The Weierstrass-Mandelbrot (W-M) function was first used as an example of a real function which is continuous everywhere but differentiable nowhere. Later, its graph became a common example of a fractal curve. Here, we first review some basic ideas from measure theory and fractal geometry, focusing on the Hausdorff, box counting, packing, and similarity dimensions. Then we apply these to the W-M function. We show how to compute the box-counting dimension of its graph, and discuss previous attempts at proving the not yet completely resolved conjecture of the equality of its Hausdorff and box counting dimensions. We also consider a surface generalization of the W-M function, compute its box dimension, and discuss its Hausdorff dimension.
Professor Denis Blackmore, Department of Mathematical Sciences, New Jersey Institute of Technology
"Fractals And The Weierstrass-Mandelbrot Function,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 13
, Article 7.
Available at: https://scholar.rose-hulman.edu/rhumj/vol13/iss2/7