Fractals keep detail no matter how far you zoom
A fractal is a shape with intricate structure at every scale, however closely you look, and its fractal dimension usually exceeds its ordinary one. Benoît Mandelbrot coined the word in 1975 from the Latin frāctus, broken, and offered a formal definition in 1982.
Scaling shows the difference. Double the sides of a filled polygon and its area quadruples, two to the power two; double a ball's radius and its volume grows eightfold, two cubed. Double every length in a fractal and its content grows by a power that need not be a whole number and usually exceeds the ordinary dimension. The Koch curve splits into four copies each a third the size, so its dimension is the number D with 3 to the D equal to 4, about 1.26.
Many fractals are nowhere differentiable. Try to measure the Koch snowflake with ever shorter rulers and the jagged pattern keeps reappearing, pulling in more tape each time, so its perimeter is infinite. A fractal curve is still one-dimensional topologically, yet it fills space more thoroughly than an ordinary line. Shapes such as the Menger sponge, whose copies match perfectly at each level, earn the label affine self-similar.
The ideas trace back to seventeenth-century thinking about recursion, then to continuous but non-differentiable functions studied in the nineteenth century by Bolzano, Riemann and Weierstrass, before computers fuelled a twentieth-century boom. Mathematicians still disagree on a formal definition. Mandelbrot's 1982 version required the Hausdorff–Besicovitch dimension to exceed the topological one; he later loosened it to a rough shape made of smaller near-copies of itself, and finally suggested using the word without a pedantic definition.
Fractals can describe processes in time as well as shapes, and they appear throughout chaos theory as strange attractors and the boundaries between basins of attraction. The public tends to know fractal art better than the mathematics behind it.
Source: Fractal
