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*."

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)

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" --> ....

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

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 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
and as usually they are not in perfect smooth shape, they are called**attractors**,*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