Math Journeys, Volume 3
Boy, Devil, Fractal, Part 3
Uncle Bob intends to show a connection between the arithmetic we've been using to analyze the Boy and the Devil problem, and the colorful and varied graphs commonly called fractals. The one in the illustration is a prominent example known as the Mandelbrot set.
Let observe some of the M-set's qualities. It is two-dimensional. We see it has black regions surrounded by a lacy border. This border separates the black interior from the banded, many-colored exterior.
Now let's revisit the original Boy and Devil problem. A boy meets a devil at the entrance to a bridge. The devil determines that the boy has some money, and offers to double the boy's money each time he crosses the bridge, as long as the boy pays the $6 toll at the far end. On first crossing the boy's money doubles and he gladly pays the toll, but after three crossings the boy is flat broke. How much did he start with?
You may have tried different starting amounts to see what possible fates awaited a boy who continues to play this game. The trails of intermediate amounts of money possessed by the Boy are known as orbits. Here are some examples.
|seed||1st crossing||toll||2nd crossing||toll||3rd crossing||toll|
In fractal lingo, the starting amounts are known as seeds. Zero dollars (going broke) seems to be the end result of seeds less than $6. We call zero an attractor of these orbits. Seeds higher than six result in the boy winning the game (getting rich). What happens to a seed of exactly $6? We double 6 to get 12 and then subtract the toll of 6, and see that after any number of crossings, the boy ends up with what he began. Six dollars is a fixed amount and the boundary between sad and happy results. Those sad orbits are called non-escaping, and so you can guess what the happy orbits are called.
The fate of various starting amounts can be mapped on a number line. The various seeds fall into bands of color that distinguish how rapidly or slowly the orbits escape (cool colors) or go broke (warm colors). In between the attractor zero and the fixed point $6 I've indicated amounts that allow the boy to cross the bridge once, twice or three times.
Fractals are graphs that keep track of orbits too. These orbits result from repetitions of a rule not unlike the Devil's rule applied to various seeds. The graph interiors, usually black, display the regions of stability, in other words, seeds that don't escape. In those regions there may be attractor points. The exterior regions are arbitrarily colored to show the ease with which seeds escape under the rule.
One difference in fractal graphs is that the seeds and the rules apply to complex numbers and are mapped onto the complex plane. Complex numbers have a real part and an imaginary one. This explains why the graphs need to be two-dimensional. The horizontal scale is the same real line we used to map the boy's outcomes. The vertical axis is scaled in multiples of the imaginary number i. To do fractal analysis one needs to understand arithmetic with these complex numbers under repetitions of a rule, and how to sort and graph the seeds. Despite its name, complex arithmetic is not that complicated. You can get a start on understanding it at
A second difference in fractals is that the boundary is infinitely complex. Zooming in on a portion of the boundary leads to more and more complexity similar to the original portion. These zooms can be seen on the internet. One with a musical redering of the story can be viewed at
For the Mandelbrot set the action takes place inside a square centered at zero and ranging from -2 to 2 horizontally, and -2i to 2i vertically. Both fractal types use rules that involve squaring the numbers. If only squaring is involved, without any adding or subtracting, the stable, non-escaping numbers lie on or inside the unit circle, green in color on our graph above. The circle's definite boundary disqualifies it as a fractal, but adding or subtracting small complex amounts to the rule, as does the Devil's bridge toll in the real example, results in an infinite zoo of fractal creatures, some looking like broccoli and others like elephants.
Mathematicians and scientists are now using fractal math to analyze the complex systems in nature such as watersheds and forests on a large scale, and circulatory and nervous systems on a minute scale. The following 7-minute vid tells one story
Questions for you. What if our Devil's rule was that he would square the amount of money each time the boy crossed? You can create a sampling of orbits of real seeds under repeated squaring. Are there non-escaping values? ... a boundary? ... a fixed point? ... an attractor?
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