# The Sierpinski Triangle

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The Sierpinski Triangle

Deep within the realm of fractal math lies a fascinating triangle filled with unique properties and intriguing patterns. This is the Sierpinski Triangle, a fractal of triangles with an area of zero and an infinitely long perimeter. There are many ways to create this triangle and many areas of study in which it appears.

Named after the Polish mathematician, Waclaw Sierpinski, the Sierpinski Triangle has been the topic of much study since Sierpinski first discovered it in the early twentieth century. Although it appears simple, the Sierpinski Triangle is actually a complex and intriguing fractal. Fractals have been studied since 1905, when the Mandelbrot Set was discovered, and since then have been used in many ways. One important aspect of fractals is their self-similarity, the idea that if you zoom in on any patch of the fractal, you will see an image that is similar to the original. Because of this, fractals are infinitely detailed and have many interesting properties. Fractals also have a practical use: they can be used to measure the length of coastlines. Because fractals are broken into infinitely small, similar pieces, they prove useful when measuring the length of irregularly shaped objects. Fractals also make beautiful art.

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The Sierpinski Triangle holds many secrets that have yet to be discovered, and I am intrigued by the triangle’s apparent simplicity that hides all of its unique properties. One of the most interesting properties of the Sierpinski Triangle is the Fractal Dimension. Although it appears to be two-dimensional on paper, the triangle is actually about 1.58 dimensional. As stated on the Chaos in the Classroom website, the formula for determining dimension is:

Log(number of self...

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...t. The Chaos Game can be applied to create other fractals and shapes, and is a major part of an entirely separate area of study: chaos theory. The fact that the Sierpinski Triangle transcends the boundaries of fractal and number theory proves that it is an important part of mathematics. Perhaps the Sierpinski Triangle still holds secrets that, if discovered, will change the way we think about mathematics forever.

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Bibliography

Crilly A.J., R.A. Earnshaw, H. Jones. Fractals and Chaos. New York: Springer-Verlag, 1991.

Devaney, Robert L. “Fractal Dimension”. 2 April, 1995. Boston University. 28 June, 2005 .

"Mandelbrot Set." Wikipedia: The Free Encyclopedia. 3 Aug 2005. 10 Aug 2004 .

Tricot Claude. Curves and Fractal Dimension. New York: Springer-Verlag, 1995.