New Dimensions

Another intriguing aspect that fractals brought to the world is that the dimensions most commonly used in mathematics (1D, 2D, 3D, 4D, etc.) are not discrete but instead are continuous. Meaning there are dimensions between the first dimension and the second, the second and the third, and so on. This idea was considered and modeled when Mandelbrot first defined fractals.

In a video by 3Blue1Brown (which is shown below), you can see this idea presented. They show examples of several objects in different dimensions (1, 2, and 3). Then scaling these objects down by a factor of two (thus the scaled object was one-half the starting scale of the original object). In doing this, they found that whatever the scale factor was, it was taken to the nth power for whatever nth dimension the object was in.

They go on further and observe what this looks like for a fractal. For their evample, the Sierpinski Triangle was used, see Figure fore. The Sierpinski Triangle, from first look, is three identical triangle placed together. When scaled by one-half, the resulting image was one of those three small triangle within the original Sierpinski Triangle. This meant that the scaled down triangle was one-thrid the size of the original. Moreover, this example did not follow what would be observed if the fractal was in the first or second dimensions. In fact, it posed the question of what dimension the Sierpinski Triangle was in.

Following this, an equation was set up. It was that \[\frac{1}{2}^n=\frac{1}{3}\] Using logarithmic properties, the dimension was found to be roughly 1.585 or more precisely, the \[log_3(2)\] It is important to note that not every fractal has the same dimension because they each have a different scale of roughness.

Aplying this back to Mandelbrot's original question of how long the Coast of Brittany was related to its dimension. It was found that the Coast of Brittany had a dimension of 1.2 and the Coast of Norway was in the 1.5 dimension. Watch the video below to dig deeper!

Use the following links to explore more of the mathematics of Snowflakes and Fractals!

Follow this link to see how Fractal Geometry impacts our world today.