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Fractals and Dimensions

Picture of a fractal and dimension
Fractals and Dimensions

Fractals are some of the most beautiful and most bizarre geometric shapes. They look the same at various different scales – you can take a small extract of the shape and it looks the same as the entire shape. This curious property is called self-similarity.
To create a fractal, you can start with a simple pattern and repeat it at smaller scales, again and again, forever. In real life, of course, it is impossible to draw fractals with “infinitely small” patterns. However we can draw shapes which look just like fractals. Using mathematics, we can think about the properties a real fractal would have – and these are very surprising.

The name “fractals” is derived from the fact that fractals don’t have a whole number dimension – they have a fractional dimension. Initially this may seem impossible – what do you mean by a dimension like 2.5 – but it becomes clear when we compare fractals with other shapes.

Fractals are very popular in mathematical visualisation, because they look very beautiful even though they can be created using simple patterns like the ones above. You can zoom into a fractal, and the patterns and shapes will continue repeating, forever.

Mandelbrot and Nature
“Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.”(Mandelbrot, 1983).


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