## The art behind Fractals

by Mohit Mayank

Fractals !?

Recursion

Self-similarity

Fractal dimensions

A fractal is a never-ending pattern that repeats itself at different scales (either visually or statistically)

They are created by repeating a simple process over and over in an ongoing feedback loop.

They have their own geometry (fractal geometry) and could have non integer dimensions, like ~1.584 dimensions!

"Clouds are not spheres, mountains are not cones, coastlines are not circles and bark is not smooth, nor does lightning travel in a straight line." - Benoit Mandelbrot

What do we mean by fractal dimensions?

Aim of this talk

1. Explore ways of creating complex but beautiful fractals

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2. Understand how trivial actions lead to non-trivial patterns

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3. Discuss their applications and presence in nature

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4. Fractals are part of a bigger animal - **revealed later** :)

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"Fractals are a paradox. Amazingly simple, yet infinitely complex. New, but older than dirt."

## ACT I: Chaos Game

a iterative random game with not so random outcome

Steps in Chaos Game

1. Choose some fixed points (vertices)Â  in a 2D space

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2. Choose one dynamic point, we will move this a lot in the game (point)

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3. By some logic, iteratively select one of the vertices (random, ..)

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4. Move the point closer to the selected vertex by some proportion of their distances (compression ratio)

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But how?

We understand by considering SierpiÅ„ski triangle

Creating a fractal is nothing but apply the same operations again and again....

f3(x, y) = (x/2 + 1/4, y/2+ 1/2)

These operations can be written as,

f1(x, y) = (x/2, y/2)
f2(x, y) = (x/2 + 1/2, y/2)

## ACT II: Lindenmayer system

a string re-writing engine dev by a biologist

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Steps in Lindenmayer system

1. Define unique characters (variables) and using them write down a string (axiom)

Ex: variable = {A,B}; axiom = "A"

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2. Define some string replacement rules, which replaces one variable with another string (rules)

Ex: Two rules (A â†’ AB), (B â†’ A)

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3. Iteratively perform string replacement and transform the string to patterns (by turtle rendering engine)

Ex: "A" --> "AB" --> "ABA" --> "ABAAB" --> ....

## ACT III: Escape-time fractals

color the space by using a formula

Mandelbrot set

1. Define a complex 2D space, where x-axis showcase real numbers and y-axis showcase imaginary numbers

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2. Pick one complex number (c) and pass its values iteratively into the formulae, Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â

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3. Color the space w.r.t. logic: black - if c doesn't explode, else other color (color showcase how quickly 'c' explodes)

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z_{n+1} = z_{n}^2 + c ; z=0

## ACT IV: More fractals!!

in nature, if not then application and finally conclusion

Fractals in Nature

Fern similar to stem which inturn similar to frond

Neuron from human cortex

A hurricane is a self-organizing spiral in the atmosphere, driven by the evaporation and condensation of sea water.

mountains peaks (its computer generated)

Fractals in Nature Part 2

Fractal river network in China

Our lungs are branching fractals

Biggest fractal ever - the spiral galaxy :)

Top view of Himalayan mountains

Koch snowflake

~1.25 dimensions!

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In the 1990s Nathan Cohen became inspired by the Koch Snowflake to create a more compact radio antenna using nothing more than wire and a pair of pliers. (Left mouse click to see the effect)

Landscape

specially mountains are very much fractals

In 1978 Loren Carpenter wanted to make some computer-generated mountains. Using fractals that began with triangles, he created an amazingly realistic mountain range. (Press R to toggle state, A to toggle animation and left mouse click to change shape)

Hilbert curve

a type of space-filling curve, rendering till order 5

Ideal for transforming 2D grids into 1D domain because of its convergence property. Increases accuracy when used in CNN's input.

The Barnsley Fern

representation of black spleenwort fern

Generate a realistic fern with just some formulas !

Menger sponge

three-dimensional generalization of the one-dimensional Cantor set and two-dimensional Sierpinski carpet

Today, antennae in cell phones are shaped as the Menger Sponge, the box fractal or space-filling fractals - as a way to maximize receptive power in a minimum amount of space (Move the mouse to rotate, mouse wheel to zoom and click to increase level)

Fractals are part of bigger picture - Chaos Theory !

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• Deterministic systems are unpredictable: take two starting states which are very, very similar to each other - over time the states will diverge and re-converge in unpredictable ways.
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• Oddly, the system is unpredictable but sometimes when observed over large period of time, it follows a pattern.
• These patterns seems to revolve around a force called attractors, and as usually they are not in perfect smooth shape, they are called a strange attractor. This is what a fractal represents!
• Ex: a healthy heart beats in a periodic (nonchaotic) pattern, and healthy brain waves are chaotic. Conversely, the dangerous fibrillation of a heart in trauma shows chaotic patterns, and the brainwaves seen during epileptic seizures are periodic. Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â Â

"Tyrannosaurus doesn't obey a set pattern or park's schedule...the essence of chaos....a shorthand is the butterfly effect" -- Ian Malcolm (Jurassic Park - 1993)

Recap

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1. We discussed the characteristics of fractals, especially the fractal dimension

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2. We saw "Iterative function system" of drawing fractals - stochastic (chaos game) and deterministic (transformation)

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3. We saw "L systems" - a string re-writing fractal maker (recursive)

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4. We saw "Escape-time system" - uses formulae at each point in space

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5. Saw additional fractals and their applications. Fractals in Nature.

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6. Connected fractals to Chaos Theory.

References

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