Draw Pentaflake Fractal

The pentaflake is a self-similar fractal with 5-fold rotational symmetry, built by recursively replacing every regular pentagon with 6 smaller copies arranged as 1 center pentagon plus 5 surrounding pentagons at the parent’s vertex directions. It is one of the simplest fractals to exhibit pentagonal symmetry – which is rare in classical tilings because regular pentagons do not tile the plane.

φ = (1 + √5) / 2 ≈ 1.618 is the golden ratio. For the surrounding pentagons to fit naturally at the parent’s vertices, each must be scaled by 1/(1+φ) = 1/φ² ≈ 0.382 of the parent. This is the unique scale where the construction has clean self-similarity. The number 1+φ ≈ 2.618 is also φ² because of the identity φ² = φ + 1.

The real count is (5 to the N+1, minus 1, divided by 4) – a geometric series, because each non-leaf level pushes 1 center pentagon and recurses 5 times for the surrounding ones. Iter 3 = 156; iter 4 = 781; iter 5 = 3,906. An older version of this tool’s stats reported 5 to the N (giving 125 at iter 3, 3,125 at iter 5) – that was wrong. The current stats line shows the real number.

The classical pentaflake’s similarity dimension is log(6) divided by log(1+φ), which works out to about 1.8617. An older version of this tool’s code used the formula log(6) / log(2+φ), which gives about 1.394 – that was wrong (factor in the denominator was off). The current code computes the correct value. The number sits between 1 (a curve) and 2 (a filled region), which matches the visual impression of a “thick but not solid” snowflake.

No – regular pentagons cannot tile the plane (the interior angle 108° does not divide 360 evenly), and this construction does not produce a true edge-sharing tiling either. The 5 surrounding pentagons are placed at the parent’s vertices, which leaves small visible gaps between the surroundings and the center. This is intentional – it is the “Sierpinski pentaflake” variant. A tighter tiling would require a different placement rule.

Each pentagon is tagged with the recursion level at which it was emitted (level 0 = leaf pentagons; level N = the very first center pentagon at the top of the recursion). Alternating by level mod 2 makes the hierarchy visible as concentric color rings – you can trace the layers like growth rings.

No. An earlier version of the spec claimed this used an L-system with axiom “F++F++F++F++F” and a rewrite rule. That was inaccurate – the code uses direct recursive subdivision (push center, recurse 5 times). The misnamed L-system would not actually produce a pentaflake at all; it would produce a single pentagon. The spec has been corrected.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – pentagon generation, drawing, PNG export, clipboard copy. You can disconnect from the internet after the page loads. No analytics, no tracking, no cookies.

Yes – free, unlimited, no signup, no watermark. Use the PNGs in lectures, papers, blog posts, or art freely. Attribution to is appreciated but not required.