Friday, 3 February 2012

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.