Paper-Folding-Fractals
Hold Infinity in the Palm of Your Hand...

Figure 4: Levels 1 through 13 of the Dragon

With each fold, the Dragon get increasingly complex, and smaller in area. Level 13 in figure 4 is barely visible as 3 or 4 pixels; yet the length of the line in level 13 is exactly the same as the length of the line in level 1. Additionally, the line never crosses on top of itself - this can be seen in the 128x magnification of level 13 shown in figure 1. By level 20, a line that was a kilometer in length would be so intricately twisted that it would cover less area then a pin point! The computer lets us both generate this complex dot, and magnify it so that we may observe all of its detail. Each step of creating this fractal is simply understood - fold a line in half and open it up to from right angles. Who would ever have imagined the beauty that emerges by just repeating these simple steps?

To see a world in a grain of sand,
And a heaven in a wildflower:
Hold infinity in the palm of your hand,
And eternity in an hour.
-- William Blake

Fractal patterns come in infinite varieties, yet there are some qualities that can be found in all fractal patterns. One such quality is being very crinkly. Being crinkly means that very long lines and very large areas have a structure that fits them into very small spaces. Another essential quality of fractals is self-similarity on very different scales: a small piece of the Dragon has the same general pattern as does a large piece of the dragon. These two qualities of crinkliness and self-similarity on different scales are also commonly found in nature, and there are many references to fractals in nature.

Consider, for example, the human circulatory system. If all of the veins, arteries and capillaries in an adult human were laid out end-to-end, then the total length would be over 50,000 miles long! That is long enough to wrap two times around the Earth's equator! It is hard to grasp that there are over 50,000 miles of plumbing in my body. That is lots of crinkliness. Furthermore, the shape and pattern of the tiny capillaries in our bodies bears a great likeness to the shape and pattern of the major veins and arteries of our bodies: the human circulatory system has self-similarity across different scales.

Next time you are outside, notice that small branches of a tree have the same patterns, same angles and shapes as large branches of the same tree. Especially in the western United States where large scale rock formations are exposed, it is not hard to find a rock with small scale fractures, lines, bumps and patterns that have the same form as the large scale fractures, lines, bumps and patterns on the formation of which the rock is a part: Self-similarity on different scales abounds.

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