Koch Snowflake Arrays

Koch Snowflake Fractal Arrays (an alternate Koch Snowflake construction method)

Another couple of non-standard ways of creating Koch Snowflake outlines. These two approaches use a ring of six corner-touching hexagons, with or without an additional seventh central copy. There's an example of the first version appearing as a fractal solid's silhouette in figure 9.5 of the book (as the tiny lower-right image). The book also has face-aligned hexagon-based Koch Snowflakes in figures 3-13 and 7-5.


Note that every smaller detail is also a Koch Snowflake, and every little remaining space between the snowflakes also progressively gets nibbled away to form yet more Koch snowflakes as you apply more iterations.

The more conventional Koch Snowflake approach gives a self-similar fractal outline, but its components are triangles. In these two cases, we go one step better – we generate the same fractal outline as before, but now the whole thing is fractal, including the interior. In theory, it's a Koch Snowflake built from nothing but other Koch Snowflakes. The shape becomes its own building-block, and its own template ... no fundamental shape exists in its self-contained geometrical universe but itself.

For more examples of Koch Curve tilings, see figures 30.4 and 31.2 in the book.

The Baird Delta

The "Delta" is is one of my new favoritest fractals. It was difficult to discover because it didin't have any obvious 3D siblings. It kinda seems to be a one-off. I haven't found any record of it being documented before, I used it on page 161 of the book (anf referred to it as just the "Delta" fractal), but it really needs a more distinctive and search-engine-friendly name than just "Delta" ... so now, unless anyone can find evidence of prior work, I think I'm going to start referring to it as the Baird Delta. :)

The shape's building-block is a solid with six identical triangular faces. You can then cut away material to produce three smaller rotated copies of the original (and repeat), or stack three of the rotated blocks together to produce a larger copy. Similar things happen with the Sierpinski Pyramid, but this is a leetle bit more subtle in that the component blocks (and the main shape) aren't quite regular polyhedra, they have to be twisted perpendicularly at each iteration, and the shape doesn't seem to have an obvious two-dimensional fractal counterpart. This makes the shape more difficult to visualise and more difficult to stumble across. The first time that you see the shape, it probably takes a bit of squinting before you realise that the three corner-pieces are identical smaller angled copies of the whole thing, rotated from the original baseline by 90 degrees, and rotated with respect to their siblings by 120 degrees.

As  you iterate, the area of each face gets progressively nibbled away until it's effectively a Koch Curve (although its a differently-angled version of the curve to the one that gets used for the Koch Snowflake).

All in all, a cool shape.