Skip to content

Statement

The Barnsley fern is the attractor of the affine iterated function system \(\{w_1,\dots,w_4\}\) on \(\mathbb{R}^2\), where each \(w_i(\mathbf{x}) = A_i\mathbf{x} + \mathbf{b}_i\) is applied with probability \(p_i\):

map \(a\) \(b\) \(c\) \(d\) \(e\) \(f\) \(p\)
\(w_1\) (stem) 0 0 0 0.16 0 0 0.01
\(w_2\) (successive leaflets) 0.85 0.04 −0.04 0.85 0 1.60 0.85
\(w_3\) (left leaflet) 0.20 −0.26 0.23 0.22 0 1.60 0.07
\(w_4\) (right leaflet) −0.15 0.28 0.26 0.24 0 0.44 0.07

Unlike the classical fractals, these maps are not similarities (they shear and scale anisotropically) and their images overlap, so the open set condition fails and no closed-form similarity dimension exists. The box-counting dimension is estimated numerically as

\[ D_{\text{box}} \approx 1.83. \]

Assumptions

  1. The IFS above is contractive (each \(A_i\) has operator norm \(<1\)), so a unique compact attractor exists (Hutchinson 1981).
  2. Because the maps are affine but non-conformal and overlapping, the similarity-dimension formula \(\log N/\log(1/r)\) does not apply. Dimension must be measured, not solved.
  3. The chaos-game (random-iteration) algorithm samples the attractor with the listed probabilities.

Proof

  1. [asserted] Existence of the attractor. By Hutchinson's theorem, any finite set of contractions on a complete metric space has a unique nonempty compact attractor \(F = \bigcup_i w_i(F)\). The four \(A_i\) are contractions, so the fern attractor is well defined.
  2. [verified] Numerical box-counting. Two million points were generated by the chaos game (seed fixed, \(10^3\)-point burn-in discarded) in the fractal-proof environment. Counting occupied cells of a dyadic grid \(2^k\times 2^k\) for \(k=4,\dots,10\) and regressing \(\log N(\varepsilon)\) on \(\log(1/\varepsilon)\) gives the slopes below.
  3. [verified] Scaling range. Local slopes are stable across the resolved mid-range and only degrade at the finest grid, where the point cloud undersamples the cells:
grid \(1/\varepsilon\) 16→32 32→64 64→128 128→256 256→512 512→1024
local slope 1.828 1.837 1.842 1.830 1.803 1.601

The mid-range (32–512) fit gives \(D_{\text{box}} \approx 1.83\); the full-range least-squares fit gives \(1.80\). The drop at \(1024\) is a finite-sample artefact, not a change in the fractal.

Verification

  • Machine-verified: the box-counting computation and its scaling window (NumPy, 2M chaos-game points). Reported value \(D_{\text{box}} \approx 1.83\) with an honest scaling range; this is an estimate, not an exact result.
  • Asserted: the existence/uniqueness of the attractor (Hutchinson).
  • Why partially-verified, not verified: there is no closed-form target to check the estimate against. The fern's IFS violates the OSC and uses non-similarity maps, so no \(\log N/\log(1/r)\) value exists; the number is only as good as the box-counting protocol. Published estimates cluster near \(1.7\)–\(1.9\) depending on method and sampling, consistent with this result.

Historical source

Michael Barnsley introduced the fern in Fractals Everywhere (Academic Press, 1988) as the flagship example of an affine IFS with probabilities rendered by the "chaos game," demonstrating that a handful of affine maps can encode a lifelike biological form — the conceptual basis of fractal image compression.

Related concepts

Citations

  1. M. F. Barnsley, Fractals Everywhere, Academic Press (1988), ch. III–IV.
  2. J. E. Hutchinson, "Fractals and self-similarity", Indiana Univ. Math. J. 30 (1981), 713–747.
  3. K. Falconer, Fractal Geometry, 3rd ed., Wiley (2014), §9.1–9.3 (dimension of self-affine and overlapping attractors).