Saturday, 30 April 2011

The Impossible Snowflake

3D Koch Snowflake

Since the 2D Sierpinski Carpet projects nicely into 3D to give the Menger Sponge, and the 2D Sierpinski Triangle similarly up-dimensions to give the Sierpinski Pyramid, it seems obvious to try to get a 3D version of another famous 2D fractal construction, the Koch Snowflake.

A number of people have probably tried this over the years, but I haven't seen anyone manage it. The snag is that the "obvious" solution doesn't work. We're taught that the Koch Snowflake outline is created by assembling triangles, but if we try to use the most obvious triangle-based solid, the tetrahedron, we fail ... starting with a single tetrahedron and adding half-scale copies to each side initially produces a six-pointed profile, but after that it all goes horribly wrong (book, page 20). People have tried offsetting the positions of the daughter pieces to try to keep the shape looking interesting, but it's kinda cheaty.

So the secret to creating this "impossible" solid is not to use a standard approach. "Step One" is to understand that the Koch Snowflake doesn't have to be made out of triangles, it can also be built from hexagons (book, Figure 3.13, page 26), and "Step Two" is to remember that the simplest Platonic Solid with a hexagonal profile is the cube  ... when viewed corner-on.

The rest turns out to be simple. Take a cube, apply a 3×3 grid to each face to divide it up into 27 smaller cubes, and throw away the eight corner-pieces. Then do the same thing for each of the smaller remaining  cubes, and repeat.

The resulting fractal solid (diagrammed as Figure 37 in the book) has a crosslike fractal pattern on each of its six faces, and shows a perfect Koch Snowflake silhouette when viewed from each of the original cube's eight corners.
A 3D Koch Snowflake ... paperweight
I have one of these that I'm using as a paperweight. It's perhaps not the prettiest of fractal solids, but I suppose that it's not bad for something that wasn't really supposed to exist.

1 comment:

  1. There is also another version of the koch snowflake in 3d-