Gosper Island

Visualization

@
$dim=2
$subspace=[s,0]
s=[0,-1,1,1]
g=[3,1,-1,2]
h0=s^2*[1,0]
h1=s^5*[0,-1]
h2=s^2*[-1,0]
h3=1
h4=[0,-1]
h5=s^2*[0,-1]
h6=s*[1,0]
A=g^-1*(h0|h1|h2|h3|h4|h5|h6)*A

Overview

The Gosper Island (also called the flowsnake or Gosper hexagon) is a self-similar fractal named after computer scientist Bill Gosper, who discovered it in the early 1970s. It is the attractor of a system of 7 affine maps, each contracting the plane by a factor of $\frac{1}{\sqrt{7}}$.

The island is a rep-7 tile: seven congruent copies of it fit together perfectly to form a scaled copy of itself. Because of this, Gosper Islands tile the entire plane.

Algebraic Structure

The construction uses two integer matrices:

The matrix $s$ is the companion matrix of $x^2 - x + 1$, whose roots are the primitive 6th roots of unity $e^{\pm i\pi/3}$. Since $|\det(s)| = 1$, the matrix $s$ is an isometry — it represents a 60° rotation in the plane. In particular, $s^6 = I$ and $s^3 = -I$.

The matrix $g$ has characteristic polynomial $x^2 - 5x + 7$, with complex roots

The map $g^{-1}$ contracts distances by $\frac{1}{\sqrt{7}}$, and $\det(g) = 7$ confirms that all 7 maps together tile exactly once: $7 \cdot \bigl(\tfrac{1}{\sqrt{7}}\bigr)^2 = 1$.

IFS Definition

The Gosper Island $A$ is the unique non-empty compact set satisfying:

where the seven maps are (with $T_v(x) = x + v$):

The powers of $s$ describe the rotational structure: $s^0 = I$ (0°), $s^1$ (60°), $s^2$ (120°), $s^3 = -I$ (180°), $s^4$ (240°), $s^5$ (300°). Together, the seven maps place copies of the island in six rotated orientations symmetric around a central unrotated copy.

Properties

• Hausdorff dimension: exactly $2$, since $7 \cdot (1/\sqrt{7})^2 = 1$ (the island has positive area)
• Lebesgue measure: positive — the Gosper Island is a solid 2D region, unlike most fractals
• Tiling property: it is a rep-7 tile; seven copies tile a $\sqrt{7}$-scaled copy
• Boundary: the boundary of the Gosper Island is the Gosper curve (flowsnake), a fractal curve of Hausdorff dimension $\log(9)/\log(7) = 2\log(3)/\log(7) \approx 1.129$; see below
• Polynomial: $x^2 - 5x + 7$ is irreducible over $\mathbb{Q}$; its roots live in $\mathbb{Z}[e^{i\pi/3}]$, the Eisenstein integers
• Moran equation: $7 \cdot r^2 = 1$ where $r = 1/\sqrt{7}$ confirms $\dim_H = 2$

Two Kinds

The Gosper Island has two combinatorially distinct variants — the 1st kind (cis) and the 2nd kind (trans). Both are rep-7 tiles built on the same Eisenstein integer lattice with the same inflation factor $\sqrt{7}$, but neither can be transformed into the other by any isometry (rotation, translation, or reflection). They are genuinely distinct tile shapes.

The 2nd kind is defined by a different arrangement of the seven $h_k$ maps combined with a reflection $r$:

where $r = \begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$ flips coordinates.

When the plane is tiled by Gosper Islands, both variants appear simultaneously — a patch of 7 copies around a centre necessarily mixes cis and trans tiles.

Boundary Dimension

The boundary $\partial A$ — the Gosper curve (flowsnake) — has Hausdorff dimension

This follows directly from the Moran equation: the neighbour graph has exactly 3 boundary pieces (the boundary of the Gosper Island consists of three distinct neighbour contacts, each at scale $1/\sqrt{7}$), so $3 \cdot (1/\sqrt{7})^{\,d} = 1$, giving $d = 2\log 3/\log 7$.

References

• Gosper, R. W. (1970s). Unpublished correspondence with Martin Gardner.
• Mandelbrot, B. B. (1982). The Fractal Geometry of Nature. W. H. Freeman.
• Gosper island — Wikipedia
• Hinsley, S. R. Self-Similar Tiles — Flowsnake / Gosper Curve (heptahextal, cis and trans variants).

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