Many fractal curves can be generated using L-systems or string-rewrite rules, in which each stage of the curve is generated by replacing each line segment with multiple smaller segments in a particular arrangement. The same technique can be extended to surfaces, where each stage is constructed by replacing each triangle with multiple smaller triangles. This Demonstration shows an analogy of the Koch curve as a three-dimensional surface.
Snapshot 1: creation of the surface begins with a single triangle
Snapshot 2: each successive iteration is created by dividing each triangle into four smaller triangles, raising the midpoint of the middle triangle, and then closing the surface by building a tetrahedron-shaped shell on the middle triangle
Snapshot 3: the bottom face of the surface is a Sierpinski sieve