McWorter's Pentigree

Visualization

@
$dim=4
$subspace=[s,0]
s=$companion([1,-1,1,-1])
g=2+s^2-s^3
A=g^-1*(s|s^3*[1,-1,1,-1]|s^9*[-1,0,0,1]|s^7*[-1,0,-1,1]|s^9*[0,0,1,0]|s*[2,-1,1,-1])*A

Overview

McWorter’s Pentigree is a self-similar fractal curve with 5-fold (pentagonal) symmetry, discovered accidentally by William McWorter during his searches for “dragon”-type fractals. It was first published in Byte magazine in 1987 and later analysed in detail by Gerald Edgar.

The name pentigree is a portmanteau of pentagon and filigree. Five copies of the pentigree fit together to form a set with exact 5-fold rotational symmetry.

Construction

Starting from the unit segment $[0,1]$, the basic motif replaces each segment with six shorter segments of length $r = \frac{3-\sqrt{5}}{2} = \frac{1}{\varphi^2} \approx 0.382$ where $\varphi = \frac{1+\sqrt{5}}{2}$ is the golden ratio. The six segments rotate by $+36°$, $+108°$, $-36°$, $-108°$, $-36°$, $+36°$ successively, with the final endpoint landing exactly at $(1,0)$.

The contraction ratio $r = 1/\varphi^2$ is determined by the constraint $r + 2r\cos 36° = 1 \implies r = \frac{1}{1 + 2\cos 36°} = \frac{1}{1+\varphi} = \frac{1}{\varphi^2}$

Algebraic Structure

The IFS lives in $\mathbb{Q}(\zeta_{10})$ where $\zeta_{10} = e^{i\pi/5}$ is a primitive 10th root of unity. The companion matrix $s$ (defined as $companion([1,-1,1,-1]) in AIFS) is the companion for $\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1$; in the projected plane $s \leftrightarrow e^{i\pi/5}$ (rotation by 36°).

The expansion matrix can be simplified using $s^5 = -1$ (a consequence of $\Phi_{10}(s) = 0$). Starting from $g = s^{-1}(s^2+s+1)$ and substituting $s^{-1} = -s^4$, then reducing $s^4 = s^3-s^2+s-1$:

$g = 2 + s^2 - s^3$

At the eigenvalue $\zeta_{10} = e^{i\pi/5}$, this projects to $2 + e^{2i\pi/5} - e^{3i\pi/5} = 2 + 2\cos(72°) = \varphi^2$ (purely real, using $\cos(108°) = -\cos(72°)$ and $\sin(108°) = \sin(72°)$). Thus $g^{-1}$ is a pure contraction by $r = 1/\varphi^2$ with no rotation, and the rotation of each map $g^{-1} \circ s^k$ is exactly $36° \cdot k$.

The six maps factor as $A = g^{-1}(h_1 \cup \cdots \cup h_6)(A)$ with:

All 4D translation vectors are in $\mathbb{Z}^4$ — no floating-point approximation. The key identity $s^5 = -1$ (since $\zeta_{10}^5 = -1$) gives $s^9 = s^{-1}$ and $s^7 = s^{-3}$, explaining the short cycle of rotations.

Hausdorff Dimension

The similarity dimension satisfies $6 \cdot r^d = 1$:

$d = \frac{\log 6}{\log(1/r)} = \frac{\log 6}{\log \varphi^2} = \frac{\log 6}{2\log\varphi} \approx 1.862$

Five-fold Symmetry

Five rotated copies of the pentigree, each rotated by $72° \cdot k$ ($k=0,1,2,3,4$), fit together without overlap to form a set with exact 5-fold rotational symmetry, sometimes called the “second form of McWorter’s pentigree” (Edgar, 1990). This packing mirrors how five equilateral triangles meet at a vertex of a regular pentagon.

Boundary Dimension

The boundary $\partial A$ has Hausdorff dimension

where $\varphi = (1+\sqrt{5})/2$ is the golden ratio and $\varphi^4 = (7+3\sqrt{5})/2 \approx 6.854$ is the expansion factor of the IFS. The value $\varphi^4$ satisfies $p^2 - 7p + 1 = 0$.

Related lives in the same field $\mathbb{Q}(\zeta_{10})$

with the same companion matrix $s$, but uses a different expansion $g = 3 - s + s^2 - s^3$, giving a different contraction ratio and a dendrite (tree-like) rather than curve-like attractor. Both are in the same algebraic family of pentagonal IFS discovered by McWorter.

References

• McWorter Jr., W. A. & Tazelaar, J. M. (1987). Creating Fractals. Byte, August 1987, 123–132.
• Edgar, G. A. (1990). Measure, Topology, and Fractal Geometry. Springer, p. 197.
• Riddle, L. Classic IFS — McWorter's Pentigree. Agnes Scott College.

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