2D fractal
Recursive log spiral
Live turntable captured from Spiralyst Lab.
This is a logarithmic spiral taught to branch: smaller spirals peel off the main arm, and smaller ones off those, curling into an organic, fern-like recursion. It captures the unmistakable gesture of a fiddlehead fern or an unfurling frond.
The Fiddlehead Unfurls
This is the logarithmic spiral made recursive. Instead of one curve winding to a point, smaller logarithmic spirals sprout at intervals along the main arm, and smaller ones sprout off those, producing an organic, fern-like curl that contains scaled copies of itself. It captures the unmistakable gesture of growing things — the fiddlehead of an unfurling fern, the frond of a palm, the curl of a chameleon's tail or a seahorse.
Because both the parent curve and the branching rule are scale-invariant — the spiral is self-similar, and each child is just a smaller version of the parent — the figure looks fern-like at every level of zoom. Children are spawned at fixed fractions along each arm and grow off roughly perpendicular to it, the way leaflets branch from a frond, then twist by a fixed amount per generation to give the whole structure its sense of flow.
It sits at the boundary between the pure spirals and the recursive trees: the smoothness and growth of the logarithmic spiral, married to the branching self-similarity of an L-system.
Every arm, parent and child alike, is a logarithmic spiral with growth rate b over a chosen number of turns.
Child spirals are placed at evenly spaced points along the parent arm, scaled by a shrink factor s, rotated to the local tangent plus 90° and a per-level twist.
In Spiralyst Lab
Spiralyst Lab draws the parent log spiral, then sprouts a chosen number of child spirals along it (each scaled by 'child scale', rotated perpendicular to the arm plus a 'twist'), recursing a few levels deep with a size floor and point budget. Colour runs along the path so the recursion reads as depth. It is gorgeous with glow and a long trail, which smear the fronds into a luminous curl.
Every parameter below is a live control — set it by hand, map it to a frequency band, or let it ride a smooth animation. These ranges are the actual in-app slider limits.
| Parameter | Range (in-app) |
|---|---|
| Arc length (turns) | 0.3 – 3 |
| Growth b | 0.08 – 0.4 |
| Branches | 1 – 5 |
| Child scale | 0.2 – 0.7 |
| Twist | -1.5 – 1.5 |
Audio-reactive by default: twist -1.2→1.2, growth 0.08→0.4. Any control can be mapped to audio or animation.
Plus the universal 2D controls every spiral type shares: density & stroke, rotation, squash, jitter, zoom & pan, glow, trails, vignette, and multi-layer stacking (count, hue offset, opacity).
Watch it in action
assets/video/fractals/11-recursive-log-spiral.mp4
Did you know: Ferns were among the very first things ever rendered as a fractal — Michael Barnsley's 1988 'Barnsley fern' used exactly this idea of a curve that contains shrunken copies of itself.