Many artists and designers use repeating patterns in their creation. The repetition creates a sense of order and comfort and often makes the design look more appealing. Some artists may take this one step further and create patterns within patterns, with designs repeating themselves inwardly or outwardly on a gradually smaller scale. This kind of design is known as a fractal.

A fractal is a self-similar geometric pattern. Figures that are similar have the same shape, but typically different sizes. If you zoom in on the fractal, you'll see a design the same shape as the original. Then if you zoom in further, you'll see more occurrences of the shape, and so on.

Some fractals are found in nature, like within flower petals, while others are created by performing the same operation to a given shape over and over. For example, think about taking a square, and cutting out four smaller squares, one from each side. The result will resemble the Purina logo, showing a five-square pattern as shown below. Now think about taking each of those squares and doing the same. Then do the same for the resulting smaller squares, and so on.

When you perform these repeated actions, or iterations, you eventually start to see some intricate patterns. The above example is known as the box fractal, and is one of the simplest occurrences of such a figure. There are many famous examples of fractals that have been studied throughout the years.

Novelist Michael Crichton used fractals when writing Jurassic Park. The book opened with an illustration of a geometric pattern, and the first page of every successive chapter added another iteration to the design. The figure began to look more and more complex, symbolizing the ongoing chaos taking place in the story.

This pattern is called the dragon fractal, but has since become known as the "Jurassic Park fractal."

Many fractals, like the box fractal, are closed figures. An unusual thing about closed-figure fractals is that they clearly have a finite area, but as the iterations continue, their perimeters get indefinitely large, seemingly to no end. Mathematician Benoit B. Mandelbrot¹ (1924-2010), who famously studied fractals, once posed the question, "How long is the coastline of Great Britain?" His answer, half-jokingly, was infinity (SOURCE: The New York Times).

The reason he said that was because if you zoom in on the coastline of the island, you'll see a bunch of jagged lines. Then if you zoom in on those, you'll see even more jagged lines comprising them, and so on. Each of those zigzags will make the perimeter longer. Even the tiniest rock, pebble, or grain of sand jutting outward or inward will add more to the perimeter. Therefore, the length of the coastline becomes practically infinite. Obviously, though, the island has a finite area.

¹ This guy was a real cut-up. He added the meaningless middle initial "B." to his name, and when people asked him what it stood for, he replied, "It stands for 'Benoit B. Mandelbrot.'" This implied that the "B." within his middle name also stood for "Benoit B. Mandelbrot," and so on. He literally turned his own name into a fractal.

Below are illustrations of some well-known fractals. Can you figure out the pattern? Sketch them on a piece of paper and draw what you think the next iteration looks like.