The really fun part of nuclear pennies is the link at the very end, Blass' paper "seven trees in one." There, he shows how since a full binary tree is either a leaf or two trees, we have T = T^2 + 1 and, solving for T, we get a 6th root of unity for the cardinality of the groupoid T. T^6 isn't isomorphic to 1, but T^7 is isomorphic to T. Now, that's nothing special, since the set of full binary trees is countable. But this isomorphism doesn't need to go deeper into the trees than a fixed amount illustrated on the page. Baez and Dolan showed how groupoids are like "fractional sets," that it makes sense to talk about algebraic structures with rational nonnegative cardinality: |X| = \sum_{equivalence classes [x]} 1/Aut(x) Many others have extended their work. Fiore and Leinster follow up the seven-in-one paper with Marcelo Fiore and Tom Leinster, Objects of categories as complex numbers, math/0212377 and show that their definition has Baez & Dolan's cardinality as a special case. Many more references here: http://www.math.ucr.edu/home/baez/counting/ -- Mike Stay metaweta@gmail.com http://math.ucr.edu/~mike