Viper

Visualization

@
$dim=2
$subspace=[s,0]
s=$companion([1,1/2])
r=$exchange()
h0=1*[0,0]
h1=1*[2,4]
h2=1*[4,4]
h3=1*[4,8]
h4=-1*[-6,-8]
h5=s*[-3,-6]
h6=s*[-1,-6]
h7=-s*[-1,2]
h8=-s*[-3,2]
A=3^-1*(h0*A|h1*A|h2*A|h3*A|h4*A|h5*A|h6*A|h7*A|h8*A)

Overview

The Viper is a 2D IFS tile with 9 affine maps and a scalar inflation factor of 3. Unlike most tiles in this collection, it does not arise from a standard crystallographic symmetry group. Instead it uses an irrational rotation angle generated by the companion matrix $s$ (AIFS: $companion([1,1/2])), whose eigenvalues lie on the unit circle but at a transcendental angle.

Because the rotation $s$ is an irrational fraction of $2\pi$, the symmetry group $\langle -1, s, s^{-1}, r \rangle$ is infinite and generates dense orientations — similar in spirit to the Pinwheel tiling.

Structure

The companion $s$ (AIFS: $companion([1,1/2])) satisfies $s^2 + \tfrac{1}{2}s + 1 = 0$, giving eigenvalues $e^{\pm i\theta}$ where $\theta = \arccos(-\tfrac{1}{4}) \approx 104.5°$ — an irrational multiple of $\pi$.

The nine maps are generated by the group $\{1, -1, s, -s\}$ (rotations by $0°$, $180°$, $\theta$, $180°+\theta$) combined with integer translations:

The IFS equation is:

The Moran equation $9 \cdot (1/3)^2 = 1$ confirms that the Hausdorff dimension equals 2 (solid tile).

Properties

• Prototile count: 1 — a single polytopal (polygonal) tile with non-fractal boundary
• Inflation factor: 3 (scalar)
• Maps: 9, using isometries from $\{1, -1, s, -s\}$ — no reflections; all tile copies are right-handed
• Symmetry: statistical circular symmetry — tile orientations are dense in $[0, 2\pi)$ due to the irrational rotation angle $\theta \approx 104.5°$
• Hausdorff dimension: 2 (solid tile, positive area)

References

• Viper — Tilings Encyclopedia

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