Thompson’s group F is a perplexing group with a number of different ways of understanding it. F can be understood algebraically, via generators and relations with an effective normal form. F can be understood combinatorially, in terms of pairs of rooted binary trees. And F can be understood analytically, as a group of piecewise-linear homeomorphisms of an interval or as a group of maps between Cantor sets. Usually tree pair diagrams used in conjunction with F are finite, but there are some applications of infinite but periodic tree pair diagrams to understanding finite index subgroups of F and to understanding the automorphism group of F. This is joint work with Jose Burillo of the Universitat PolitÃ¨cnica de Catalunya.