Draw Multiflake Fractal

“Multiflake” is a generic name for an N-flake – a fractal built by replacing every regular N-sided polygon with one smaller center copy plus N smaller copies arranged at the parent’s vertex directions. The construction generalizes named fractals like the hexaflake (N=6) and pentaflake (N=5). The tool covers N from 3 (triflake) through 12 (dodecaflake) using a single parameterized subdivision rule.

The actual count is the geometric sum (N to the L+1 minus 1) divided by (N minus 1) – for example hexagon iter 3 = 259, hexagon iter 5 = 9,331, octagon iter 5 = 37,449. An earlier version of this tool’s FAQ claimed (N+1) to the L (which would give 343, 16,807, 59,049 respectively). That was a math error inherited from misreading the recursion: the center polygon pushes 1 piece at the current level and recurses N (not N+1) times. The current tool reports the real count in the stats line.

The displayed similarity dimension is log(N+1) divided by log(1 / scale-factor), using the actual size-scaling used in the render: 1/2 for triangle, 0.6 for square, 1/3 for pentagon and above. This formula is exact for N=6 (hexaflake → log 7 over log 3 = 1.7712, matching the classical published construction). For other N the rule is a recognizable approximation of the polyflake family – the reported number is the dimension of the parametric construction this tool implements, not necessarily a canonical published N-flake. The stats line marks N=6 as “classical hexaflake” and other N as “non-classical approximation”.

With scale 0.6 and surrounding distance 2/3 of the parent radius, the four surrounding squares for N=4 overlap with each other and with the center – that is intentional for the implementation rule but means the “fractal” has dimension above 2 (it covers more than 2D area). The classical Sierpinski carpet (which is also a square-based fractal but uses a 3×3 grid minus center, 8 copies at scale 1/3) has dimension log 8 over log 3 ≈ 1.8928 and is what most “square fractal” references mean. This tool’s quadflake is a different construction.

A figure has N-fold rotational symmetry if rotating it by 360 divided by N degrees produces an identical image. Triangle has 3-fold (120 degrees), hexagon has 6-fold (60 degrees), octagon has 8-fold (45 degrees). The “1 center + N at vertex angles” rule preserves the parent’s N-fold symmetry at every level, so the whole fractal inherits it. This same symmetry shows up in snowflakes (6-fold), pentagonal flowers (5-fold), and Islamic geometric tilings (8 and 12-fold).

The classical hexaflake uses scale 1/3 with surrounding hexagons placed so they tile contiguously with the center hexagon – this is the unique scale where the 7 hexagons fit perfectly without overlap or gap. The pentaflake’s classical construction uses a different scale (1 over (1+phi) where phi is the golden ratio) so the 6 pentagons tile correctly; this tool uses 1/3 for pentagons too, which produces a similar-looking but non-tiling figure. Octaflake, heptaflake and higher do not have widely-published canonical constructions; this tool draws them with the same 1/3 rule for consistency.

Each polygon is tagged with the recursion level at which it was emitted. Level 0 = the leaf polygons (smallest); level L = the very first center polygon (largest, at the top of the recursion). Alternating by level mod 2 makes the hierarchy visually obvious – you can trace the layers like growth rings.

No. The tool loads three static files (HTML, CSS, JS) and then runs entirely in your browser – polygon 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.