen Kokoska
Math 335

2. Background 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...

3. Apollonian Circle Problem (p.333)
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

4. Apollonian Circle Problem
Case of three circles becomes the most difficult
up to 8 solution circles can exist

5. Kissing Coins Problem
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 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

6. Eppstein’s Construction
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.

7. Eppstein’s Construction
Reason behind why this construction works deals with inversion of circles

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

9. Works cited:
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