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:
- Using your ruler, draw a horizontal line across the page, a few inches from the bottom.
- 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.
- Using your ruler, find and mark the midpoint of each side of the triangle.
- Connect the three midpoints to form a new set of triangles. Shade the center (upside-down) triangle.
- For each of the three unshaded triangles, mark the midpoint of each side.
- Repeat steps 4 and 5 until your triangles get too small to divide.
- 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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