The logo for Ballantine Ale and Beer consists of three almost circular rings, named "Purity", "Body" and "Flavor". Purity lies under Body, Body lies under Flavor, and Flavor lies under Purity. Experiment shows that the three rings can not be pulled apart, but it is clear to the eye that if any one of the rings were to disappear, then the other two could easily be pulled apart. John Milnor, in his senior thesis at Princeton (published in the Annals of Math in 1954 after he graduated), invented a "triple linking number" whose nonvanishing for the three Ballantine rings proved mathematically that they could not be pulled apart. Last spring, Paul Melvin from Bryn Mawr gave a seminar talk on a paper he wrote with Blake Mellor, which presented a very geometric way to compute Milnor's triple linking number. After Paul's talk, I read his paper with Blake and found it to be both beautiful and inspiring. My talk on Tuesday will be about this paper, with plenty of examples worked out to illustrate it. The talk is intended to be understandable by those who did not hear Paul's talk, but also to contain stuff of interest to those who did.