Examining the World of Fractals

1. Examining the World of Fractals

2. Myles Akeem SingletonCentral Illinois Chapter National BDPA Technology Conference 2006Los-Angeles, CA

3. Content of presentation • Introduction to fractals • L-systems/Production rules • Plant images • Turtle geometry • Conclusion

4. Introduction to fractals • 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”

5. Example of self-similarity

6. Koch Snowflake iterations

7. Julia set graphic

8. Introduction to L-systems • 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:

9. Production rules • 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

10. Deterministic, context-free Lindenmayer system (D0L system) Rule 1: a → ab Rule 2: b → a • b → a • a → ab • ab → aba • aba → abaab • abaab → abaababa

11. Ben Hesper and Pauline Hogeweg • 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

12. Koch Island example“F → F + F - F - FF + F + F - F” F → move forward one, drawing + → rotate clockwise by a given angle - → rotate counterclockwise by a given angle

13. Plant images • Adding a cursor stack • system branching is gained • Allows for the creation of plant-like images • Mimics the structure of trees, bushes and ferns

14. Push/pop operations at work Angle 45 Axiom F F = F [ + F ] F

15. Variables, constants, start words, and rules • 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

16. Turtle geometry • 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)

17. <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 Turtle path example

18. Conclusion • 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