Computer rendering of fractal curves and surfaces

Fractals are a class of highly irregular shapes that have myriad counterparts in the real world, such as islands, river networks, turbulence, and snowflakes. Classic fractals include Brownian paths, Cantor sets, and plane-filling curves. Nearly all fractal sets are of fractional dimension and all are nowhere differentiable. Previously published procedures for calculating fractal curves employ shear displacement processes, modified Markov processes, and inverse Fourier transforms. They are either very expensive or very complex and do not easily generalize to surfaces. This paper presents a family of simple methods for generating and displaying a wide class of fractal curves and surfaces. In so doing, it introduces the concept of statistical subdivision in which a geometric entity is split into smaller entities while preserving certain statistical properties.