Pascal's Fractals

55. Triangle Tribulations

Pascal's Fractals

One of the simplest geometric patterns in Pascal's triangle (see "Pascal's Patterns") turns out to be an example of one of the more important geometric shapes in modern mathematics: a fractal. In a fractal, each part is made up of scaled-down versions of the whole shape.

When you shade the even numbers (multiples of 2) in Pascal's triangle, the resulting design resembles a special type of fractal called a Sierpinski triangle. This fractal consists of triangles within triangles in a pattern such that smaller triangles contain the same pattern as the larger triangles.

TRY IT!

Draw a Sierpinski triangle.

You will need:

• pencil and paper
• ruler
• protractor

What to do:

1. Using your ruler, draw a horizontal line across the page, a few inches from the bottom.
2. Use your protractor to draw a 60 degree angle from each end of your horizontal line pointing toward the middle. Extend the angle rays to form an equilateral triangle.
3. Using your ruler, find and mark the midpoint of each side of the triangle.
4. Connect the three midpoints to form a new set of triangles. Shade the center (upside-down) triangle.
5. For each of the three unshaded triangles, mark the midpoint of each side.
6. Repeat steps 4 and 5 until your triangles get too small to divide.
7. Compare your result with the pattern you got from shading the even numbers in Pascal's triangle (see "Pascal's Patterns").

First two stages in creating a Sierpinski triangle.

See also "Fractals in Pascal's Triangle."

Answers:

The first four stages in creating a Sierpinski triangle.

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