Sierpinski Triangle

The Sierpinski Triangle is a fractal made of equilateral triangles. The midpoints are connected to make a new, smaller triangle inside of the larger one and that new triangle is then removed from that generation, and the process continues in the remaining triangles. The removed triangle(s) will always be pointing down while the rest will be pointing up. The number of triangles pointing up will increase by a factor of three in each new generation.

Fractals are self-similar, the parts are the same as the whole. As you zoom in on a fractal and look at a smaller piece of it, it will have the same shape and pattern as the whole and as every other part. A simple process, repeated over and over again, makes a complex fractal! Common fractal patterns found in nature include branches, ferns, Romanesco broccoli, and snowflakes.

The Sierpinski Triangle is named for Polish mathematician Wacław Sierpiński who first described the fractal in 1915. But the design was used much earlier and can be found in cathedrals in Italy dating to the 13th century.

Make one at home

1. Print the black line PDF of a triangle or draw a large equilateral triangle—a triangle with all sides the same length
2. Find the midpoint of each side of the triangle and mark with a dot.
3. Use a ruler to connect the three midpoints, making a new equilateral triangle in the middle and upside down inside the first one. Color this triangle.
4. This should give four triangles—three pointing up and one pointing down. Repeat the process of finding the midpoints and connecting them for each triangle that points up.
5. Continue repeating the process (coloring only the triangles that point down) until you run out of space.
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