Speaker
Andrew Krieger
Abstract
Consider the following problem in geometry: Given two non-intersecting circles A and B, how many ways are there to draw a chain of n circles C1 , C2 , ... , Cn with the property that each Ci is tangent to A, B, and also to Ci−1 and Ci+1 (indices taken mod n). This is Steiner’s Porism. We have to be lucky enough to get one good configuration (depending on the radii of A and B), but if we do, it turns out that there
are infinitely many such configurations. The easiest case is when the two circles are concentric, but using inversive geometry, we can reduce any other arrangement to this easy case.
So what is inversive geometry? It makes use of the simple map on R2 (or higherdimensional Euclidean space) given by inversion relative to a circle (or higherdimensional sphere). One simple case of an inversion map is the map z ≥ 1/z , which is just inversion relative to the circle {z : |z| = 1} in the complex plane. General geometric inversions of this sort allow for an easy proof of Steiner’s Porism and other problems. I will give a brief introduction to inversive geometry, with the proof of Steiner’s Porism
as the motivating application.
For more information, see the What Is... ? Seminar page.