Sierpinski triangle

A Sierpinski triangle or Sierpinski gasket, named after Wacław Sierpiński, is one of the most famous fractals, meaning smaller details look exactly like the full thing.

Constructions

To make one, start with a triangle. Remove the center part, or copy 3 times and arrange those copies in the shape of a triangle, either way works. If you keep repeating this, you get a shape that looks like this:

The Chaos Game

• Make a triangle.
• Choose any point in the triangle.
• Draw a line between that point and any corner of the triangle.
• Find the midpoint of the line.
• Draw a line from that point and repeat the process.

The process should look like this:

Pascal's Triangle

Pascal's Triangle is a special arrangement of numbers where every number is the sum of the ones above it.

The first 11 rows of Pascal's Triangle.

If you color in every even number (2, 4, 6, 8, 10, 12, 14, etc) in pink and every odd number (1, 3, 5, 7, 9, 11, 13, etc) in red, the result should look like the Sierpinski triangle.

Sierpinski Triangle Media

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  Sierpinskitriangle

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  Generated using a random algorithm

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  Sierpiński triangle in logic: The first 16 conjunctions of lexicographically ordered arguments. The columns interpreted as binary numbers give 1, 3, 5, 15, 17, 51... (sequence [{{fullurl:OEIS:{{{id}}}}} {{{id}}}] in OEIS)

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  Sierpinski triangle evolution

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  Animated construction. Click to enlarge.

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  Sierpinski triangle evolution square

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  Animated creation of a Sierpiński triangle using the chaos game

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  Arrowhead construction of the Sierpiński gasket

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  An approximation to a Sierpiński triangle obtained by shading the first 25 (32) levels of a Pascal's triangle white if the binomial coefficient is even and black otherwise

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  Sierpiński pyramid recursion (8 steps)