Mod Tables

Patterns in mod tables

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Chaos Game and the Hexagon

For a hexagon the chaos game forms the same pattern at 1.5 as it does 0.5.

Chaos Hexagon 1.5

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Chaos Game Path Plotter Version 4

Chaos game Path Plotter V4

Rhombic Inner Chaos Game

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The Chaos Game and PHI

The outer chaos game shows some interesting patterns with phi as a magnitude.

Triangle:
Chaos Game Background

Square:
Chaos Game Square Phi Magnitude Background

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3D Shapes Painted by Sections of the Outer Chaos Game

Square with magnitude range 1.4-1.5:
Chaos Square 3d Tiles (1.4-1.5)

Pentagon with magnitude range 1.3-1.4:
Pentagon Chaos Game Strands

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Personal Website

I have setup a personal website koozdra.ca.

There is a preliminary demo of the Chaos Game Path Plotter.

http://koozdra.ca/chaosgame/main.html

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More Interesting Patterns Hidden in the Chaos Game

UPDATE: Interactive version Chaos Game

I was curious to see what would happen If generated chaos games in ascending magnitudes and drew them together. Typically a pattern emerges from 0.5 to 1.5 magnitudes depending on the number of attractors. I split up the spectrum into two parts 0.5 to 1 and 1 to 1.5. This is done because values above 1 draw into the middle of the attractors. This interferes with the internal pattern.

Three Attractors:
Inner:
Chaos Triangle Inner
Outer:
Outer Square
Four Attractors:
Inner:
Chaos Square Inner

Outer:
Choas Square Outer

Five Attractors:
Inner:

Outer:
Chaos Game Past 1

Nine Attractors:
Outer:
9 Symmetry Chaos Game 1-1.48

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175

175

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1.5 Magnitude Pentagon Chaos Game

If at each iteration in the chaos game the current point is moved 1.5 times the distance to a randomly selected pentagonal vertex attractor, this image is created.
1.5 Chaos Game

This image seemed familiar to me. I had created a magnetic shape that looked very similar.
Interlocked Penta Quadruplets

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Adding a Little More Chaos to the Chaos Game

The Chaos Game defines a set of attractors and an iterative process that moves a point towards a random attractor by a specified magnitude.

This image is generated by having five attactors, the points of a pentagon. At each iteration the current point moves towards a point on the pentagon by PHI (0.618) times the distance to the attractor.
Chaotic Pentagonal Fractal (PHI)

I wanted to see what would happen if the magnitude of attraction was also chaotic. At each stage of the algorithm a random attraction magnitude is chosen between [0.5,1.5].
Moving Target Chaotic Pentagonal Fractal

More Resources:
Mathworld Chaos Game Page
Wikipedia Chaos Game Page

[Update] Processing code to generate the shapes above:

var attractors = [];
var current = [random(800),random(800)]

for(var i = 0; i < 6; i++){
attractors.push([cos(i * 2 * PI/5) * 300, sin(i * 2 * PI/5) * 300]);
}

var attraction = 0.5;

void draw(){

translate(width/2, height/2);
rotate(PI/2);
stroke(0);

for(var i = 0; i < attractors.length; i++){
rect(attractors[0], attractors[1], 5,5);
}

for(var i = 0; i < 100; i++){
point(current[0],current[1]);
var target = attractors[Math.floor(Math.random() * (attractors.length-1))]
current[0] = current[0] - (current[0] - target[0]) * attraction;
current[1] = current[1] - (current[1] - target[1]) * attraction;
}

attraction = Math.random()+.5;

}

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