West End Number Theory Seminar: Stefan Erickson

Abstract:

Apollonius's Theorem states that given three mutually tangent circles, there are exactly two circles which are tangent to all three.  Apollonian circle packings are produced by repeatin the c onstruction of mutually tangent circles to fill all remaining spaces. A remarkable consequence of Descartes' Theorem is if the initial four tangent circles have integral curvatures, then all of the circles in an Apollonian circle packing will have integral curvatures. This process results a sequence of integers with fascinating arithmetic properties.