Fractals…like onions, they have layers

A fractal is a geometric construction that is self-similar at different scales.  In other words, a fractal shape will look almost, or even exactly, the same no matter what size it is viewed at.  If you search for fractals, you will see examples in art, nature, even in vegetables.

It remains, however, an unintuitive concept. But let us look at a famous example of a fractal: the Sierpinski Triangle.

There are various ways to construct the Sierpinski Triangle. The first step in the geometric construction, for example, involves splitting a triangle up into four other triangles and removing the middle one.  Then split each sub-triangle into four more triangles and remove each of the middle ones, and so on.

The area of a Sierpinski triangle is zero because the area remaining after each iteration is clearly 3/4 of the area from the previous iteration, and an infinite number of iterations results in zero!

When we look at the finished Sierpinski Triangle, we can zoom in on any of these three sub-triangles, and it will look exactly like the entire Sierpinski Triangle itself.  In fact, we can zoom in to any depth we would like, and always find an exact replica of the Sierpinski Triangle.

What is the significance of fractals? They have numerous implications in mathematics and topography, as well as in computer graphics applications. But mostly they serve as a conversation starter at cocktail parties!

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Pamela Hellig