More Mathematics . . .

Patterns

The natural world is awash with detail. Even in the bleak lanscape of a frozen tundra, patterns can be found at all scales, whether in the gentle curve of a wind swept serac or in the intricate crystalline structure of a snowflake. It is fortunate for us that many of the observable patterns in nature form as the result of a process. A step by step series of instructions to be carried out. Some processes are more complicated than others. Ice crystal formation is a fairly simple process when compared with the fluid dynamics relevant to the creation of clouds, for example.

In science one tries to develop theoretical models to predict these patterns, thereby understanding the process, and then tests these predictions against experiement. Often these predictions use mathematics, and during the development of the full model, many interesting questions arise. An example is pictured schematically at right.

In the picture at right, suppose for the sake of

argument that the line segment in Stage 0 of the figure is 1 meter long. The next stage, Stage 1, is produced from the previous stage by first dividing the line in Stage 0 into three equal pieces of length 1/3 the original size, then removing the middle third and inserting the tent of an equilateral triangle.

Stage 2 is obtained from Stage 1 by applying the above process to each of the four straight line segments in Stage 1. And we continue... If you want to draw Stage n you simply apply the process to the previous stage, Stage n-1 . But, of course, you need to know all the stages prior in order to do this. The result is a sequence of drawings becoming more comlex the higher the stage number, but still looking somewhat like the previous members of the sequence. You can see in the figure that already at Stage 4, the drawing is quite complex with much detail. In fact, if you continued the construction further you might say that stages 4,5,6,7,... don't look that much different from one another, and you'd be right. Of course they are different fundamentally, but at the scale we've drawn them, we can't see much difference.

What is a limit?

Niels Fabian Helge von Koch (1870-1924) was a swedish mathematician who first played with the figures we are discussing. He noticed that as the stages progressed, the figures seemed to "settle down" to a figure not that much different from that in Stage 4, as we've observed. He asked the question, "What happens to the figures if we continue the process indefinitely?" In other words, suppose that you set you set your little, artistically gifted, sister to work making drawings according to the process, could you tell the difference between Stage 1,000,000 and Stage 5,555,679, or further? Does this sequence have a limit?

In fact, this sequence of drawings does have a limit, in a technical sense, and that limit is called "von Koch's Curve." What's interesting, is that if you arrange 3 copies of the curve along the edges of an equilateral triangle, you get the figure at left. That's a pretty good resemblence to the perimeter of a snowflake, if you ask me.

To answer the question at the beginning of this section, a limit is an object that a sequence of similar objects settles down to at the "end" of the sequence. That's the rough idea, anyway.

Limits are very important and sometimes difficult concepts, because they don't always exists, but are key to the study of Calculus. Usually, one begins exploring limits of sequences of numbers.

What is the length of von Koch's curve? The only way to answer such a question is using limits. Here's a guide:

1. Recall that the line segment in Stage 0 was 1 meter long. Follow the process and compute the length of Stage 1, remember that each straight segment is the same length. Compute the lengths of the next few stages (you made need a calculator for this). Can you see a pattern?
2. Find a formula for the length of Stage n. Check your formula against those you previously computed.
3. What happens to the lengths as n becomes very large? Do the lengths settle down to a particular number? How are they behaving? Your answer should make you feel a little uneasy if you've never done this before.

Don't panic. There are three possible answers to the command: "Find the limit of this sequence (of numbers)." The limit exists and is finite. The limit exists and is infinite. The limit does not exist. Notice that existence is an important part of the answer. An example where the limit does not exist can be found in the sequence

1, -1 , 1, -1, 1, -1, . . .

The sequence does not continue to grow indefinitely to large unmanageable numbers, it merely flips back and forth between 1 and -1 for eternity. So no matter how many stages out in the sequence you go, there is always a "big" difference between Stage n and Stage n-1.

Sierpinski's Gasket

Another interesting sequence of figures is the one depicted at right. It begins with a triangle. The one depicted is a right iscoseles triangle. Assume again for the sake of argument that the length of the base and height are equal to 1. The triangle can be nicely divided into four congruent triangles, all similar to the original. Removing the central triangle from Stage 0 produces Stage 1. To get to Stage 2, we divide each of the blue triangles in Stage 1 as before, and remove each of the central triangles. Continue this process to generate the next stage, and so on.

This sequence again appears to settle down to a "final" figure. The limit of this sequence is called Sierpinski's Gasket, named for Waclaw Sierpinski (1882-1969), a polish mathematician who made great contributions to the foundations of mathematics.

Here is another interesting problem. Compute the area of Sierpinski's Gasket. Again a job for limits.

1. Find the area for the first few stages, establish a pattern. Use a calculator to estimate the area of several more stages.
2. Find a formula for the area of Stage n. Check it against your previous computations.
3. Find the limit.

Chaos Game

The Chaos Game begins requires a sheet of paper, a die, a pencil and a ruler, and a lot of free time. Fortunately with

computer technology, we can reduce all these requirements to a computer, and a few mouse clicks, as we'll see below. First: the setup.

1. Draw a fairly large triangle on a sheet of paper. Randomly mark a number from 1 to 6 at each vertex of the triangle being careful not to use any number more than once. Assign the remaining numbers to the vertices in the same way so that each vertex has two numbers from a die.
2. Plot a point at random anywhere on the paper.
3. Roll the die. Locate the vertex with that number.
4. Using the ruler, mark a point half way between the previously marked point and the vertex.
5. Repeat steps 3 and 4 many, many, times.

Think about this for a moment. This is a completely random process. What do expect should happen? It sounds to me like this will produce a page full of dots similar to what crazed toddler with a crayon could accomplish with less effort. You might be surprised if you actually carry it out. Try it. The links below lead to a German website with a java applet that plots the points for you with each mouse click. In fact one link lets you do 100 rolls of the die with each click, for those of you with little time. Try it a couple of times with different starting points and see what you get after a few thousand tosses. Enjoy!

(To get the idea)