The art of asking the right questions in mathematics is more important than the art of solving them. Georg Cantor Georg Cantor (1845-1918) was one of the first mathematicians to consider problems involving infinite processes. But he couldn't go very far with them. He didn't have a computer. You have! Roll over, Georg. Iteration is the process of taking a value, doing something with it (called applying an algorithm) and finding an answer. This answer is then put back into the algorithm and the process is repeated, over and over, ad infinitum, for ever and ever.

Iteration is a different form of mathematical exploration. You are used to solving equations. The real world has those equations applied over and over forming a continual process which is never completed. Weather patterns do not result in a final weather. Cloud formations do not lead to a solved cloud.

Equations are not always solved - sometimes they just go on and on being applied.

Mathematics can, and should, reflect this reality.

The first computers were so expensive, they weren't used for the extravagance of exploring ideas. They were needed to solve existing problems. Now computers are cheap enough to become playthings. Mathematical playthings.

It is with the advent of the personal computer we can be allowed this indulgence. You can explore mathematical ideas which were beyond the reach of the most gifted mathematicians only a few decades ago.

This has become what has been referred to as The New Science: Chaos Theory. The mathematics associated with it has become the foundations of the most visually beautiful field of all: Fractal Geometry. James Gleick created a cult with his best selling book, Chaos: the making of a new science. The interest in iterative processes has soared since then. Exploring mathematical processes can be exciting - if you can eliminate the drudgery. So let the computer do the tedious calculations, leaving the you free to do the thinking and the exploring. Computers are great at repetitive calculations by the million. As a team, student plus computer, you have the means to explore areas of mathematics which were only toyed with until computing power came to the desktop. Why study iteration? Apart from being fun, iteration matches the way the real world works in many situations. It is used to model real systems such as population growth and weather patterns. Iteration can be extended to generate the beautiful patterns known as fractals which are now so popular on posters and tee-shirts. Relax a little. Play a little. Feel free to explore, to make up ideas, to create your own problems and delve into a mathematical way of thinking. Sometimes you will only be able to pose the questions and start on the exploration of possibilities without reaching a firm conclusion. But that's fine. That's the New Science.

You may be uncomfortable with guessing and submitting incomplete work. If you have a wild idea, which may be really interesting to explore, but you think may be totally "wrong", go with it all the same. Play on! It is time for mathematics to become an experimental science - and the time is now! Iteration is a process which involves repeating the same steps over and over again. The answer to the calculation is then fed back into the equation and becomes the starting point for the next calculation. It is the basis of the mathematics they call fractal geometry and the science they call chaos theory.

The Algorithm Square the number then take the answer and square it then take that answer and square it and  continue to iterate under the function, squaring the number. Do NOT round off values.  Slight differences can cause radical changes.

An Example Start with a number, say x = 1.234 square it: x2=1.5227560 Iterating means putting the result of the calculation back in for x: x is now taken to be 1.5227560  it is squared again, to give 2.3187858 which is squared again to give 5.3767677 and so on - which will eventually go to infinity. Another Example What about - 0.45? -0.45 2 = 0.2025 0.2025 2 = 0.0410062 and then: 0.0016815, 0.0000028 and so on to 0.

Try the squaring function for other numbers, say 2.3 and 1.7.
Include the tools you used (pencil and paper, calculator, computer with Excel, programming) and any output. Try the squaring function again, but start with a number smaller than 1, say 0.5678 Did it go to 0?
Can you predict what will happen to any number when iterated under the squaring function? Say, -4.56 or 0.9758?
What happens to -1, 1 and 0 under the iteration of the squaring function?

Can you generalise your findings for the squaring algorithm by giving the rules for different groups of numbers?

For example, can you generalise what will happen for all numbers over 1?
All numbers between 1 and 0?
What about -1, 1 and 0 - the special cases?
Can you generalise about all negative numbers?
Now for some other functions which offer some real surprises.
For each of the functions given, explore what will happen if the function is applied over and over and over - that is, it is iterated.
Remember, you will need to try all types of numbers. Numbers less than 1 will often be the most interesting.

What do you get when you iterate the following functions? Some go weird!

a. Square root of x

b. 1/x

Reminder: check for decimal values less than 1, and for negative values.

The answers for the next few are not simple. They could take the rest of your life! Be very careful you don't just rush and miss the weird and wonderful things which happen. Don't forget to:

Try numbers above and below 1, fractions or decimals, and negative as well as positive numbers.

c. x² - 1

d. x² - 2

e. 2x² - 1

What conclusions could you draw about these functions under iteration?
Make sure you explore them fully: numbers above and below 1, fractions or decimals, and negative as well as positive numbers.

Attach any written figures, spreadsheets, program code or computer output which have been used in the course of these explorations.