About

A canopy fractal is a binary branching structure where a single trunk splits recursively into two sub-branches at each node. The geometry is governed by the split angle θ, the branch length ratio r, and the recursion depth n. Small changes in θ produce dramatically different topologies: at 60° the canopy resembles a deciduous oak, while 25° produces a narrow conifer. The total number of terminal branches equals 2ⁿ, so a depth of 14 yields 16,384 endpoints. This tool computes exact trigonometric positions for every segment and renders them on a hardware-accelerated Canvas surface. An optional jitter parameter perturbs each angle by a seeded pseudo-random offset, producing organic asymmetry without losing reproducibility.

Fractal dimension of the canopy boundary depends on r and θ. When 2 ⋅ r ⋅ cos(θ) < 1, branches do not overlap and the Hausdorff dimension is ln(2)ln(1 / r). When branches overlap the structure self-intersects, creating denser fill patterns. This generator approximates but does not clip overlapping geometry. All rendering is deterministic for a given seed value.

Formulas

Each branch segment is computed recursively. Given a starting point (x₀, y₀), current angle α, and current length L, the endpoint is:

x₁ = x₀ + L ⋅ cos(α)

y₁ = y₀ − L ⋅ sin(α)

At each node, two recursive calls are made with updated angles:

α_left = α + θ + ε

α_right = α − θ + ε

where ε is a jitter value drawn from a seeded PRNG in the range [−J, +J]. The next branch length is:

L_next = L ⋅ r

The total number of terminal segments at depth n:

N_terminals = 2ⁿ

The fractal (Hausdorff) dimension when branches do not overlap:

D = ln(2)ln(1 / r)

Where: θ = split angle (degrees), r = branch length ratio (0 < r < 1), n = recursion depth, J = maximum jitter angle, ε = random perturbation per branch, L = current branch length, α = current branch angle from vertical, D = fractal dimension.

Reference Data

Preset Name	Angle θ	Depth n	Branch Ratio r	Jitter	Terminal Branches	Visual Character
Symmetric Oak	45°	12	0.70	0°	4,096	Balanced deciduous crown
Narrow Conifer	20°	13	0.75	0°	8,192	Tall, narrow spruce
Wide Elm	70°	11	0.65	0°	2,048	Broad spreading canopy
Wind-swept	35°	12	0.68	8°	4,096	Organic, asymmetric
Minimal Sapling	40°	6	0.72	0°	64	Simple, few branches
Dense Thicket	30°	15	0.73	3°	32,768	Very dense fill
Right-leaning	55°	10	0.66	12°	1,024	Asymmetric, windblown
Fractal Dust	90°	14	0.50	0°	16,384	Cantor-set boundary
Golden Angle	137.5°	10	0.72	0°	1,024	Phyllotaxis-inspired spiral
Binary Symmetric	60°	10	0.50	0°	1,024	Self-similar, non-overlapping
Coral Fan	25°	14	0.80	5°	16,384	Dense, fan-shaped
Lightning	15°	14	0.78	15°	16,384	Jagged, electric discharge
Weeping Willow	80°	11	0.74	6°	2,048	Drooping, wide arcs
Asymmetric Bonsai	50°	9	0.60	10°	512	Irregular, sculpted look
Maximum Density	35°	16	0.71	2°	65,536	Near-solid canopy fill

Frequently Asked Questions

At depth n, the generator draws 2ⁿ⁺¹ − 1 line segments. At n = 14 that is 32,767 segments. At 16 it reaches 131,071. Canvas 2D stroke calls are not batched by default, so each lineTo/stroke pair incurs GPU overhead. For depths above 14, expect 200-500 ms render time on mid-range hardware.

The Hausdorff dimension is D = ln(2) / ln(1/r). At r = 0.5, D = 1.0 - the canopy boundary is a curve. At r = 0.7, D ≈ 1.94 - nearly area-filling. At r ≥ 0.75, branches overlap heavily and the formula is only an upper bound.

The seed initializes a Mulberry32 pseudo-random number generator. The same seed always produces the same jitter pattern, so a specific organic-looking tree is reproducible. Changing the seed while keeping all other parameters fixed produces a structurally similar but visually distinct tree. Setting jitter to 0° disables the PRNG entirely and yields a perfectly symmetric fractal.

Self-intersection occurs when 2 ⋅ r ⋅ cos(θ) ≥ 1. For r = 0.7, the critical angle is θ_crit = arccos(1/(2 × 0.7)) ≈ 44.4°. Below this angle, branches from adjacent nodes overlap, producing denser visual patterns.

The fractal is generated entirely by your parameter choices and rendered client-side. No copyrighted assets are involved. The output is a mathematical object parameterized by your inputs. You own the resulting image. Export at the highest resolution your display supports by maximizing the browser window before downloading.

If r is very low (below 0.4) and depth is modest (below 8), terminal branches become sub-pixel in length and invisible. Similarly, θ near 180° causes branches to fold back on the trunk. The generator does not clip these degenerate cases. Adjust r above 0.5 or increase depth to compensate.