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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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Examining the World of Fractals

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Examining the world of fractals

Examining the World of Fractals

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”

Example of self similarity

Example of self-similarity

Koch snowflake iterations

Koch Snowflake iterations

Julia set graphic

Julia set graphic

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


    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



  • 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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