---
type: Fractal Proof
title: Barnsley Fern — Box-Counting Dimension (Numerical)
description: The Barnsley fern is the attractor of a four-map affine IFS with overlapping, non-similarity maps; its dimension has no closed form and is estimated numerically at D ≈ 1.83.
resource: https://tyson-swetnam.github.io/fractal-notebooks/okf/proofs/barnsley-fern-dimension/
tags: [ifs, dimension-theory]
result: "box-counting D ≈ 1.83 (numerical; no closed form)"
concept: ../concepts/ifs/barnsley-fern.md
historical_source: Michael Barnsley, "Fractals Everywhere" (Academic Press, 1988)
verification_status: partially-verified
sources:
  - id: history-chapter
    resource: https://tyson-swetnam.github.io/fractal-notebooks/foundations/history/
    title: "Chapter 1: History of Fractal Mathematics"
generated:
  by: human:tswetnam
  at: "2026-07-13T00:00:00Z"
verified:
  - by: human:tswetnam
    at: "2026-07-13T00:00:00Z"
status: stable
---

# 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

* [Barnsley Fern](../concepts/ifs/barnsley-fern.md)
* [Iterated Function System](../concepts/ifs/iterated-function-system.md)
* [Box-Counting Dimension](../concepts/dimension-theory/box-counting-dimension.md)

# 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).