SCI-FUN's Christmas Card
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The Koch Snowflake
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The Koch snowflake belongs to a more general class of shapes known as fractals. (See below for more information.)
The curve was originally described in a 1906 paper by the Swedish mathematician Niels Fabian Helge von Koch, and was one of a series of curves which horrified nineteenth- and early twentieth-century mathematicians, by being seemingly intractable to analysis (by being everywhere continuous and nowhere differentiable, for example). The curve is self-similar: if we zoom in on it at any point (and to any depth), it retains the same (or at least broadly the same) structure and shape. (This distinguishes it from a simpler curve such as a circle, which looks more and more like a straight line as we zoom in.) |
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This snowflake curve is fairly simple to analyse, and we can investigate one seemingly impossible property below: that it can have an infinitely long perimeter, yet enclose a finite area. First, the perimeter. | |||||||||||||||||||||
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The Perimeter
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So the perimeter of the Koch snowflake can be as large as we want, for any number of iterations. (A shorthand is to say that it "becomes infinite", but that's a little vague.)
But what about its area? |
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The Area
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We see that the area of the Koch snowflake is finite, tending towards 1.6 times the size of the original triangle, while the perimeter becomes infinite!
This is a characteristic of many fractal shapes: they have an infinite perimeter, but enclose a finite area. |
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Biological Fractals | |||||||||||||||||||||
An interesting link can be found at Fractals in Physiology. Fractal shapes appear in many other living forms; plant growth in particular seems to be dominated by such rule sets. An interesting set of links on fractals can be found in its (incomplete) Wikipedia entry. (Wikipedia should often be taken with a pinch of salt, but in this case there's some useful commentary.) |
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Go back to the snowflakes page
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Consider an infinite sum of terms, each of which is a multiple of the previous one, as follows: |
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Go back to the snowflakes page
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