FRACTALSOriginal 2D and 3D fractal artbyTommy Von“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 |