Cross Menger (Jerusalem) Cube Fractal

I’m sure I’m doing this wrong, but here goes: Start with a square.

square

Next, create a shape by scaling down the original square and moving the copies to the four corners of the original.

four smaller squares

Finally, fill in the gaps along the edges with smaller scaled versions of the squares.

square with cross-shaped hole

That’s the shape we want. The hole looks kind of like something called a Jerusalem cross—I don’t completely see it myself—hence one of the names for the overall figure.

If we choose a scaling factor between the original square and the smaller ones as k = (√2 − 1), and the factor between the original square and the smallest to be k2, since k + k2 + k = 1, each iteration fits in the previous one’s space.

k plus k squared plus k equals 1

After that, repeat the process of scaling and translating, and the limiting figure is the Cross Menger square, or Jerusalem square, depending on whom you ask.

Jerusalem square iterations 0 through 4

I say that I might be doing it wrong because half of the figures out there in the world don’t use just the two different scales with the previous figure, as here (and the Cube de Jérusalem page on Wikipedia), but instead mix in different scales with the two previous figures, if that makes sense. For example, see the pages here and here.

Jerusalem square alternate, iterations 0 through 4

In this style, the crosses added at each step are all the same size. Live and let live. Here are inverse figures of the previous, in case you’re interested:

Inverse of previous

In either case, you can do the same thing in three dimensions: Start with a cube, and then for each iteration make eight smaller copies in the original cube’s corners, and twelve even smaller copies between those. Repeat the process as long as you can stand it, and the limiting figure is the Cross Menger cube, or Jerusalem cube.

Jerusalem cube iterations 0 through 3

Or here’s the other style, if you prefer.

Other style, zero through three

I think these are the inverses of the previous, but I’m in kind of a hurry and need to verify later:

Inverse of previous, I'm pretty sure

The Cross Menger/Jerusalem cube is different from similar-looking fractals such as the Menger sponge in that the ratio between each iteration’s components is an irrational number, (√2 − 1). The Menger sponge pieces, on the other hand, have a (rational) middle ninth punched out each time. Remember when we thought that was cool?

A natural extension is to pick a side length so that s + s2 + s3 + s = 1.

k plus k squared plus k cubed plus k equals 1

(Previous figure lists the parts in reverse order, if you couldn’t tell.)

That gives us a fractal that looks like a houndstooth pattern:

houndstooth iterations 0 through 4

The 3-D version is disappointing: four of the cube’s sides have nice rotational symmetry, but the other two (look at the right side) are just diagonally symmetric, which ruins the effect. No surprise, since there are three faces but only two sizes of neighbors, but still.

houndstooth 3D iterations 0 through 3

Let’s see where this goes. Here’s an obvious next step of the houndstooth fractal:

second-order houndstooth, iterations 0 through 4

Keeping the largest corner copies (size k) in place, the middle parts k2, k3, and k4—in addition to {2,3,4} above—can also be ordered as {2,4,3} and {3,2,4}. Here’s {2,4,3}:

And here’s the vaguely bat-shaped {3,2,4}:

It gets boring from there. The version with k + k2 + k3 + k4 + k = 1 is too similar to the previous version—see those four white dots?—to be much fun to look at:

Designed and rendered using Mathematica.

Copyright © 2014 by Robert Dickau.

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