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.
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-similar pieces)
= Log(magnification factor)
A square can be broken into four self-similar pieces that are each half the size of the original square. Using this formula:
A square comes out to be a two-dimensional object, but it doesn’t work out so cleanly with the Sierpinski Triangle. When you divide the triangle into pieces whose sides are half the size of the original, you get three self-similar triangles, so the formula works out:
But how can this be possible? The picture of the triangle is two dimensional. But remember, this is only a depiction of the triangle. The real Sierpinski Triangle cannot be drawn because it is infinitely detailed.
Another interesting aspect of the Sierpinski Triangle is its creation. There are several ways of producing an image of the Sierpinski Triangle. The first and simplest method is by taking an equilateral triangle and connecting the midpoints to create four triangles with sides that are half the size of the original. Disregard the middle triangle and repeat these steps with the newly created three other triangles. The Sierpinski Triangle begins to appear.
Of course, this would only be a...