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Sierpinski's Gasket
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Sierpinski's Gasket
Click on the graphic above to generate a new image.

Sierpinski's Gasket is an example of a fractal. A fractal is an object that is self-similar across multiple scales. If you were to zoom in on an area of the image above, it would not look considerably different. Fractals have amazing properties: some fractals are infinite in area, and some are infinite in perimeter. This isn't necessarily exceptional, until you realize that fractals are finitely bounded. What this means is you can have a circle of known area and known perimeter that contains a fractal of infinite area and/or infinite perimeter inside itself! This is possible because fractals are not entirely one-dimensional, but rather they vary between 1 and 2 dimensions.

Sierpinski's Gasket is especially cool because it produces a pattern familiar to some students of algebra. Remember polynomial expansion from school?

(1 + x)0 = 1
(1 + x)1 = 1 + x
(1 + x)2 = 1 + 2x + x2
(1 + x)3 = 1 + 3x + 3x2 + x3
(1 + x)4 = 1 + 4x + 6x2 + 4x3 + x4
...
(1 ± x)n = 1 ± nx⁄1! + n(n - 1)x2⁄2! + ... (x2 < 1)

If you disregard the x's, and focus on the coefficients, you can create a diagram known as Pascal's Triangle.

                1
             1    1
           1    2    1
        1    3    3    1
      1    4    6    4    1
    1   5    10   10   5    1
  1   6    15   20   15   6   1
1   7   21   35   35   21   7   1

Pascal's triangle is interesting for a number of reasons, one of which involves combinatorial mathematics. How many ways are there to choose two objects from a collection of five unique objects? Let's say the objects are the letters A,B, C, D, and E. The possible combinations of two of these letters (where order does not matter) are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. There are 10 total combinations. Did you notice that by counting five down and two over in Pascal's Triangle you find a 10?

What do you get if you color the odd numbers in Pascal's Triangle black?

                1
             1    1
           1    2    1
        1    3    3    1
      1    4    6    4    1
    1   5    10   10   5    1
  1   6    15   20   15   6   1
1   7   21   35   35   21   7   1

A little hard to see in the text, so I'll draw it with symbols:

                •
             •    •
           •         •
        •    •    •    •
      •                   •
    •   •              •    •
  •        •         •        •
•   •   •    •    •    •    •   •

Yes, there it is again! At least to me, that's unbelievable. Using a very simple pattern you can create Sierpinski's Triangle, which relates to some fundmental sets of numbers in mathematics.

If you're interested in this kind of thing, a neat book (and a quick read) is Introducing Fractal Geometry by Nigel Lesmoir-Gordon, Will Rood and Ralph Edney.

If you're interested in the source code that generates the fractal, it's here. The applet source code is here.

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