Fractals and The Cat in the Hat

Akhlesh Lakhtakia
September 01, 1993

Sometime in July 1984, my colleague Russell Messier came in for a chat and ended up introducing me to fractals. Being a materials scientist, he, of course, tried to explain fractals in terms of the growth of thin films. But, not being a materials scientist, I had considerable difficulty in visualizing fractals in SEMs and TEMs of pyrolytic graphite films.

Fractals were sufficiently intriguing, nevertheless; the beautiful computer-generated images of the Mandelbrot set were too intricate to leave me unaffected. I ordered a copy of Mandelbrot's The Fractal Geometry of Nature, marveled at the colorful pictures, and tried to read that book. I do not mind acknowledging here, with apologies to Prof. Mandelbrot, that I found it very taxing; in fact, I have not yet finished reading it.

Enlightenment came in September 1985. Since I had recently become the proud father of a baby girl, I decided to pay a visit to a children's bookstore and reacquaint myself with Dr. Seuss and his hilarious belly-contorting creations. Imagine my surprise when I leafed through The Cat in the Hat Comes Back and the mystery went out of fractals in a trice.

The hierarchical nature of the 27 cats in this tale is very apparent. Assuming that cat A and its hat are the exact replicas, respectively, of the CAT and ITS HAT, and so on, there must be scaling laws for the sizes of the cats. One could in that case deduce a similarity dimension for the cats, say as the ratio log (volume of cat M's tail—volume of cat N's tail) log (area of cat M's paw—area of cat N's paw) Of course, several such similarity dimensions based on different anatomical parts of the cats could be found! Would they all agree, necessarily? That leads to the question of self- similarity versus self-affinity, or, in other words, to the recently emerged concept of multifractality.

Cat Z is very interesting: It is too small to see. The CAT, on the other hand, is very much observable, but there is definitely an upper limit on its size. This brings us to distinguish between strictly mathematical fractals and the more commonly observed natural fractals.

One can also think of the 27 cats as the members of a time- series which folds into a strange attractor. In this case, cat Z would be the strange attractor, for no other cat followed it. Let us now finally look at the pink material. The analogy of the pink snowy terrain with the basin of a strange attractor appears to be very attractive.

I will take my leave with the following thought: There are fractals galore waiting to be discovered in the chaotic works of Dr. Seuss. Enterprising readers may devote some time and energy to discovering these juvenile fractals; in time to come, the books of Dr. Seuss, probably much to his own surprise, may turn out to be pedagogical tools for training future generations of fractal researchers.

Condensed with permission from the Journal of Recreational Mathematics, Vol. 22 (3), 161-64, 1990

Last Updated September 01, 1993