Robinson Triangles

Visualization

@
$dim=4
$subspace=[s,0]
s=$companion([1,-1,1,-1])
r=$exchange()
g=s-s^4
h1=s^4*[-1,0,-1,0]
h2=s^2*r*[-1,0,0,0]
h3=s^9
h4=s^6*r*[0,-1,0,-1]
h5=s^3*[1,-1,0,-1]
A1=g^-1*(h1*A1|h2*A1|h3*A2)
A2=g^-1*(h4*A1|h5*A2)

Overview

The Robinson triangles are a pair of isosceles triangles — the acute triangle $A_1$ (angles 36°–72°–72°) and the obtuse triangle $A_2$ (angles 36°–36°–108°) — that generate the Penrose P2 aperiodic tiling (the “kite and dart” tiling) through self-similar substitution.

This entry uses a Generalized IFS (GIFS) with two interleaved attractors.

The Two Triangle Types

Substitution Rules

Each triangle decomposes into smaller copies at ratio $\tau^{-1}$ where $\tau = \frac{1+\sqrt{5}}{2} \approx 1.618$ is the golden ratio:

$g A_1 = h_1(A_1) \cup h_2(A_1) \cup h_3(A_2)$

$g A_2 = h_4(A_1) \cup h_5(A_2)$

The expansion factor $g = s - s^4$ in the rational space corresponds to scaling by $\tau$ on the projected plane.

Algebraic Structure

Like the Pentadendrite, the Robinson triangles live in $\mathbb{Q}(\zeta_{10})$ with companion matrix $s$ (defined as $companion([1,-1,1,-1]) in AIFS) for $\Phi_{10}(x) = x^4-x^3+x^2-x+1$.

The additional matrix $r$ ($exchange() in AIFS, the anti-diagonal 4×4 identity) represents reflection in the projected plane, needed because maps $h_2$ and $h_4$ are orientation-reversing.

All five maps are similitudes with the same contraction factor $1/\tau$.

GIFS Attractor Equations

The AIFS uses two coupled equations — this is a GIFS (Generalized IFS):

A1=g^-1*(h1*A1|h2*A1|h3*A2)
A2=g^-1*(h4*A1|h5*A2)

The notation h*A means the image of the set $A$ under map $h$. The unique solution $(A_1, A_2)$ is the pair of compact Robinson triangles.

Connection to Penrose Tilings

By translating and rotating copies of $A_1$ and $A_2$ according to the substitution rules, one obtains an aperiodic tiling of the entire plane. The tiling is aperiodic — it has no translational periodicity — yet exhibits 5-fold local symmetry at every point.

Properties

• Contraction ratio: $1/\tau \approx 0.618$ (both triangle types)
• Symmetry: dihedral group $D_5$ (same as the Pentadendrite)
• Aperiodicity: no translational period; related to quasicrystal diffraction patterns
• Dimension: 2 (both $A_1$ and $A_2$ are solid triangles with non-empty interior)

References

• Grünbaum, B. & Shephard, G. C. (1987). Tilings and Patterns. Freeman, Fig. 10.3.14.
• Mekhontsev, D. (2019). An algebraic framework for finding and analyzing self-affine tiles and fractals. §9.4.
• Penrose tiling — Wikipedia

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