Another one of the most interesting books I have read is Chaos and Fractals , New Frontiers of Science, Peitgen, Jurgens, Saupe; Springer-Verlag, (1992). It is probably dated by now – may even be later editions, but when I first encountered it I was so impressed that I bought it for our high school library. It turned out that I was the only one according to the school library records who signed the book out to read, so when the library was renovated and books were culled, I got it.
The book is remarkable for its illustrations and photos, but is also quite exhaustive and thorough in its exposition of what chaos and fractals are, their connections and the many practical applications of each to science and other arenas of life.
I attempted to give a brief explanation with examples of fractals in one of my annual high school addresses at graduation exercises. An excerpt from the 1996 remarks follows:
One of the terrific things about teaching and learning is that it is very much a matter of human interaction. As a result there is never really any limit to the surprises, the nuances and the insights we can gain from one another. When we spend time together in the classroom, on the soccer field, or at organization meetings, we are not only learning what is in our textbooks, but we are also learning about other people and about ourselves.
I’ve learned a lot about many of you in the last few weeks leading to graduation that I never knew or suspected. We won’t embarrass [name of student] any more with what we learned about his childhood at the banquet the other night. We’ve all learned what tremendous amount of musical and creative talent we have in this class – musicians, singers, composers [names of some students] – the list could go on and on. It wasn’t all easy to put together. It took cooperation and willingness to accommodate one another.
We can all learn and we all have something to teach – because each and every one of us is a source of tremendous influence if we are prepared to let ourselves be so and to accept one another as being so. When [Dr. M] spoke at our banquet the other night, one of the things he did was share with us some ideas he had read in books that he found impressive. We can all do that and we should do it more often.
One of the fascinating things I have been learning about recently are fractals. Fractals are kind of a new thing [this is 1996] – they are nature’s way of dealing with geometry – a different way of using math. I don’t have time to really delve into fractals – for which you are thankful, I’m sure, but I can give you some examples to make my point.
If I were to draw a line, say, this long… and I asked how many points are in that line what would you say. Do any of you know – maybe all you remember from [Mr.G’s ] geometry class are his wonderful jokes. There are an infinite number of points in this line. If I cut the line in half – how many – an infinite number. Cut it in half again – still an infinite number of points.
I asked [SG – name of student] a few months ago to check out fractals on the internet. He had never heard of them but he did check it out and he came back and told me that such things definitely existed because he had found 600 websites under that heading. I thought that was pretty good. Several hundred years ago a mathematician and philosopher by the name of Descartes said, “I think, therefore I am.” This has now been updated by that somewhat less famous computer hacker and philosopher, [SG] – “It’s on the internet, so it exists.”
One way of understanding what fractals are is to ask a simple question like – How long is the coastline of PEI? [This is a variation of the more famous question asked by B.B. Mandelbrot – How long is the coast of Britain? in a paper published in Science, in 1967] Now the coastline of PEI is a fractal and the answer to the question is – it depends. It depends on how you go about measuring it. If you measure the coastline very roughly you can get one answer. If you measure every bay and point you get a different greater length. And if you take a meter stick laying it end to end from East Point round North Cape and back again you would find that the answer gets really complicated. How far up the Souris River do we go before we cross it and continue on? How closely do we follow all the crooks and galleys made by the surf as we travel around the shore. Maybe the coastline of PEI is infinite – because it is a fractal.
People are fractals. You think you know someone. Maybe you heard about them from a friend. Or maybe you met them briefly and you had a first impression. Then you get to know the person a little better, and you find out things you never suspected. First impressions, stereotypes – these are rough ways of measuring people. But people are fractals, and when we take the time to talk and to listen we discover all kinds of fascinating things about each other.
I keep encountering the very complicated and fractal nature of you people all the time. I may have to deal with a student who is always late – simple – some detentions. Then I find out why this student is always late – and I see this student is carrying a weight to school that I doubt I could carry to school at all. How long is the coastline of PEI? Long enough to spend a lifetime exploring and enjoying. How much creativity and potential to do great things do you young people have – well it’s the same answer – enough to fill a lifetime – there is really no limit to it.
Graduates of 1996, I hope you will always be sensitive to the infinite possibilities in others and in yourselves. Each and every one of you has a life of great promise and we want to see you fulfill if. God be with you.
As is apparent from the examples in the speech, the fractal concept can be applied in a wide variety of areas. For Mandelbrot and others, including myself, it can also apply to the universe itself when one looks at the many hierarchies observed – such as galaxies, clusters of galaxies, and superclusters. In spite of the presence of fractals in so many aspects of nature, it is a concept that is not really part of the standard paradigm (usually referred to as the Standard Model or (Lambda Cold Dark Matter model) in cosmology.