Jerusalem Cube

Visualization

@
$dim=3
k=2^(1/2)-1
t=k+k^2
s=1-k^2
A=(k|[0,t,0]*k|[t,0,0]*k|[t,t,0]*k|[0,0,t]*k|[0,t,t]*k|[t,0,t]*k|[t,t,t]*k|[k,0,0]*k^2|[k,s,0]*k^2|[0,k,0]*k^2|[s,k,0]*k^2|[k,0,s]*k^2|[k,s,s]*k^2|[0,k,s]*k^2|[s,k,s]*k^2|[0,0,k]*k^2|[0,s,k]*k^2|[s,0,k]*k^2|[s,s,k]*k^2)*A

Overview

The Jerusalem cube is a 3D fractal whose cross-sections parallel to the coordinate planes are all Jerusalem crosses. It is constructed using 20 maps at two different contraction ratios derived from $k = \sqrt{2} - 1$.

Algebraic Structure

The key parameter is $k = \sqrt{2} - 1 \approx 0.414$, derived from the diagonal of the unit square. Define:

The 20 maps split into two groups:

• 8 corner maps (scale $k$): placed at the eight corners of the cube at positions in $\{0, t\}^3$.
• 12 edge maps (scale $k^2$): placed along the 12 edges of the cube.

The Hausdorff dimension $d$ satisfies the Moran equation:

Letting $u = k^d$ gives $12u^2 + 8u - 1 = 0$, with positive root $u = \tfrac{-2 + \sqrt{7}}{6}$, yielding:

Properties

• Every cross-section perpendicular to a coordinate axis is a Jerusalem cross pattern.
• The fractal has the full cubic symmetry group $O_h$ (order 48).
• Uses two distinct contraction ratios ($k$ and $k^2$), making it a non-homogeneous self-similar set.

References

• Jerusalem cube — Wikipedia
• Jerusalem cross — Wikipedia

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