Title: Paperfolding, automata, and rational functions

Alf van der Poorten, Centre for Number Theory Research, Sydney

Advertisement: The sequence of creases --- the valleys and ridges --- obtained in a sheet of paper by repeatedly folding it in half (and finally unfolding it) is surprisingly complicated and interesting. Learn how this simple sequence contains unexpected patterns, gives rise to a fractal tessellation of the plane and produces frightening dragon curves. Bring along a sheet of paper (not for taking notes, but to fold).

Abstract: The act of folding a sheet of paper in half, and iterating the operation, places in that sheet a sequence of creases appearing as valleys or ridges. Coding these appropriately yields a sequence $ (f_h)$, the paper folding sequence, with generating function $f(X)= \sum_{h\ge1}f_hX^h$, the paperfolding function. It turns out to be easy to notice that $f(X)$ satisfies a functional equation of a kind first studied by Mahler some seventy five years ago. Moreover, viewed as defined over $\mathbb F_2$, the field of two elements, the paperfolding function is algebraic --- it satisfies a polynomial equation over $\mathbb F_2(X)$. It's also easy to see that the paperfolding sequence is `automatic'; it is generated by binary substitutions.