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
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Using this formula:

Log(4)

= Log(2)

Log(22)

= Log(2)

2Log(2)

= Log(2)

= 2

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:

Log(3)

= Log(2)

≈ 1.58496

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. 3

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 true Sierpinski Triangle if you were to repeat these steps an infinite number of times. So, realistically, this method only creates an image that resembles the Sierpinski Triangle.

The next method of Sierpinski Triangle creation is by using the Chaos Game. The Chaos Game is a method that uses randomly picked points to build an image of the triangle. To

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