Examining the World of Fractals

Myles Akeem SingletonCentral Illinois Chapter

National BDPA Technology Conference

2006Los-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

• 1975 - 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• 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

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, drawingf → 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 workAngle 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 forwardnB - “n” steps backaR - turn a degrees rightaL - turn a degrees leftConstants = {nF, nB, aR, aL, Stop}Variables = {, , , ...}Start = (none)

Turtle path example<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, drawingF → move forward, without drawingnF → “n” steps forwardnB → “n” steps backaR → turn “a” degrees rightaL → 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

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