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Fractals and Dynamics

This post gives an introduction to the fascinating topic of fractals and dynamics and describes what to expect in the upcoming course on this- starting Aug 28, 2020. Click here to know more about Jerry and register for the upcoming course.

Imagine beings from another world looking down on Earth from far above its surface and searching for signs of civilization. What characteristics would distinguish ‘artificial’ features from natural ones?

For one thing, human-made objects often combine the basic shapes that you learn about in geometry class: ‘pure’ forms from classical geometry such as lines, planes, circles, and spheres.

At one time, humans even believed that they had found evidence of life on Mars when they thought they saw objects that reminded them of canals.

Natural objects on the other hand have a special kind of complexity that arises from the dynamic physical processes that create them. The shapes of clouds, trees, river basins, mountains, lightning, and even the circulatory and nervous systems of living beings are not typically described well by classical geometry.

In the late 20th century, computer technology had developed to a point that it became practical to explore the ways in which natural processes can lead to geometric figures known as fractals. The resulting mathematical ideas have given us profound new perspectives on both the processes themselves and their physical manifestations as fractals. We began to realize that the simple act of repeating a process or rule can lead to surprisingly complex outcomes.

Unfortunately, the mathematics that we learn in school usually includes very little of these new understandings. The GenWise Fractals and Dynamics course will increase your awareness and knowledge of these mathematical ideas.

An example of a fractal

Interestingly, when you first study fractals, you often begin with familiar (classical) geometric shapes. The resulting images may or may not look ‘natural’ to you, but they certainly seem to have some of the complexity that you find in nature. For example, suppose that you begin with a line segment.

Now remove the middle third and replace it by two segments of the same length as shown here.

If you imagine repeating this simple process forever, you ultimately obtain a fractal known as a Koch curve.

Although this (approximate) picture of a Koch curve may look a little too perfect to be natural, its complexity may suggest certain forms that you have seen in nature.

Note that even if you start with the same shape (a line segment in this case), changing the rules of the process in small or large ways may create dramatically different results. (You can even insert a little randomness if you like!) In fact, as we will see in the class, it is ultimately the dynamical process, not the initial image, that usually determines the final result. The rules used to create this tree-like fractal may be hard to see, but they are probably much simpler than you would expect!

An example with numbers

As the preceding examples illustrate, dynamical processes like the ones that create fractals involve choosing a starting condition and then applying a rule or set of rules (often very simple ones!) over and over. You can apply such processes to numbers as well as shapes.

For example, suppose you start with the number 1.1 and apply the rule ‘square the number’ (that is, multiply it by itself) over and over. The result will soon grow very large!

1.1 1.21 1.4641 2.14358881 4.59497298636 21.1137767454 . . .

Suppose now that you change the starting number to 1 (a decrease of only 0.1) and apply the same rule. You will simply get the number 1 over and over!

1 1 1 1 1 1 1 . . .

Finally, try starting with 0.9. The number now gets smaller and smaller each time you square it, getting closer and closer to 0.

0.9 0.81 0.6561 0.43046721 0.185302018885 0.034336838203 . . .

These examples illustrate an important phenomenon in dynamics: in some instances, small changes in the initial conditions (the number that we start with in this case) may lead to dramatically different outcomes. Why? Because as you repeat the rule, the effects can accumulate and magnify at each stage.

The Butterfly effect

The phenomenon above is referred to as sensitivity and is a hallmark of dynamical processes. It is popularly known as the butterfly effect’ — a term that comes from the title of a 1972 talk by Edward Lorenz, a founder of the study of dynamics: ‘Predictability: Does the flap of a butterfly’s wing in Brazil set off a tornado in Texas?’

Have you ever wondered why it is so difficult to predict weather accurately? Although humans have become much better at doing this in recent years, it is very difficult to obtain reliable predictions over even a moderate time span. Consider the following scenario described by James Gleick in his book, Chaos-

Suppose the Earth could be covered with sensors spaced one foot apart, rising at one-foot intervals all the way to the top of the atmosphere. Suppose every sensor gives perfectly accurate readings of temperature, pressure, humidity, and any other quantity a meteorologist would want. Precisely at noon an infinitely powerful computer takes all the data and calculates what will happen at each point at 12:01, then 12:02, then 12:03, … The computer will still be unable to predict whether Princeton, New Jersey, will have sun or rain on a day one month away. At noon the spaces between the sensors will hide fluctuations that the computer will not know about, tiny deviations from the average. By 12:01, those fluctuations will already have created small errors one foot away. Soon the errors will have multiplied to the ten-foot scale, and so on up to the size of the globe.

This scenario suggests that while dynamics offers a powerful framework for creating predictive mathematical models, it can also have surprising limitations. And these limitations are not due merely to our insufficient knowledge but to the very nature of the physical processes themselves!

Applications

Dynamics are useful in studying weather, the environment, financial systems, and a host of

other phenomena. And since dynamical processes lead to fractal images, they are also useful in art and design. For example, movie and game producers can use software that relies on fractal geometry to create artificial landscapes like the one shown here. Armed with a few new algebraic and geometric concepts combined with some simple software tools, you can create meaningful mathematical models and beautiful fractal images of your own. The GenWise Fractals and Dynamics course offers you an opportunity to do this. And you will learn some fascinating mathematics along the way!

85+ children have attended Jerry's GenWise courses on mathematical thinking since April 2020. They have fallen in love with exploring mathematical thought processes in an environment that is simultaneously challenging and supportive. 15 of the children have taken 2 or more of his courses.

Below are 2 quotes from Sheela, an international school parent from Singapore-

On his first course on 'Geometric Designs': Jerry has been very encouraging and made my son enjoy Geometry. He was reluctant to join this course but Jerry's style of teaching and positive encouragement for merely attempting the problem has changed his attitude towards the subject

On his second course on 'Groups & Symmetry': My son was not interested in the topic when signing up, but he joined as he had enjoyed Jerry's previous course and was willing to try this out. He enjoyed the course and attempted all questions whole heartedly. However, he did have questions on its practical usage. Jerry did give some pointers about it. But the icing on the cake was the last session where Jerry got a guest speaker who spoke in length and showed the practical usage in molecular chemistry. My son found it very fascinating and exciting. Excellent course. Thank you!