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 dth iteration contains N+1 points, where N=2^d, and can be obtained by setting b(0)=b(N)=0, letting


and looping over n=d to 1 by steps of -1 and m=0 to N-1 by steps of 2^n.


See also

Batrachion, Hofstadter-Conway $10,000 Sequence, Monsters of Real Analysis, Nowhere Differentiable Function, Pathological, Stolarsky-Harborth Constant, Weierstrass Function

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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.

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Blancmange Function

Cite this as:

Weisstein, Eric W. "Blancmange Function." From MathWorld--A Wolfram Web Resource.

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