Fractalverse

A tour in the wonderland of deterministic, random, and natural fractals with Python.

Fractals are patterns that are generated by repeating themselves. They are recursive i.e. starting with a simple rule that is replicated at every level of the fractal. Fractals can be infinitely complex, yet created from a simple rule.

The long-term plan for this project is to be open source so that people can add more and more beautiful fractals to the mix. I'd like everybody to play and potentially build their little toy in the playground.

List of fractals

Cantor Set

In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties.

It is an infinite set defined by German mathematician Georg Cantor (1845-1918) in 1883.

Note to myself: A version of the set was defined earlier by Irish mathematician Henry John Stephen Smith in 1875.^[1]

The basic set is an infinite set of points in the unit interval [0,1]. The design may have been influenced by Egyptian architectural designs, as Cantor had a keen interest in Egypt. For example in the figure over the side which shows the biological recursion of the lotus flower atop the columns of the Temple of Dendur.

Variations

• 1D generalized symmetric Cantor set and (1/4, 1/2) asymmetric Cantor set
• 2D Cantor dust
• 3D Cantor dust
• Cartesian product of the von Koch curve and the Cantor set

Julia Set

Named after the esteemed French mathematician Gaston Julia, the Julia set expresses a complex and rich set of dynamics, giving rise to a wide range of breathtaking visuals. The set lies on the complex plane, which is a space populated by complex numbers. Each point you see in the image above corresponds to a specific complex number on the complex plane with value: z = x + yi, where i = √-1 is the imaginary unit.

Smith–Volterra–Cantor set

Takagi or Blancmange curve

Dendrite Julia set

Boundary of the Rauzy fractal

contour of the Gosper island

Fibonacci word fractal 60°

Boundary of the tame twindragon

Hénon map

Triflake

Koch curve

boundary of Terdragon curve

2D L-system branch

Julia set z2 − 1

Apollonian gasket

5 circles inversion fractal

Quadratic von Koch island using the type 1 curve as generator / Minkowski Sausage

Douady rabbit

Vicsek fractal

Quadratic von Koch curve (type 1)

Quadric cross

Quadratic von Koch curve (type 2)

Weierstrass function

Boundary of the Dragon curve

Boundary of the twindragon curve

3-branches tree

Sierpinski triangle

Sierpiński arrowhead curve

Boundary of the T-square fractal

a golden dragon

Pascal triangle modulo 3

Sierpinski Hexagon

Fibonacci word fractal

Attractor of IFS with 3 similarities of ratios 1/3, 1/2 and 2/3

32-segment quadric fractal (1/8 scaling rule)

Pascal triangle modulo 5

Ikeda map attractor

50 segment quadric fractal (1/10 scaling rule)

Pinwheel fractal

Sphinx fractal

Hexaflake

Fractal H-I de Rivera

A self-affine fractal set

Pentaflake

Monkeys tree

Sierpinski carpet

Boundary of the Lévy C curve

Penrose tiling

Boundary of the Mandelbrot set

Contributing

There are two main ways to contribute - improving the existing fractals or adding new ones.

References

1. Cantor, G., “Uber unendliche, lineare Punktmannigfaltigkeiten”, On infinite, linear point-manifolds (sets), Mathematische Annalen, 21, pp.545-591 (1883).
2. Smith, H.J.S, “On the integration of discontinuous functions”, Proc. of the London Mathematical Society, 6, pp.140-153 (1874/75).