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Category Graphic Generators

Polygon Sides Base polygon shape

Recursion Depth: 4 Higher = more detail (slower)

Scale Factor: 0.33 Bump height relative to segment

Bump Angle: 60° Negative = inward bumps

Rotation: 0° Rotate entire fractal

Stroke

Fill

Background

Stroke Width: 1.5px

Fill Mode

Filled Stroke Only Gradient

Presets:

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About

Cesaro polyflakes belong to a family of self-similar fractals constructed by recursive edge subdivision. The generation process starts with a regular n-gon and applies the Cesaro curve transformation to each edge: every straight segment is replaced by a bent polyline with a controlled "bump" whose height is determined by the scale factor s. After d iterations, the boundary explodes into 3^d segments per original edge, creating intricate snowflake-like patterns. The fractal dimension D depends on the subdivision ratio and lies between 1 and 2 for most configurations. Incorrect parameter choices produce either visually uninteresting blobs (scale too low) or computationally intractable vertex counts (depth too high). This generator computes vertex coordinates using affine transformations in a Web Worker, preventing browser freezes even at depth 7 with 10,000+ vertices.

Real-world applications include procedural texture generation for games, laser-cut artwork templates, and educational demonstrations of iterated function systems. The classic Koch snowflake is a special case: a 3-sided polyflake with scale factor 1/3 and bump angle 60°. Cesaro variants generalize this by allowing arbitrary bump angles and polygon bases. Note: at depth 8 with a hexagon base, the generator produces over 1.5 million line segments - expect render times of several seconds.

Formulas

The Cesaro polyflake is generated by recursively applying an edge-replacement rule to a regular polygon. Each line segment AB is replaced by a three-segment polyline A→P→Q→B, where P and Q lie at fractional positions along the segment and the midpoint is displaced perpendicular to the original line.

M = A + B2

The bump point C is computed by rotating the vector from M perpendicular to segment AB and scaling by factor s:

C = M + s ⋅ |AB| ⋅ n

where n is the unit normal to segment AB. The perpendicular direction is computed via 90° rotation:

n = 1|AB| ⋅ (−dy, dx)

For the classic Cesaro curve with bump angle θ, the intermediate points P and Q are positioned at fractions along AB:

P = A + t ⋅ (B − A), Q = A + (1 − t) ⋅ (B − A)

where t = 12 ⋅ (1 + cosθ). The fractal dimension D of the resulting curve is:

D = log(4)log(2 + 2cosθ)

where D = fractal dimension, θ = bump angle in radians. For Koch snowflake (θ = π/3), this yields D ≈ 1.2619. Vertex count after d iterations starting from an n-gon:

V_d = n ⋅ 4^d

Reference Data

Configuration Name	Polygon Sides (n)	Scale Factor (s)	Bump Angle (θ)	Fractal Dimension	Visual Character
Koch Snowflake	3	0.333	60°	1.2619	Classic triangular snowflake
Cesaro Triangle	3	0.5	85°	1.7848	Spiky, aggressive peaks
Quadratic Koch	4	0.333	90°	1.5	Square-based snowflake
Cesaro Square	4	0.4	70°	1.6309	Rounded square pattern
Pentagon Flake	5	0.382	72°	1.4404	Golden ratio harmony
Hexagonal Flake	6	0.333	60°	1.2619	Honeycomb snowflake
Cesaro Hexagon	6	0.45	55°	1.5812	Dense organic pattern
Heptagonal Flake	7	0.35	51.43°	1.3562	Seven-fold symmetry
Octagonal Flake	8	0.333	45°	1.2619	Eight-pointed star
Cesaro Octagon	8	0.5	67.5°	1.7712	Highly intricate rosette
Nonagonal Flake	9	0.3	40°	1.1949	Nine-fold subtle pattern
Decagonal Flake	10	0.35	36°	1.3219	Ten-pointed complexity
Anti-Snowflake	3	0.333	-60°	1.2619	Inward-facing bumps
Shallow Cesaro	4	0.15	80°	1.0875	Nearly smooth polygon
Deep Cesaro	5	0.6	50°	1.8617	Maximum complexity
Minimal Triangle	3	0.1	30°	1.0269	Barely perturbed triangle
Dodecagonal Flake	12	0.333	30°	1.2619	Twelve-fold mandala
Asymmetric Cesaro	6	0.4	75°	1.6781	Irregular organic feel
Low-Depth Sharp	4	0.5	90°	1.5	Bold angular at depth 2
High-Depth Smooth	8	0.2	45°	1.1292	Subtle detail at depth 6

Frequently Asked Questions

A scale factor s below 0.15 produces bumps too small to be visually distinct. Additionally, if the bump angle θ is very shallow (under 20°), the displacement becomes negligible. Increase scale factor to at least 0.25 and bump angle to 45° - 90° for pronounced fractal features.

Vertex count grows exponentially as V = n ⋅ 4^d. A hexagon at depth 7 produces 6 × 16,384 = 98,304 vertices. At depth 8, this reaches 393,216 vertices. The generator uses a Web Worker to prevent UI freezing, but canvas rendering of this many line segments requires 2 - 5 seconds on typical hardware.

A negative bump angle inverts the displacement direction, creating inward-facing bumps instead of outward ones. This produces "anti-snowflake" patterns where the fractal boundary intrudes into the polygon interior rather than expanding outward. The classic example is the anti-Koch curve with θ = −60°.

The scale factor s determines bump height relative to segment length. Higher scale factors create more pronounced bumps, which increases the fractal dimension D toward 2 (space-filling). The dimension formula D = log(4)/log(2 + 2cosθ) shows that bump angle is the primary dimension driver. Scale affects visual density but not the mathematical dimension directly.

Yes. The mathematical algorithms (Cesaro curve, Koch construction) are public domain. The specific output image you generate is your creation. Export at high resolution via the SVG option for vector-based scaling suitable for print production, laser cutting, or CNC routing. SVG export preserves exact vertex coordinates without rasterization artifacts.

When the scale factor s exceeds approximately 0.5 and the bump angle is large (over 80°), the bump points displace far enough to cross adjacent segments. This is mathematically valid but produces overlapping geometry. For non-intersecting results, keep s ≤ 0.45 and θ ≤ 75°.