One of Occupy Math’s readers said that a friend of hers, who has seen Occupy Math’s fractals both in the blog and on Facebook, asked her what a fractal is. She tried to explain, but found it a bit difficult. This is because it is a bit difficult to explain, but Occupy Math will take a shot at it in today’s post. The actual answer is simple.
A fractal is an object with a fractional number of dimensions.
In other words the number of dimensions of a fractal can be more than one but less than two. Since that sounds completely absurd, lets work through an example. An initiator-generator fractal starts with a shape, called the initiator, and then uses a rule for replacing lines with other lines, called the generator. The Koch snowflake starts with a triangle as its initiator. Its generator works like this.
What it does is to split the line being replaced into three equal pieces and then replace the middle piece with two equal pieces that are a 60-degree angle to its original direction. Every line is replaced in this fashion in each step of generating the Koch snowflake. Here are pictures of the initiator and the first five “replace everything” steps. The actual Koch snowflake is the result of doing replacement steps an infinite number of times.
Each time we do a replacement step, each line in the picture is replaced with four lines that are, together, one-third longer than the line being replaced. This means that the perimeter of the snowflake is multiplied by 4/3 with each step. If we keep doing this forever, the perimeter grows without limit and becomes infinite. On the other hand, it would not be hard, based on the pictures, to see that the figure never grows outside of the circle that touches the points of the six-pointed star that is the first step after the initiator.
This means the Koch snowflake has infinite perimeter and finite area.
This is something that is impossible for a normal figure from plane geometry, and so the Koch snowflake is paradoxical! It took a while, but in 1918 Felix Hausdorff came up with a quantitative method of computing the dimension of objects called, appropriately, the Hausdorff dimension. His idea for computing dimension resolves the paradox posed by the Koch snowflake. Its boundary is so twisty, after an infinite number of line replacements, that the dimension of the boundary is no longer 1 (which is the dimension of any line or smooth curve). Instead the dimension is Log(4)/Log(3) or about 1.26. It is just a little more than one dimensional. The idea of fractional dimensions was forced on mathematics by the need to resolve paradoxes that appeared when we tried to calculate what would happen after an infinite number of replacements by things like the Koch generator.
Initiator-generator fractals are just one of many types of fractals. Generalized Sierpinski fractals are another interesting type. The starting point for understanding these is the Sierpinski triangle which Occupy Math has carefully generated and colored in the following fashion.
This fractal is really easy to generate. Starting at one corner of the triangle you do the following step over and over. Pick any corner, move half way there, and plot a point. You always move from the last point you plotted. This is a randomized algorithm (because of the random selection of the next corner to move toward) but it fills in the fractal triangle pretty quickly. Occupy Math assigned the colors red, blue, and green to the corners and changed the plotting color by averaging the current color with the color of the corner chosen. This is what causes the shading and lets you see, a little more clearly, how the triangle is generated. The Hausdorff dimension of this fractal is log(3)/log(2) or about 1.585 so its more than half-way to being two dimensional.
Occupy Math has a bad case of “hacker” and immediately wondered what would happen if we used a square the same way the triangle was used. The result was a little sad; the square fills in to make a simple 2-dimensional shape. The fourth corner was assigned “yellow” to let us see how it fills in with color.
The original algorithm for the triangle and the square both let you pick any corner to be next. What happens if we limit which square can be next? This diagram shows one possible restriction.
If you picked a corner, your next corner can be anything except the square clockwise from you. If these restrictions are imposed the fractal that arises is the one shown below. Not a two-dimensional object.
Here are some more restricted square fractals.
The algorithm to make these fractals retains the step “move half-way toward the corner you picked”. There is no particular reason to move half way. Here is an animation where we move from 1/20th of the way to almost all the way one for the “Menu Border” fractal above. As we do this it goes from zero dimensional (just points) to two dimensional (the blob). This means that by choosing the right “move toward” distance we can get any dimension from 0 to 2. Cool, no?
Occupy Math makes pretty pictures with fractals, but it may be worth noting they occur in nature and engineering. Your blood vessels are not only fractal but a fractal that damps out the powerful pulse of your heart to prevent damage. Your lungs are a fractal that maximizes gas exchange. Fractal anodes (still in the lab) are the key to cell-phone batteries that recharge in a few minutes. Clouds, trees, and leaf-veins are all well modeled by fractals.
To repeat the answer at the beginning of the post – fractals are objects with fractional dimensions. Occupy Math has gathered many of them together in his fractal taxonomy. If you have encountered cool fractals, let Occupy Math know! Comment or tweet pointers to or names of these cool fractals.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics