Complexity_ A Guided Tour - Melanie Mitchell [52]
FIGURE 7.2. Top: Large-scale aerial view of Ireland, whose coastline has self-similar (fractal) properties. Bottom: Smaller-scale view of part of the Irish coastline. Its rugged structure at this scale resembles the rugged structure at the larger scale. (Top photograph from NASA Visible Earth [http://visibleearth.nasa.gov/]. Bottom photograph by Andreas Borchet, licensed under Creative Commons [http://creativecommons.org/licenses/by/3.0/].)
FIGURE 7.3. Other examples of fractal-like structures in nature: A tree, a snowflake (microscopically enlarged), a cluster of galaxies. (Tree photograph from the National Oceanic and Atmospheric Administration Photo Library. Snowflake photograph from [http://www.SnowCrystals.com], courtesy of Kenneth Libbrecht. Galaxy cluster photograph from NASA Space Telescope Science Institute.)
1. Start with a single line.
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2. Apply the Koch curve rule: “For each line segment, replace its middle third by two sides of a triangle, each of length 1 / 3 of the original segment.” Here there is only one line segment; applying the rule to it yields:
3. Apply the Koch curve rule to the resulting figure. Keep doing this forever. For example, here are the results from a second, third, and fourth application of the rule:
This last figure looks a bit like an idealized coastline. (In fact, if you turn the page 90 degrees to the left and squint really hard, it looks just like the west coast of Alaska.) Notice that it has true self-similarity: all of the subshapes, and their subshapes, and so on, have the same shape as the overall curve. If we applied the Koch curve rule an infinite number of times, the figure would be self-similar at an infinite number of scales—a perfect fractal. A real coastline of course does not have true self-similarity. If you look at a small section of the coastline, it does not have exactly the same shape as the entire coastline, but is visually similar in many ways (e.g., curved and rugged). Furthermore, in real-world objects, self-similarity does not go all the way to infinitely small scales. Real-world structures such as coastlines are often called “fractal” as a shorthand, but it is more accurate to call them “fractal-like,” especially if a mathematician is in hearing range.
Fractals wreak havoc with our familiar notion of spatial dimension. A line is one-dimensional, a surface is two-dimensional, and a solid is three-dimensional. What about the Koch curve?
First, let’s look at what exactly dimension means for regular geometric objects such as lines, squares, and cubes.
Start with our familiar line segment. Bisect it (i.e., cut it in half). Then bisect the resulting line segments, continuing at each level to bisect each line segment:
Each level is made up of two half-sized copies of the previous level.
Now start with a square. Bisect each side. Then bisect the sides of the resulting squares, continuing at each level to bisect every side:
Each level is made up of four one-quarter-sized copies of the previous level.
Now, you guessed it, take a cube and bisect all the sides. Keep bisecting the sides of the resulting cubes:
Each level is made up of eight one-eighth-sized copies of the previous level.
This sequence gives a meaning of the term dimension. In general, each level is made up of smaller copies of the previous level, where the number of copies is 2 raised to the power of the dimension (2dimension). For the line, we get 21 = 2 copies at each level; for the square we get 22 = 4 copies at each level, and for the cube we get 23 = 8 copies at each level. Similarly, if you trisect instead of bisect the lengths of the line segments at each level, then each level is made up of 3dimension copies of the previous level. I’ll state this as a general formula:
Create a geometric structure from an original object by repeatedly dividing the length of its sides by a number x. Then each level is