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# Self-Similarity

## One part of an object can be enlarged or shrunk to look like the whole object.

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Fractals Everywhere

Credit: Jon Sullivan
Source: http://en.wikipedia.org/wiki/File:Fractal_Broccoli.jpg

Geometric patterns can be beautiful and fun to look at, but why do we study them in school? It turns out that fractals are all around us! From vegetables to weather formations, these intricate yet simple patterns can be found in natural structures almost everywhere.

#### Why It Matters

In the 1970s, mathematician Benoit Mandelbrot introduced the new field of fractal geometry. He discovered that natural formations such as clouds, plants, mountain ranges, and even arteries in the human body follow a mathematical pattern that is not at all random but actually very organized. Fractal geometry produces patterns using self-similarity. This means that parts of the pattern can be enlarged or shrunk and still look like the original pattern! Think about the parts of a tree from its trunk, to its many branches, to even more twigs with many leaves. With infinite iterations, or repetitions, the pattern can continue forever with each small part resembling the pattern you started with.

Credit: Solkoll
Source: http://commons.wikimedia.org/wiki/File:Dragon_trees.jpg

Mandelbrot's discovery of a mathematical "recipe" for natural structures was exciting for computer graphic artists. Fractal patterns with triangular iterations are used to create digital images of mountain ranges and other natural formations, such as the computer-generated trees above. In 1982, they were used for the first time on the big screen to create an entire planet landscape in Star Trek II: The Wrath of Khan. The study of fractal geometry has helped us understand structures that at first seem chaotic, such as fern leaves or a network of millions of air sacs in human lungs. These structures, when studied from a geometric approach, begin to appear very organized—organized enough for humans to recreate the natural patterns on a computer!

#### Explore More

Learn more about the importance of fractals in man-made designs in the following video. What did Ron Eglash learn about African cultures using his knowledge of fractal geometry?

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