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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.

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myles akeem singleton central illinois chapter

Myles Akeem SingletonCentral Illinois Chapter

National BDPA Technology Conference 2006Los-Angeles, CA

content of presentation
Content of presentation
  • Introduction to fractals
  • L-systems/Production rules
  • Plant images
  • Turtle geometry
  • Conclusion
introduction to fractals
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”
introduction to l systems
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
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
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
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
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
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
Push/pop operations at work

Angle 45

Axiom F

F = F [ + F ] F

variables constants start words and rules
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
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
<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
conclusion
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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