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Abstract

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.

Author Bio

Anthony Zaleski is a senior at the New Jersey Institute of Technology who expects to earn his BS in applied mathematics Spring 2013. Afterwards, he plans to pursue a PhD in mathematics at Princeton, NYU's Courant Institute, Brown, or Rutgers.He was first encouraged to write a paper by Dr. Denis Blackmore, one of his professors at NJIT. When he said that one of his interests was fractals (which he had long appreciated from an artistic standpoint but had never studied in an analytical context), Dr. Blackmore suggested a pursuit of this topic. The Weierstrass-Mandelbrot function was an appropriate sub-field since Dr. Blackmore had already done research concerning its surface generalizations. Anthony wishes to thank Dr. Blackmore for the helpful comments he offered, especially during the final stage of revising his work.

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