One of the principal features of mathematical fractal structures is that scaling produces two or more copies of each stage, and this scaling is carried to infinitesimal detail.
In the natural physical world this can happen over only a limited range of scales. The atoms making up a fern do not look like ferns. For that matter, the cells of a fern do not look like ferns, either.
The tetrahedron we build illustrates this limited scaling range in a dramatic way, though with a variation:
This example serves another purpose as well. In one of the first years of our fractals course, all the mathematical fractals happened to be subsets of the plane (more-or-less a consequence of comparatively primitive 3-dimensional graphics software available in our student lab), while all the natural fractals - cauliflower, broccoli, Queen Anne's lace, crumpled paper - were objects inhabiting 3-dimensional space. The final examination included the problem to characterize the differences between natural and mathematical fractals. The solution we expected was, "Mathematical fractals are exactly self-similar on all scales, while natural fractals are approximately self-similar over a limited range of scales." While some students gave approximately that answer, many more wrote, "Mathematical fractals live on pieces of paper or computer screens, natural fractals live in the 3-dimensional world." While true for the examples presented that semester, this is not the point we want to make. The Sierpinski tetrahedron is a physical representation of a mathematical fractal that inhabits 3-dimensional space.
Return to Sierpinski Tetrahedron Project.