The 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.

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.
 

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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.

Moorish-styled Tiles

'Granada' range tiles, FiredEarth.com

I think these tiles are absolutely gorgeous. They’re based on classic Moorish tiling artwork from Spain. If you know the Alhambra in Granada (widely regarded a representing the pinnacle of Islamic geometrical art), you’ll know roughly what to expect (M.C Escher made a pilgrimage to the Alhambra to study its designs before embarking on his series of famous tesselated artworks, and some of the Alhambra's designs also seem to show up in the work of Roger Penrose).

The borders between the smaller protruding glazed shapes are recessed and unglazed, so that each tile looks as if it's been assembled from a mosaic of smaller and more complicated tile shapes. It's a nice surface.

The Granada range of tiles is made in Spain and sold by Fired Earth. They also do other North African and Middle Eastern designs without the mosaic effect, like the Andalucia and Anatolia ranges.

(Thanks to Mark at FiredEarth Brighton for providing background info and letting me take the photos)

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!