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MAT 331 Fall 2023 Project

The box-counting dimension of the Weierstrass function

The Weierstrass function on [0, 1] is defined by

where b is an integer greater or equal to 2 and 0 < ↵ < 1. It is a famous example of a continuous function that is nowhere di↵erentiable. You may have encountered it in MAT319/320. The graph of this function is also an example of a “fractal” set. This project concerns numerically estimating the box-counting dimension of the graph. The box-counting dimension is also commonly called the Minkowski dimension.

The basic idea involves covering a set with an  grid. Then, counting how many boxes from that grid a set hits. As n grows and the grid squares get smaller, more squares will hit the set. The number of squares often grows like a negative power of n, say Nn ⇡ nd, and d is called the box-counting dimension. For example, the box-counting dimension of a line is 1 and the box-counting dimension of a square is 2. To compute the dimension of the graph of the Weierstrass function, we take the logarithm of this equation and solve for ↵ to get α = log(Nn)/ log(n) (the base does not matter as long as you use the same base for both logarithms).

(1) Take b = 2 and α = 1/4, 1/2, 3/4. Plot the graph of f2,α on [0, 1] for each of these choices.

(2) Let Nn(f) be the number of boxes from the standard  grid of squares in the plane that are hit by the graph of f. Show that

where  Write a Matlab function that takes f and n as inputs and returns the value of the sum on the right.

(3) The box-counting dimension of the graph of f is defined as

Estimate this limit for the Weierstrass function f2,α for several α values, say α =, .2, .3,...,.9. Plot your estimates. Can you formulate a conjecture for what the box-counting dimension is as a function of α? Since fb,α is given by an infinite series, you will have to replace the infinite sum by a finite sum in your experiments. Taking k = 50 or 100 should give good results for n ≤ 1, 000, 000.

(4) Repeat your experiments for other values of b, say b = 3, 4. For each α, does the box-counting dimension seem the same as before, or does it change when b changes?

Remark: The dimension of the graph of the Weierstrass function is discussed in Chapter 5 of “Fractals in Probability and Analysis”; a PDF of this book is available at:

https://www.math.stonybrook.edu/~waterman/Fall23_MAT331/OtherMaterial/fractalbook_final.pdf