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© Copyright 2002, Jim Loy
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Above left we see the first four orders of the Koch curve (drawn using Fractint and Paint Shop Pro), discovered by Helge von Koch. Sometimes, a straight line segment is called the first order. And then the four images above left are the next four orders. You can probably see how each order is built from the previous one. Above right we see the third order Koch island (or snowflake), made up of three Koch curves. Below, is the fifth order Koch curve, magnified four times.
The sixth order Koch curve (below) looks much like the fifth order, except that each tiny point is indistinct. It's hard to tell what is going on. Actually it is made up of many tinier points. But the resolution of the graphic image is inadequate to show points that small. The actual Koch curve (and island) is the limit of infinitely many orders. It looks like the picture below, again with inadequate resolution.
You may have noticed that the Koch curve is very self-similar (see Fractals and Self-Similarity). Various parts of it (the infinite order version) are identical to larger and smaller parts. So, each point that you see in the fifth order curve becomes a very convoluted portion of the curve in higher orders.
How long is the Koch curve? Well, it is easy to see that each order is 4/3 times longer than the previous order. After infinitely many orders, the length would then be infinite. This relates to Mandelbrot's question, "How long is the coast of Britain?" Britain is essentially a fractal, much like a Koch island. And the length of its coastline varies, depending upon the size of the ruler used. The smaller the ruler, the longer the coastline. With the Koch curve, the ruler gets smaller and smaller, as we get to higher and higher orders. And the length of the curve grows without bound.
Benoit Mandelbrot is often considered the inventor of fractals. He coined the term "fractal," and essentially originated the field of fractals as a useful branch of mathematics. But many people created fractals, long before Mandelbrot. The Koch curve is one of them. In fact, Euclid's use of polygons with more and more sides, to approach a circle, has a lot in common with fractals.
Zooming in on the Koch Curve: I made
this movie using Geometer's Sketchpad and
Jasc's Animation Shop.