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The Sierpinski Triangle is a famous fractal that forms a geometric pattern as the midpoints of the sides of a triangle are connected, creating four more triangles the sides of which are then connected, and so on.
 

Try this Java demonstration by Jacobo Bulaevsky, part of Cynthia Lanius' fractals lessons site.

If you color the odd numbers in Pascal's triangle and shade out all the other spaces, you will see Sierpinski's triangle.

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References

1. Cynthia Lanius: The Sierpinski Triangle, Sierpinski Meets Pascal
2. Eric's Treasure Trove of Mathematics: Sierpinski Sieve
3. Jeremy Baer: Explore Patterns in Pascal's Triangle

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