Alt.Fractals

Jerusalem Square

The Jerusalem Square is the "shadow" of the Jerusalem Cube. The ratio between the sizes of a piece of the shape and its copies is an "irrational" number,  1 : ( 1 + root 2 ), or ~2.414213562... .
The square root of two (1.4142...) is the diagonal corner-to-corner distance across a square that has sides of length "one", so if you take a square and butt it up against a copy rotated though 45 degrees, you get the length of the side of the next size up.



As with Fibonacci packings, if you start with a proportion that's way off (say, you try to build the shape with two initial sets of squares of ratio 1:2), then the shape converges on the correct ratio by itself as you add more iterations.
 

---

If you're wondering where the name comes from, here's the Jerusalem Cross as used in the heraldic shield of the Kingdom of Jerusalem (1099 -1291) : 
The shield is notable in heraldry for its "illegal" use of precious metal-on-metal colours (gold on silver), which is said to symbolise the idea that the Knights considered themselves above conventional laws. However, it may also be a reference to the quantity of precious metals that went into the area during the period, and which the Knights Templar then hauled away with them when they scarpered.

The design persists in the Vatican-based Equestrian Order of the Holy Sepulchre of Jerusalem (1099-), and alternative versions of the Jerusalem Cross, with different proportions and embellishments, appear in other organisations' logos as a reference to the Kingdom, including the masonic orders and the National Flag of the Country of Georgia.

Jerusalem Cube

The Jerusalem Cube fractal is a little odd. Although it seems simple enough — it's just a cube repeatedly penetrated by crosses — for it to work properly, the ratios of the cube and sub-cubes don't have whole-number integer, or even fractional integer ratios. We're talking irrational numbers, here, and while you might expect irrationals to show up when you're assembling shapes at funny angles, in this case, they appear when we connect simple cubey blocks together, face-to-face.

It can't be built using a simple integer grid, and that's probably why you probably haven't come across it before. Where the Menger Sponge can be visualised as the result of applying discrete logic within a simple "base three" number system, the Jerusalem Square and Jerusalem Cube correspond to the same sorts of orderly processes being performed on number systems that aren't based on integers.