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.

A Koch Snowflake Sponge

Koch Snowflake Sponge

After the little triumph of the Koch Snowflake Solid, here's a better one: this beastie doesn't just give you the Koch Snowflake profile, it's also punctured by an infinite number of little Koch-snowflake-shaped copies, as holes.

It's a sponge!

The method's pretty much the same as before: divide your source cube into a 3×3×3 grid and delete the eight corners, but then also delete the central cubelet.
When you carry out the very first division it seems that deleting the centre shouldn't change the shape in any way, because the centre cube is totally isolated from the outside world by its six face-adjacent neighbours, but as the number of iterations increases, the cubelets get their corners progressively nibbled away and you get to peek through the holes created by the missing corners into the central void (and out of the other side).

At each iteration the sponge opens up a new set of hexagonal holes, and the existing holes open out and grow more Koch-Snowflakey detail.

This is a way cooler shape than the last one ("It's a sponge!").

And again, for the ultimate verification, I have a real one sitting on my desk made out of plastic.

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.

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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.

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!