Sunday, 17 April 2011

The Diamond Eye Fractal

Starting the fractal tiling process ...
(click any image to enlarge)

The "Diamond Eye" is a fractal that I snuck into the book at the last minute as a small pagespace-filler without a title or figure number (on page 31). As a result, it's probably too small to see properly.

At first sight, this fractal looks like a fairly intricate (but unsignificant) crystal growth pattern with two competing seed types (in this case, clusters of horizontally- and vertically-aligned diamonds).
Iterations 3-5
We start with a horizontal diamond-shaped space, and add our first piece, a vertical diamond smaller than the original by a ratio of the square root of three (1.732-something). It wedges exactly across the centre of the space (top diagram, middle), and then we can’t go any further, so we switch to the second configuration. Scaling down by another factor of root[3], we can fit two horizontal diamonds into the left and right corners of the original space (top, right). Switching back to vertical mode, we can then wedge in four more smaller copies of the shape, and switching back to horizontal again lets us shove in a further eight. As the number of iterations increases we end up with a single solid mass of vertical diamonds growing out from the centre, competing with a hollow shell of horizontal diamonds growing in from the perimeter.
Iterations 6-8

Here’s what you end up with when you’ve carried out so many stages that you effectively have a single, solid,  frozen block (click to enlarge).
Fractal Rhombic Mesh
The important thing here is the shape of the boundary between the two “crystal” types. You can’t really see it too well in the above diagram, so we’ll colour the horizontal and vertical diamonds differently to emphasise the boundary.
Rhombic Koch Snowflake (interior and exterior)
Aha! And this is when we realise that what this “dual diamond” construction is really doing is sneakily growing a Koch Snowflake (I would have put "Rhombic Koch Snowflake" as this post's title, but it would have given away the punchline).

The internal network of cross-crossing diamond-ey shapes are still a bit distracting, so we’ll delete one of the two components. We’ll delete all the vertical diamonds and leave just the horizontals.
Rhombic Koch Snowflake (exterior only)
Yep, definitely a Koch Snowflake!

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