Examining the World of Fractals

Myles Akeem Singleton Central Illinois Chapter National BDPA Technology Conference 2006 Los-Angeles, CA

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

Introduction to fractals Fractal –Geometric –Self-similar –Has fractional dimension Categorized under chaos science - fractal geometry Benoît Mandelbrot defined the term fractal from the Latin fractus, “broken” or “fractured”

Example of self-similarity

Koch Snowflake iterations

Julia set graphic

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:

Production rules 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

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

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

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

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

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

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

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)

Turtle path example → 5F 90R → 5f → 5F 90R → 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 denotes the part of the turtle's trail that is not specified Moves are represented by the transactions Turtle graphic generated

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