**Fractals**

Fractals & History

Fractals were given their name in 1973

the "Father of Fractals" is Benoit Mandelbrot

Mandelbrot was introduced to mathematics by his two uncles when he was a young boy.

Fractals in Real Life

Fractals are used in mathematics by set that typically displays self-similar patterns, which may mean they maybe exactly the same at every scale or nearly the same while being used in mathematics.

**Fractals in Math**

**By: Carrie McClellan, Kaylee Burchett, Caleb York, J.R. Rich, & Haley Huckelby**

Other mathematicians that worked with fractals include: Aristid Lindenmayer, Michael Barnsley.

Blaise Pascal connected fractals with Algebra 1 and Binomial Expansion.

Helge von Koch wrote a paper in 1906 to prove his Koch Snowflake.

Examples of Works

Benoit Mandelbrot: He focused on how things such as islands could look smooth if you were zoomed out a lot and rigid if you zoomed in on them.

Aristid Lindemnmayer is most known for his L-systems. L-systems are used to rewrite complex objects.

Fractals are present in other specific places in things besides mathematics. Like graphs, nature, and DNA sequences.

There are other fields of fractals being used besides in mathematics, like science, art, technology, etc.

1. Lightning.

2.Leaf.

3. Snowflake.

Fractals in Nature!

The Koch curve fractal is one of the most used methods in mathematics, and in nature. Which all three are used to create the the snowflake method fractal of many kinds of properties.

Literally speaking, Fractals belong to the geometry or trigonometry branch of math.

What's a Fractal?

Any pattern that reveals complexity as enlarged...

To Produce a Fractal...

Self-Similar Figures

Michael Barnsley is popular because of his book Superfractals. Barnsley brings the story up to date by explaining how IFS have developed in order to address this issue. New ideas such as fractal tops and super IFS are introduced

Blaise Pascal is known for Pascal’s Triangle.

To build the triangle, start with "1" at the top, and then continue placing numbers below it in a triangular pattern. Each number is the two numbers above it added together (except for the edges, which are all "1")

Helge von Koch wrote a paper in 1906 to prove his Koch Snowflake. Koch was born January 25, 1870 in Stockholm Sweden. His father was Richert Vogt von Koch, who had a military career, and his mother was Agathe Henriette Wrede. Von Koch attended a good school in Stockholm, completing his studies there in 1887. He died March 11, 1924 in Stockholm, Sweden.

why?

Because...

even though Fractals are their own thing its all about what their composed of. Fractals are made up of geometric shapes such as squares and triangles as well as lines

1.) Have knowledge of complex numbers.

Complex numbers:

combination of a real number & an imaginary one

real numbers- basically all numbers

imaginary numbers- possibly a number but could be a letter or etc. but when the square root of it is found it equals a negative one

Example: an i is an imaginary number, and by definition i squared is -1

it is a way to put an x & a y into one number

2.) Know how to perform mathematical processes with complex numbers

adding

subtracting

dividing

multiplying

etc.

3.) You go through several iteration processes to create a fractal.

iterations- a repetitive process of utterance

The more iterations you go through the better quality the image will be.

To start you put your complex number into a formula then take the result and put it back into a formula. This would be one iteration.

You continue this process over and over until you create a fractal

The term self-similar can be used to describe fractals.

Fractals are, in fact, self-similar figures.

This means they are figures composed of many more of the same figure to get the figure created.

Some examples would include the Mandelbrot, the Julia sets, and self-affine fractals.

The Mandelbrot set are different nonlinear transitions which create the most famous fractals.

Then there's the Julia Sets which are relatives to the Mandelbert Set.

Self-affine fractals which scale different amounts in x and y directions to produce fractals that are self-affine.

No matter the scale factor of the fractal it will always look the same because it is a huge replica of the beginning shape.

This picture explains how to find the area of the Koch snowflake. This picture only goes through the first three literations but this continues forever

-Fractals are a form of art, but they also relate to mathematics

-Alice Kelley is famous for her fractals.

She is both a mathematician and an artist.

Different computer softwares are used to create fractals

1. make a koch curve ( a line divided up into 3 equal parts)

2. take the middle line and form an equilateral triangle

3. remove the base of the triangle

4. repeat the process

Making a Koch Snowflake