Wednesday, 30 November 2011

The Jerusalem Square

Jerusalem Square
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.

one plus root two

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)
Jerusalem Cross
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.