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Examining the World of Fractals

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Examining the World of Fractals

Myles Akeem SingletonCentral Illinois Chapter

National BDPA Technology Conference 2006Los-Angeles, CA

- Introduction to fractals
- L-systems/Production rules
- Plant images
- Turtle geometry
- Conclusion

- Fractal
- Geometric
- Self-similar
- Has fractional dimension

- Categorized under chaos science - fractal geometry
- 1975 - Benoît Mandelbrot defined the term fractal from the Latin fractus, “broken” or “fractured”

- Fibonacci
- Thu-Morse
- Paperfolding
- Dragon curve
- Turtle graphics
- Branching
- Bracketed
- Several biological forms are branched, fragmented, or cellular in appearance and growth
- Example where a trunk emerges from a branch:

- 1968 - biologist Aristid Lindenmayer invents the L-system formula
- Used as a grammar to model the growth pattern of a type of algae
- Set of production rules:
Rule 1: a → ab

Rule 2: b → a

Rule 1: a → ab

Rule 2: b → a

- b → a
- a → ab
- ab → aba
- aba → abaab
- abaab → abaababa

- Two of Lindenmayer’s graduate students
- Tested to see if L - systems could resemble botanic forms
- Images controlled by special characters would draw an image onto a screen
F→move forward one, drawing

f→move forward one, without drawing

+→rotate clockwise by a given angle

-→rotate counterclockwise by a given angle

[→push into stack

]→pop from stack

F→move forward one, drawing

+→rotate clockwise by a given angle

-→rotate counterclockwise by a given angle

- Adding a cursor stack
- system branching is gained
- Allows for the creation of plant-like images

- Mimics the structure of trees, bushes and ferns

Angle 45

Axiom F

F = F [ + F ] F

- Variables - symbols denoting replaceable elements
- Constants - symbols denoting fixed elements
- Start words - define how the system begins
- Rules - define how to replace variables with other variables or constants

- Form of Logo programming
- Created 1967 at BBN, a Cambridge research firm, by Wally Feurzeig and Seymour Papert
Grammar:

nF - “n” steps forward

nB - “n” steps back

aR - turn a degrees right

aL - turn a degrees left

Constants = {nF, nB, aR, aL, Stop}

Variables = {, , , ...}

Start = (none)

<Path>→ 5F 90R <Path>

<Path>→ 5F 90R <Path>

<Path>→ 5F 90R <Path>

<Path>→ 5F 90R <Path>

<Path>→ 5f <Path>

<Path>→ 5F 90R <Path>

<Path>→ 5F 90R <Path>

<Path>→ 5F STOP

Production rules:

F→ move forward, drawing

F→ move forward, without drawing

nF→ “n” steps forward

nB→ “n” steps back

aR→ turn “a” degrees right

aL→ turn “a” degrees left

<Path> denotes the part of the turtle's trail that is not specified

Moves are represented by the transactions

Turtle graphic generated

- Fractal uses
- Model many different objects and shapes
- Scientific modeling
- Creating graphic designs for clothes
- Multimedia
- 3-D artwork

- Music pioneers of this research are learning how to apply the application of fractals to create new styles of music
- Uses a recursive process
- Algorithm is applied multiple times to process its previous output
- Provides very abstract musical results
- Becoming one of the most exciting fields of new music research

- The limits of fractal will continue to stretch