Exercises

1. Explain why fractals in nature are self-similar only over a limited range. Is there an upper limit or lower limit or both?
2. Many movies use miniature models in their special effects. Discuss how the human brain is tricked into thinking that those models are life sized and how this relates to the concept of characteristic length and/or self-similarity.
3. Look at natural objects (or pictures of them) that are frequently referred to as fractal. Do you notice the self-similarity on multiple length scales?
4. Can you draw a unique tangent at any point of a Koch curve? Why not? What is the mathematical terminology to describe such curves?
5. Suppose the initiator of the Koch curve was 2cm long. Approximately how long would the Koch curve be after 100 iterations? If the curve was a piece of thread that could be stretched to its full length, could you use it to tie the Earth to its Moon ?
6. Take a piece of paper $2$cm by $2$cm. Make a hole in it big enough for you to push a glass of water through it. (Hint: This question is not really about fractals but is used to illustrate that you can have long lengths in small areas as in the Koch snowflake).
7. What is the area of the Koch snowflake if the initiator was an equilateral triangle of unit length on each side?
8. What are the topological and Euclidean dimensions of the Koch curve?
9. Starting from the definition of self-similarity dimension, derive the explicit expression for the self-similarity dimension of a fractal. What is the base of logarithms used in the equation?
10. Calculate the self-similarity dimensions of the Koch curve using the scales $s=1/9$ and $s=1/27$.
11. Give examples of objects that are self-similar over a wide range of scale but which do not have fractional dimensions.
12. Show that the length of the initiator that remains once a Cantor set is formed is zero. Convince yourself that the Cantor set is a fractal, by showing that it is self-similar and also by computing its similarity dimensions and comparing that with its Euclidean and topological dimensions.
13. (a)Determine the topological, Euclidean and self-similar dimensions of the quadratic Koch curve (Minkowski sausage), the Sierpinski carpet, and the Menger Sponge.
    (b) Determine the area and perimeter of the Sierpinski carpet.
    (c) Determine the volume and area of the Menger sponge and compare the result with that of a real sponge.
14. Search for the program FRACTINT on the Internet and use it to explore various fractal patterns.
15. Search for two examples of fractals in your environment that are almost exact, and two examples of those that are random.
16. Do you think the fractal dimension of a two-dimensional projection of a tree is more or less than 2? Why?
17. Look at the pictures of branching air passages in the lung, the branching of blood vessels in the human body, or the folds on the surface of the brain.
    (a) Suggest reasons why nature would have designed such fractal structures, that is, what is to be gained by adopting such a structure? Also suggest reasons for the fractal structures of trees. (Hint: See exercise (13) above).
    (b) Where are the instructions for growth in biological systems encoded?
18. Measure the fractal dimension of a long coastline given in your Atlas using the box-counting method.
19. The coastline in the above exercise may be approximated by a set of straight lines each of fixed length, and its total length estimated by the sum (the "structured walk technique"). Determine the number $N(r)$ of straight-line segments of length $r$ that are required to approximate the figure. Plot the results on a log-log plot and determine the fractal dimension of the coastline by this method. Compare your result with the box-counting method.
20. There exist many online web simulations of random walks. Explore some of the one and two dimensional versions. See for example Ref.[7].
21. Explore the DLA simulation at Ref.[8]
22. Look at the graphs in Mandelbrot’s paper on finance [Mandelbrot, Scientific American Feb 1999, pg.73]. Can you make out which of the graphs represent real data and which fractal simulations? Can one make money out of this? See also the discussion in Ref.[9]
23. Widen your horizons by browsing through the articles in Refs[5, 6].