Mandelbrot calculated that fractals have a fractional dimension between 1 and 2. By studying fractals, mathematicians have a whole new geometry for describing the universe, beyond the boundaries of Euclidean geometry. But very little in nature is so regular. Just as with algebra, where we continued to bang our heads against the need to understand and solve nonlinear equations in order to model most universal systems, we continue to bang our heads against the need to understand irregular shapes using a geometry that could accommodate the complexities of these shapes. Fractals mirror these irregular shapes, thereby allowing us to study and understand nature by understanding fractals.
Fractals are the place where math, science and art come together.
They allow us to view the complexity of chaos and order. They please us visually and excite our imaginations. That is why there is an abundance of sites on fractals. I've only provided the smallest list of fractal galleries. However, I suggest you start with the Math Forum's listing on fractals.