Jerusalem Cross

Visualization

@
$dim=4
$subspace=[s,0]
s=$companion([1,0,0,0])
g=1+s-s^3
h0=[0,1,0,0]
h1=[0,0,0,1]
h2=[0,-1,0,0]
h3=[0,0,0,-1]
h4=[0,-1,-1,-1]
h5=[-1,-1,0,1]
h6=[0,1,1,1]
h7=[1,1,0,-1]
A=(g^-1*h0|g^-1*h1|g^-1*h2|g^-1*h3|g^-2*h4|g^-2*h5|g^-2*h6|g^-2*h7)*A

Overview

The Jerusalem Cross fractal is a self-similar plane fractal with 4-fold (square) rotational symmetry. Its shape echoes the heraldic Jerusalem Cross — a large central cross with four smaller crosses in the quadrants — and is generated by an eight-map IFS in a 4D algebraic setting.

The attractor has Hausdorff dimension $\approx 1.786$, the positive root of $x^2 - 4x - 4 = 0$. It does not fill the plane — it is a proper fractal set.

Algebraic Structure

Like the Ammann–Beenker tiling, the Jerusalem Cross lives in $\mathbb{Q}(\zeta_8)$ with the same companion matrix $s$ (defined as $companion([1,0,0,0]) in AIFS), whose eigenvalues are the primitive 8th roots of unity $e^{ik\pi/4}$.

The key difference is the expansion matrix $g = 1 + s - s^3$, which corresponds to a different element of $\mathbb{Z}[\zeta_8]$ and therefore a different inflation geometry. The $subspace=[s,0] projection selects the dominant complex pair (eigenvalue index 0), giving the 2D rendered attractor.

Eight Maps at Two Scales

The attractor satisfies:

$A = g^{-1}(h_0 \cup h_1 \cup h_2 \cup h_3)(A) \;\cup\; g^{-2}(h_4 \cup h_5 \cup h_6 \cup h_7)(A)$

with two groups of four maps:

Maps $h_0$–$h_3$ are pure 4D translations (no matrix factor beyond $g^{-1}$), displaced along the four coordinate-like directions of the rational space:

$h_0 = [0,1,0,0], \quad h_1 = [0,0,0,1], \quad h_2 = [0,-1,0,0], \quad h_3 = [0,0,0,-1]$

These four displacements correspond to $\pm s$ and $\pm s^3$ in $\mathbb{Z}[\zeta_8]$, forming a cross-like arrangement with 4-fold symmetry.

Maps $h_4$–$h_7$ provide four additional corner pieces:

$h_4 = [0,-1,-1,-1], \quad h_5 = [-1,-1,0,1], \quad h_6 = [0,1,1,1], \quad h_7 = [1,1,0,-1]$

The 4-fold rotational symmetry is visible in the pairing $h_0 \leftrightarrow h_2$, $h_1 \leftrightarrow h_3$, and $h_4 \leftrightarrow h_6$, $h_5 \leftrightarrow h_7$ (each pair related by the element $s^2$ of $\mathbb{Z}[\zeta_8]$, corresponding to a 90° rotation).

Comparison with Ammann–Beenker

Both the Jerusalem Cross and the Ammann–Beenker tiling use the same rational space $\mathbb{Q}(\zeta_8)$ and the same companion matrix $s$, but differ in:

Note: $g = 1 + s - s^3$ and the Ammann–Beenker $g = s - s^3 + 1$ are the same element of $\mathbb{Z}[\zeta_8]$; the visual difference arises from the different sets of maps and the single vs. two-attractor structure.

References

• Jerusalem cross — Wikipedia

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