Design Thousand of Unique Fractal Icons

Generation of Fractal Icons by Hyperbolic Iterated Function Set

• Overview of this Fractal Generator
  • This free online software allows to generate fractal icons with diverse shapes, colors and symmetry
  • To generate a fractal, you simply push the red button Get Fractal
• Predefined Fractal Configurations
  • You can select some pre-defined configurations of fractals - Free, Surprise, Divinity, Stars, Vortex, Entropy -
  • The Surprise configuration is set by default
  • The configuration Free resets the fractal to a simple Sierpinski triangle
  • You can manually set the shape of the fractal at any moment (see below)
• Symmetry Properties (Zn Groups) of Fractals Icons
  • You can select the Zn symmetry group from 3 to 9
  • The default Fool means that the Zn is randomly chosen between 3 and 9
  • The Wheel of Fortune allows to set a random Zn of high rank, chosen between 30 and 60
  • You can manually set the Zn symmetry of the fractal at any moment (see below)
• Manual Settings: → Projection, Z-Pole, Affine Parameters, Iterated Function Set
  • The Projection button allows to choose if linear projections - linear transformation with a determinant equal zero - are also randomly allowed or not - it is on by default
  • The Z-pole option allows to add a linear transformation with a zero fixed point to any set of transformations
  • The Affine Parameters and Iterated Function Set buttons open the windows where you can manually set the parameters of the function sets
  • Each linear transformation matrix is decomposed in the product of scale, rotation, stretch and shear matrix
  • The Scale parameter is an homogeneous scale scalar which will be multiplied by the identity matrix
  • The Stretch parameter allow to indicate a degree of non-homogeneous scale in the x and y direction: its value is rearranged in the function k=(1+s)/(1-s) and then in a unitary diagonal matrix with k and 1/k as eigenvalues
  • The Rotation angle defines a rotation matrix operating on the left of the scale matrix
  • The Shear angle defines a rotation matrix operating on the right of the scale matrix
  • With the symmetry slider, you can manually select the Zn symmetry of the set
  • With the nb-ifs slider you can manually set the total number of different function per each symmetry mode in the set - default is 3
• Plotter Settings: → Colors, Axis, Export
  • The colors in the fractal icon image are proportional to the probability that a given point is visited during the iterations
  • Clicking the Colors button opens a window where you can select the color palette of the fractal icon
  • You can either select one of the three pre-defined color palettes at the bottom of the window, or define your own color palette
  • You define a color palette by choosing the hue, the saturation and the lightness of the two extreme colors of the palette
  • The program will then adjust the intermediate colors proportionally to the gradient between the two extreme colors you defined
  • By activating the axis option, you ask the program to plot also the coordinate axis of the space and the fixed points of each function in the iterated function set
  • The fractal icon images are plotted on a html canvas element: such element doesn't allow to copy and export the image
  • By clicking on the Export button, you convert the image into a png image: you can then right click on the image and save the image anywhere on your computer
  • Note that the image is exported without any background: you will need to add the background on any other application where you wish to display the exported image
• Drawing: → New, Redraw, Repaint
  • The New button generates a new fractal icon with the same settings as the current one: the current one is erased
  • The Redraw button draw the icon again from the beginning
  • The Repaint button allows to apply new color settings to the existing fractal icon
• Add Iterations: → +10k, +100k, +1M, +10M
  • The initial fractal icon is plot after 10,000 iterations of the function set: this should prevent users with low calculation capacities (eg. smartphones) to be locked in too long waiting times
  • The initial fractal image is usually of poor quality, but allows you to identify nice icons
  • You can then add iterations to get a high quality icon and to finetune the fractal image
  • Note that a fractal is defined as the result of an infinity of iterations, what is obviously impossible on a computer
  • Usually you get most of the image after 1M iterations, but sometimes the image yet improves significantly up to 100M iterations
  • Be ready to wait if you launch fractal iterations over 100k without a powerful enough computer
• IFS data (Mathematicians)
  • For the mathematicians, we plot in the IFS data all the most significant parameters of the iterated function set
  • A first matrix show (i) the number of times a point has been hit by an iteration divided in percentiles, and including the minimum and maximum of such hits numbers, (ii) the total number of hits (pixels points) and of iterations realized, (iii) an estimate of the mathematical ball-dimension of the fractal and (iv) the maximum radius of the fractal
  • In the left G-matrix you find the scale, stretch, rotation, shear, radius and phase of the fixed points, if the mirror was on - if mr=0 you get a projection, if mr=-1 you get a reflection -, the determinant and the trace of the transformation
  • In the right G-Matrix, for the same transformation, you find the standard parameters of the linear matrix (G), the translation parameter t and the applied probability of appearance of a transformation at each iteration
• Brief Mathematical Hints for Fractal Generation
  • An IFS is a set of affine transformations
  • Each of those affine transformation is a contraction - the absolute value of the scales in the x and y direction are less than unity - A set of affine contractions is called an Hyperbolic IFS
  • It is proven that any hyperbolic IFS has a unique set of fixed points, called the attractor A of the HIFS
  • Starting with a random complex number z (or a point in the plane) and applying iteratively a transformation randomly selected within the HIFS, after a transitory period (eg. 500 iterations) the application inevitably hits the attractor
  • Continuing to apply iteratively a transformation randomly selected within the HIFS, the program shows the attractor of the HIFS, coloring the pixel proportionally to the hits
  • The attractor of the HIFS is the fractal image
• Selected Online References: → Affine Transformations
  • The introduction - Utrecht Uni
  • Practical decomposition of 2D matrix
  • Lecture - Facta Universitatis Nis
• Selected Online References: → Iterated Function Sets
  • Introductory by Agnes Scott College
  • Lecture by Natoli - UChicago
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