A fractal is a geometric shape that exhibits self-similarity — it looks similar at every level of magnification. Fractals have fractional (non-integer) dimensions. Examples include the Koch snowflake (dimension ≈ 1.262), Sierpinski triangle (dimension ≈ 1.585), coastlines, and tree branching patterns. Fractals are generated by repeating a simple process infinitely, creating structures of infinite complexity from simple rules.

How is fractal dimension calculated?

Fractal dimension (Hausdorff dimension) is calculated using the formula D = log(N) / log(S), where N is the number of self-similar pieces and S is the scaling factor. For the Koch snowflake: each segment is replaced by 4 copies scaled by 1/3, so D = log(4)/log(3) ≈ 1.262. For the Sierpinski triangle: 3 copies scaled by 1/2, giving D = log(3)/log(2) ≈ 1.585. These non-integer dimensions capture how the fractal fills space.

What is the Mandelbrot set?

The Mandelbrot set is the set of complex numbers c for which the iteration zₙ₊₁ = zₙ² + c (starting from z₀ = 0) does not diverge to infinity. Points inside the set are typically colored black, while points outside are colored by how quickly they escape (escape velocity). The boundary of the Mandelbrot set is infinitely complex — zooming in reveals ever more intricate structures, including miniature copies of the whole set.

Fractal Explorer

Explore self-similar structures: Koch snowflake, Sierpinski triangle, fractal trees, Barnsley fern, and the Mandelbrot set. Adjust iterations and parameters to see infinite complexity emerge.

Fractal

Presets

Iterations = 4

Branch Angle = 25°

Max Iterations = 50

Zoom = 1.500

1.500

Properties

Fractal	--
Dimension	--
Segments	--
Property	--
Iterations	--

The Math Behind It

Self-Similarity

Fractals are shapes that repeat at every scale
Fractal dimension (Hausdorff) differs from integer dimensions
Koch snowflake: D = log4/log3 ≈ 1.262
Sierpinski triangle: D = log3/log2 ≈ 1.585
Each iteration multiplies complexity — finite area, infinite perimeter

Mandelbrot Set

Defined by the iteration z_n+1 = z_n² + c
Points are colored by escape velocity — how fast |z| exceeds 2
The boundary has fractal dimension 2
Connected with an infinitely complex boundary
Zooming reveals infinite detail — mini copies of the full set appear

Fractal Explorer

Fractal

Properties

The Math Behind It

Self-Similarity

Mandelbrot Set

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Parametric Curves

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