McWorter's Pentadendrite

Fractal dimension: 2log(6)/log(6+√5) ≈ 1.700

Boundary dimension: ≈ 1.0421

Visualization

Open in IFStile ↗

AIFS program

@
$dim=4
$subspace=[s,0]
s=$companion([1,-1,1,-1])
g=3-s+s^2-s^3
h1=[0,0,0,0]
h2=s^2*[0,0,0,-1]
h3=[1,0,1,0]
h4=s^6*[-2,1,-2,2]
h5=s^8*[-1,1,1,0]
h6=[2,-1,1,-1]
A=(g^-1*h1|g^-1*h2|g^-1*h3|g^-1*h4|g^-1*h5|g^-1*h6)*A

Overview

McWorter’s Pentadendrite is a self-similar dendrite (tree-like fractal) with pentagonal symmetry, discovered by William McWorter as an L-system fractal and described in Edgar (1990). It has a connected exterior and six-fold self-similarity.

Algebraic Structure

The pentadendrite lives in the algebraic number field $\mathbb{Q}(\zeta_{10})$ , where $\zeta_{10} = e^{\pi i/5}$ is a primitive 10th root of unity. Its minimal polynomial over $\mathbb{Q}$ is the cyclotomic polynomial

$\Phi_{10}(x) = x^4 - x^3 + x^2 - x + 1.$

The companion matrix $s$ (defined as $companion([1,-1,1,-1]) in AIFS) represents multiplication by $\zeta_{10}$ in a 4-dimensional rational space. After projection via $subspace=[s,0], it acts as a rotation by $\pi/5 = 36°$ in the plane.

The expansion matrix $g = 3I - s + s^2 - s^3$ has eigenvalue $6 + \sqrt{5}$ on the projected plane (the scaling factor of the IFS).

Self-Similar Structure

The pentadendrite is the attractor of six maps:

$gA = h_1(A) \cup h_2(A) \cup h_3(A) \cup h_4(A) \cup h_5(A) \cup h_6(A),$

where each $h_i$ is an affine map in $\mathbb{Q}^4$ :

Map	Definition
$h_1$	$\mathbf{x} + [0,0,0,0]$
$h_2$	$s^2(\mathbf{x} + [0,0,0,-1])$
$h_3$	$\mathbf{x} + [1,0,1,0]$
$h_4$	$s^6(\mathbf{x} + [-2,1,-2,2])$
$h_5$	$s^8(\mathbf{x} + [-1,1,1,0])$
$h_6$	$\mathbf{x} + [2,-1,1,-1]$

Properties

Hausdorff dimension: $\dfrac{2\log 6}{\log(6+\sqrt{5})} \approx 1.700$
Symmetry group: dihedral group $D_5$ (10-fold dihedral symmetry)
Topology: dendrite (tree-like, connected, no interior)
Intersection structure: pieces meet at single points or along Jordan curves of dimension $\dfrac{2\log 3}{\log(6+\sqrt{5})} \approx 1.042$

Boundary Dimension

The boundary $\partial A$ has Hausdorff dimension

\dim_H(\partial A) = \frac{2\log 3}{\log(6+\sqrt{5})} \approx 1.0421.

Here $p = 6+\sqrt{5}$ is the squared norm of the expansion $g$ on the projected plane, and $\lambda = 3$ is the eigenvalue of the boundary substitution (the boundary pieces triple under one application of $g$ ). The value $p = 6+\sqrt{5}$ is the larger root of $p^2-12p+31=0$ .

Complementary Attractor

There is a second attractor living in the complementary eigenplane of $s$ (selected by $subspace=[s,1]). The two attractors together fill a region with full pentagonal symmetry.

McWorter’s Pentigree lives in the same field $\mathbb{Q}(\zeta_{10})$ with the same companion matrix $s$ , but uses expansion $g = s^2+s+1$ (contraction $1/\varphi^2$ ) instead of $g = 3-s+s^2-s^3$ . The pentigree is a fractal curve (dimension ≈ 1.862), whereas the pentadendrite is a dendrite (dimension ≈ 1.700). Both were discovered by McWorter.