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Infinite Detail

Let’s start with a game.

Rules:

Play for a while! What shape emerges?

Mini-Me

The image that would emerge from the game that started this chapter, if you played the game on the previous page long enough, is the image in Figure 4.1 (a). Note, you’d have to play long and accurately enough to see this image emerge. Creating the image point by point can be difficult and, indeed, time consuming. This image is what is called a fractal and known as Sierpinski’s triangle.

Let’s look at an important feature of this triangle. Notice how the image contains 3 copies of the larger image. There is one at the top and two along the bottom. This can be more readily seen when each copy of the larger image is colored differently as seen in Figure 4.1 (b). Magnifying an object and seeing similarities to the whole is an important property of fractals. An object with self-similarity has the property of looking the same as or similar to itself under increasing magnification.

Broccoli exhibits properties of self-similarity. That is, smaller stalks look like the larger stalks of broccoli—albeit at a different scale. In what ways does a piece of broccoli exhibit this property? Will it exhibit such a property under any arbitrary amount of magnification? How about Serpinski’s triangle? Does it exhibit properties of self-similarity? Will it under any arbitrary amount of magnification? Sierpinski’s triangle is created after an infinite number of loops of our previous game. As such, we could find copies of Sierpinski’s triangle within the shape under any magnification.

Now, with a starting image and a photocopier or graphics program, let’s create Sierpinski’s triangle. We’ll start with a picture of pop singer Beyoncé as seen in Figure 4.3.

To create the image, we create the following loop. To begin, let the current picture be the image of Beyoncé seen on the left in Figure 4.3.

The image in Figure 4.4 is what we get after one loop of these steps. This doesn’t look much like Sierpinski’s triangle. So, we perform the loop again.

Now, we get the image on the left in Figure 4.5 and another pass through the loop produces the image on the right.

It’s a matter of taste as to when to stop. Two more loops through our steps produced the images in Figure 4.6. How’d we do turning Beyoncé into Sierpinski’s triangle? Keep in mind that the first loop used 3 copies of the starting image. The second loop used 9 = 3². The third loop used 3³ = 27. So, the image on the right in Figure 4.6 used 3⁵ = 273 copies of Beyoncé. After 10 iterations, we have 59,049 copies of the pop singer.

You will get the same results with any image. Select your favorite ones and give it a try.

Dicey Island

In this section, we extend the idea of fractals to create the landscape of a distant planet. First, let’s create another fractal by performing the following steps with a pencil and paper:

Let’s perform the first iteration of these three steps together. Again, we begin with a straight line as seen below

Then, we divide the line segment into three segments of equal length, draw an outward pointing equilateral triangle that has the middle segment as its base, and remove the line segment that is the base of the triangle. This produces the shape below containing 4 line segments.

Continuing in this way produces the images in Figure 4.7 (a) and then (b). If we iterate infinitely many times, we produce the fractal called Koch’s curve as seen in Figure 4.7 (c). An infinite number of steps isn’t visually necessary since it becomes difficult for most (if not all) of us to produce very many steps by hand. So, we have a computer perform finitely many iterates and plot the resulting curve. Even with a computer, there comes a point when successive iterates no longer produce distinguishable changes in the resulting image. Such detail can only be seen through zooming into the image.

Let’s explore this idea to create a coastline. Filling the upper and lower regions with blue and green creates the image in Figure 4.7 (d). There is a lot of structure. To create something more organic, let’s add randomness.

We will start with a square located at (0, 0), (16, 0), (16, 16) and (0, 16) as seen in Figure 4.8.

Now, we follow these steps:

2.  Roll a die and if you roll

3.  Roll the die again and if you roll

4.  Repeat steps 2 and 3 to find dy.

5.  Your new midpoint will be (x_m + dx, y_m + dy)

For instance, I produce the following:

Therefore, my square becomes the image in Figure 4.9 (a). Repeating the process but now for a roll of 1 to 3 in step 2 dx = 1 and a roll of 4 to 6 dx = 2, I got the image in Figure 4.9 (b).

Such shapes are fractal coastlines. Movies use these ideas to create the landscapes of distant planets. The images are a bit more realistic if we use a computer rather than a die and at stage k (where the square is stage 0), let dx and dy be a random number r chosen such that:

Two islands created with this approach are seen in Figure 4.10 (a) and (b).

Finally, I can take such an image and essentially wrap it around a sphere, as seen in Figure 4.11. This creates the fractal landscape for our distant planet. You can make your own landscapes with just a ruler, a die and some graph paper.

Fractal landscapes can also be extended to 3D, creating the topography of distant planets. To tour through a gallery produced by a leader in the field of fractal landscapes, search the internet for fractal images by Ken Musgrave. An example of his work appears in Figure 4.12 (a). Figure 4.12 (b) contains an image produced at a later time after fractal landscape technology had advanced.