Examples for

Fractals

A fractal is an object or quantity that exhibits self-similarity on all scales. Use Wolfram|Alpha to explore a vast collection of fractals and to visualize beautiful chaotic and regular behaviors. Examine named fractals, visualize iteration rules, compute fractal dimension and more.

Line-Replacement Fractals

Compute properties regarding fractals created by repeatedly applying iteration rules on curves.

Draw a fractal based on iterated line replacement:

Nowhere-Differentiable Functions

Ask about continuous functions that are nowhere differentiable or ask for the value at a particular point.

Plot an approximation to a nowhere-differentiable function:

Evaluate a nowhere-differentiable function at a point:

Fractals in 3D

Examine fractal behavior in three dimensions.

Draw the Sierpinski tetrahedron:

Draw the Menger sponge:

Shape-Replacement Fractals

Compute properties regarding fractals created by repeatedly applying iteration rules on shapes.

Draw fractals based on replacement of shapes:

Draw fractals by repeatedly adding smaller figures:

More examples
Space-Filling Curves

Perform various iterations whose limiting behaviors lead to space-filling curves.

Plot an approximation to a space-filling curve:

Specify the number of iterations to use:

Other Fractals

Explore various types of fractals.

Plot a curlicue fractal:

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  • Mandelbrot & Julia Sets

    Compute and visualize Mandelbrot and associated Julia sets.

    Plot a Julia set:

    Plot the Mandelbrot set:

    Plot the Multibrot set of exponent d:

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