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
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