Blancmange Function
February 28th, 2008 at 11:26 pm (Afternoon Tea)
The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway $10,000 sequence. The first six iterations are illustrated below. The dth iteration contains N+1 points, where N=2^d, and can be obtained by setting b(0)=b(N)=0, letting
b(m+2^(n-1))=2^n+1/2[b(m)+b(m+2^n)],
and looping over n=d to 1 by steps of -1 and m=0 to N-1 by steps of 2^n.