Lets Make a Difference!

We study the behaviour of iterations of the difference operator delta on streams over {0,1}. In particular, we show that a stream sigma is eventually periodic if and only if the sequence of differences sigma, delta(sigma), delta(delta(sigma)), …, the `delta-orbit’ of sigma as we call it, is eventually periodic. Moreover, we generalise this result to operations delta_d that sum modulo 2 the elements of each consecutive block of length d+1 in a given 01-stream. Some experimentation with delta-orbits of well-known streams reveals a surprising connexion between the Sierpinski stream and the Mephisto Waltz.