**Fractals**

**By Stephen Jevons, Vahid Kheirollah, and Nick Paras**

Images

1: http://www.fractalsciencekit.com/fractals/large/Fractal-Mobius-Dragon-IFS-04.jpg

2: http://www.fi.muni.cz/~xpelanek/IV104/jaro10/fraktaly/kapradi.jpg

3: http://clowder.net/hop/Keplrfrct/Octbubbl.gif

4: http://fractalfoundation.org/wp-content/uploads/2010/05/kochprog440.jpg

5: http://math.bu.edu/DYSYS/chaos-game/sierp-det.GIF

6: http://math.bu.edu/DYSYS/chaos-game/sierp-det.GIF

7: http://upload.wikimedia.org/wikipedia/commons/d/de/Menger_sponge_(Level_0-3).jpg

8: http://paulbourke.net/fractals/fracintro/fracintro47.gif

9: http://upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Sierpinski_triangle_evolution_square.svg/680px-Sierpinski_triangle_evolution_square.svg.png

10: http://www.ex-tempore.org/Pastille/past.ex4.jpg

11: http://fractalfoundation.org/OFCA/JefDuncanLightning.jpg

12: http://www.fractalsciencekit.com/fractals/large/Fractal-Mobius-Dragon-IFS-10.jpg

13: http://www.enchgallery.com/fractals/fractal%20images/fractal%20flame%20enchasketch.jpg

14: http://4.bp.blogspot.com/-ZXbpMyvYLrY/UMTWbnS8ZaI/AAAAAAAAATA/H9Ae0MyUjmE/s1600/fractal-landscape.jpg

15: http://thereisnocavalry.files.wordpress.com/2012/08/dscn7165.jpg

Bibliography

Burger, E. B., Starbird, M. 2013. The Heart of Mathematics: An Invitation to Effective Thinking (4th ed). Hoboken: John Wiley & Sons, Inc.

Fractal Foundation. What are Fractals? 13 April 2014. <http://fractalfoundation.org/>

Fractals in nature and applications.

13 April 2014. <http://kluge.in-chemnitz.de/documents/fractal/node2.html>

What are Fractals?

Fractals are infinitely repeating patterns. Each smaller section is a copy of the entire shape.

Self-Similarity

Having the same pattern under varying scales.

Infinite Replacements:

Koch Curve

Begin with a line. Then split it into 4 equal sections. Continue splitting the segments.

Infinitely continued additions of line segments.

Each line of the triangle

is split into 4 same size pieces.

Start

4 Segments

16

64

256

1024

Koch Snowflake

Sierpinski Triangle

Begin with a triangle.

Replace the middle triangle with an empty triangle. Notice the self-similarity with each sub-triangle.

Barnsley Fern Zoom:

Collages

Take an image and make various copies of different sizes. Use that collage and add it to a new collage of the same image at even smaller sizes. Continue creating collagception.

Collage of Triangles

Using the collage rules, we can see

the Sierpinski triangle emerging from

other images. If we start with a square and continue to split it into 3 separate squares we see the Sierpinski triangle emerge.

The Chaos Game

Uses in Real Life

Fractals show us how small repetitions can lead to very large outcomes. Fractals can be used to model complex patterns in astronomy, nature, and computer science.

Take an equilateral triangle and label the three vertices. Start at one and then choose another at random. Place a dot halfway between the two vertices. Choose a new vertex and repeat the process. A pattern will begin to emerge.

Synthetic and Imaginary Fractals

Natural Fractals

Menger Sponge

Start with a cube. "Punch out" all the cores of the cube. This leaves 20 sub-cubes. Continue this to get 400 and then 160,000 sub-cubes.