Jerusalem Menger Cube

Inspired by Robert Dickau
Graphics by Paul Bourke
July 2014

Level 1

Level 2

Unlike the traditional Menger sponge this variation had irrational scaling on each iteration. If the zero'th iteration, a cube, is of unit 1 then the 8 corners of the Jerusalem Cube are cubes of side length sqrt(2)-1. The 12 smaller units are cubes of side length (sqrt(2)-1)2. This is the general scaling law to iterate from the n-1 iteration to the n'th iteration, noting (sqrt(2)-1) + (sqrt(2)-1)2 + (sqrt(2)-1) = 1.

Level 3

Level 4

Another way to imagine the construction is to repeatedly remove (negative extrusion) Greek crosses from the faces of a cube. Note the Greek crosses at each scale pass right through the final Jerusalem cube.

Level 5 (Click for higher resolution)

Level 6 (Click for higher resolution)

On each iteration the number of geometric primitives increases by a factor of 20, the same as the traditional Menger sponge. The compression ratio is sqrt(2)-1, the fractal dimension is 2.529 ...

Level 7 (Click for higher resolution)

Sample PovRay file: scene.pov

Interactive virtual model.