Fractal Geometry
“In mathematics, fractals are infinitely complicated abstract objects used to describe and simulate naturally occurring objects. Fractals commonly exhibit similar patterns at increasingly small scales, also known as expanding symmetry or evolving symmetry. If this replication is exactly the same at every scale, as in the Menger sponge, it is called a self-similar pattern. Fractals can also be nearly the same at different levels, as illustrated here in small magnifications of the Mandelbrot set.
One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.”
- Wikipedia