Pascal's Triangle and Modular Exploration

*Sierpinski Triangle*

### Marianne Parsons

The Sierpinski triangle is a fractal described
in 1915 by Waclaw Sierpinski. It is a self similar structure that
occurs at different levels of iterations, or magnifications. We
can use Geometer's Sketchpad to construct these types of triangles,
and then compare them to the pattern of Pascal's Triangles.

Below, a pattern has begun by finding the *midpoints*
of the line segments of the largest triangle. Then, by connecting
these midpoints smaller triangles have been created. This pattern
is then repeated for the smaller triangles, and essentially has
infinitely many possible iterations. This is called Sierpinski's
triangle.

The Sierpinski triangle generates the same
pattern as mod 2 of Pascal's triangle. That is to say, the even
numbers in Pascal's triangle correspond with the white space in
Sierpinski's triangle. In fact, Pascal's triangle mod 2 can be
viewed as a self similar structure of triangles within triangles,
within triangles, etc.

See how this compares to Pascal's Triangle
in mod 2!

View the GSP construction and
tool for this figure, and adjust the number of iterations
shown.

The triangle below was generated from iterations
by *trisecting* the line segments that make up the
largest triangle. Does this new iteration correlate to the pattern
found in Pascal's triangle mod 3?

See how this compares to Pascal's Triangle
in mod 3!

View the GSP construction
and tool for this figure, and adjust the number of iterations
shown.

This triangle began by dividing the line segments
of the largest triangle into *four* equal parts. Then,
once new triangles were drawn the pattern was repeated. While
this does not correspond to Pascal's triangle mod 4, it is interesting
to compare this triangle to the ones above!

See how this does not correspond to Pascal's
Triangle in mod 4!

View the GSP construction
and tool for this figure, and adjust the number of iterations
shown.

For more information on Sierpinski Triangle,
visit these web sites

Serendip: The Magic Sierpinski Triangle

MathWorld: Sierpinski Sieve

Wikipedia: Sierpinski Triangle

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to Essay 1: Pascal's Triangle and Modular Exploration