Blancmange Function

The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name derives from the resemblance of its first iteration to the shape of the dessert commonly made with milk or cream and sugar thickened with gelatin.

The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway $10,000 sequence. The first six iterations are illustrated below. The th iteration contains  points, where , and can be obtained by setting , letting

and looping over  to 1 by steps of  and  to  by steps of .

REFERENCES:

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 111-113, 2007.

Dixon, R. Mathographics. New York: Dover, pp. 175-176 and 210, 1991.

Peitgen, H.-O. and Saupe, D. (Eds.). "Midpoint Displacement and Systematic Fractals: The Takagi Fractal Curve, Its Kin, and the Related Systems." §A.1.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 246-248, 1988.

Takagi, T. "A Simple Example of the Continuous Function without Derivative." Proc. Phys. Math. Japan 1, 176-177, 1903.

Tall, D. O. "The Blancmange Function, Continuous Everywhere but Differentiable Nowhere." Math. Gaz. 66, 11-22, 1982.

Tall, D. "The Gradient of a Graph." Math. Teaching 111, 48-52, 1985.

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 16-17, 1991.