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https://github.com/0x48piraj/fractalverse

A tour in the wonderland of deterministic, random and natural fractals with python.
https://github.com/0x48piraj/fractalverse

animation chaos fractal fractal-algorithms fractal-rendering fractals math mathematics matplotlib modeling nonlinear playground python vizualisation

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

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# 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]](#references)



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).