Draw Koch Fractal

Both render Koch snowflakes. This tool: shape-mode explorer – pick curve / snowflake / anti-snowflake AND set custom angle 1-89°. Stats show fractal dimension only when angle = 60° (the rigorous Koch). Sister tool draw-frosty-fractal: snowflake-only at 60° but 4 render styles (Solid, Outline, Gradient, Ice-crystal with glow). Pick this one for shape variety and angle exploration. Pick that one for ice-style visualization at the canonical 60°.

Constructed by Helge von Koch in 1904 as the first published example of a continuous curve that is nowhere differentiable (no defined tangent at any point – sharp corners everywhere). Start with an equilateral triangle; replace the middle third of each edge with two sides of a smaller equilateral triangle bump pointing outward. Repeat for every new edge. The limit has INFINITE perimeter enclosing finite area = (8/5) · A₀ where A₀ is the original triangle’s area.

Koch curve: one Koch L-system rendered as an open zigzag. Snowflake: 3 Koch curves rotated 120° apart, joined at endpoints – closed equilateral-shaped boundary with bumps pointing OUTWARD. Anti-snowflake: same construction but bumps point INWARD, creating a concave star. All three have identical fractal dimension (log(4)/log(3) ≈ 1.2619 at 60°) and identical perimeter-multiplier (4/3 per iteration).

Per-iteration perimeter multiplies by 4/3 (each edge becomes 4 edges of 1/3 length each). So total perimeter = P₀ × (4/3)^n → ∞. But area added each iter shrinks geometrically (each iter adds smaller-and-smaller bumps): iter 1 adds 3 small triangles totaling (3/9)·A₀, iter 2 adds 12 even-smaller totaling (12/81)·A₀, etc. Geometric series converges to (3/5)·A₀ of added area, total limit area = (8/5)·A₀. Hence “infinite perimeter, finite area” – the most famous fractal paradox.

At 60° you get classic Koch with provable dim log(4)/log(3) ≈ 1.2619. At OTHER angles, the rule F → F+F–F+F still runs but the geometry no longer matches an equilateral triangle bump – the “bump” becomes a different isoceles triangle. The resulting fractal has a DIFFERENT (and harder-to-compute) dimension. We mark the stats line as “dim N/A (non-60°)” rather than report a wrong value. Try 30° for sharp pointed bumps, 80° for very shallow zigzags.

Axiom: F. Rule: F → F+F--F+F. F = move forward and draw. + = turn right by the chosen angle. - = turn left by the chosen angle. Each iteration replaces every F with the 4-instruction sequence, so segment count grows as 4^iter. Snowflake mode wraps 3 of these strings with 120° rotations: <koch>--<koch>--<koch>. Anti-snowflake wraps with ++- between curves so bumps face inward.

Iter N snowflake has 3 × 4^N edges. Iter 7: 49,152 segments. Iter 8: 196,608. Each segment is a lineTo Canvas call. At ~200k operations, render time becomes seconds-long and the L-system character string itself exceeds 100k chars. Iter 7 is the comfortable ceiling. Visually, iter 6-7 are nearly indistinguishable from each other at typical canvas sizes – segments fall below 1 px and merge.

A function has a derivative at a point if you can draw a unique tangent line there. The Koch curve has CORNERS everywhere – at every point in the limit set there’s an infinite stack of right-angle bends. So no point has a tangent. This was shocking in 1904 – before Koch, mathematicians believed continuous functions were “mostly” differentiable (smooth lines with at most isolated kinks). Koch (and earlier Weierstrass in 1872 with a different curve) proved otherwise. Today we know “most” continuous functions in a measure-theoretic sense are nowhere differentiable.

Yes – 100% client-side. L-system expansion and Canvas drawing run in your browser. No parameters or images leave your device. Verify via your browser’s Network tab.