I’ve written a lot about geometry and topology on this blog, and (given my plans for the future) I don’t intend to stop. I got a specific request this week to write about fractal geometry, and appropriately enough, last week also marked the birthday of Helge von Koch (born January 25, 1870), the Swedish mathematician who did early work in fractal geometry, long before the field had its modern name.
A word we throw around a lot in science and math is “dimension”: a line is one-dimensional, a square is two-dimensional, a cube is three-dimensional, and so forth. But is it that simple? What does “dimension” really mean, and are things always so clear? Mathematician (and egotist) Benoit Mandelbrot posed a simple-sounding question: How long is the coastline of Great Britain? The answer: it depends on the resolution of your map or surveying tools! Real coastlines wiggle, the leaves of plants are jagged, smooth-looking mountains are craggy—if you examine each one closely enough, and the complexity only increases with magnification. In other words, something that may appear one-dimensional (the perimeter dividing sea from land) may fill space like a two-dimensional object. That’s the heart of a fractal: something whose dimension is not an integer number, but rather a fraction.
As with many other topics in math and science, fractals are easiest to introduce using examples, so let’s start with von Koch’s “snowflake”. Begin with an equilateral triangle—a triangle where all three sides are the same length. For simplicity, say each side has a length of 1, though it doesn’t matter what units “1” has (maybe meters, centimeters, inches, feet, leagues (I’ve been rereading Tolkien), or whatever). On each side, cut the middle third and swing it out, as shown in the picture on the right. Then add a new segment to make the whole shape into a star of David. (Actually, there are several ways to think about constructing a von Koch snowflake, but I like this one: it makes it a little easier to visualize and to calculate the length of the sides.) The new shape has a longer perimeter and larger area.
For an ideal von Koch snowflake in the mathematical sense, you would repeat this process forever: cut the middle third from each segment, swing it out like a gate, add a strut to make a new triangular protrusion, and so on. The figure below shows the first few steps in the process.
(See also this animated .gif of the process.) Of course, even with a computer and a high-resolution monitor, you’ll never be able to display an ideal von Koch snowflake, but we can still understand what its properties are. Each new addition makes the perimeter and the area of the shape increase, but already we can see something interesting going on: while the perimeter keeps growing and growing, the whole snowflake is still contained in more or less the same space. In other words, the area enclosed by the perimeter is finite…but the line bounding it is infinitely long! You can see this in the image to the right: each additional line makes the perimeter longer, so adding an infinite number of segments eventually makes the boundary infinite. However, no matter how many segments you add, the snowflake will never extend past the circle: it will always have finite area.
The weird combination of finite area with infinite perimeter is what makes the von Koch snowflake a fractal. It’s neither one-dimensional nor two-dimensional in the usual sense, but lies somewhere in between: its fractal dimension is 1.26. The method for calculating fractal dimension is beyond what I want to cover in this post, but people have estimated values for real-world examples, such as coastlines, fern fronds, and so forth.
You can make other similar shapes: try the Peano curve, the Sierpinski triangle, the Menger sponge, and the Cantor dust. One thing all these shapes have in common is self-similarity: if you zoom in on one part of the fractal, it looks as complex as the whole thing. The coastline of an island remains jagged whether you’re looking at it from the air, at ground level, or through a magnifying glass. Coastlines and other real fractals may not have exactly the same patterns on each level of magnification, but the complexity at the heart of self-similarity remains.
A von Koch snowflake is a mathematical toy, but as the Mandelbrot coastline example shows, real fractals do exist. In fact, you can make one right now: take an ordinary piece of printer paper, which is two-dimensional with a well-known area: 93.5 square inches if you live in the United States. (The paper has thickness, so it isn’t properly two-dimensional , but that’s not particularly relevant here.) Now wad the paper up into a ball, trying as best as you can to make it spherical. However, it’s not actually a sphere or even a true three-dimensional object—it’s a two-dimensional surface curling back on itself. Its fractal dimension ends up being about 2.5: somewhere between a true sphere and the original flat piece of paper.
An upcoming guest post will look at fractals created using viscous fluids, and given that it’s me, no doubt I will revisit this subject again. The beauty of geometry goes beyond the abstract, and fractals are perhaps the most aesthetically pleasing of all geometric shapes.