Fractals

Weird Dimensions

and

Procedural Landspaces

NAME: IShan Bhanuka

ID: 2016A7ps0075p

Sierpinski Gasket

Generated using fractal trees

Koch snowflake

Made up of equilateral triangles

ELEGANT FRACTALS

BUT WHAT ARE THEY?

  • selF SIMILAR SHAPES

  • recursive structures

  • shapes with non-integral dimension?!

  • all of the above!!

SAME STRUCTURE RE-APPEARS at different scales

MATHEMATICALLY, "some" quantity remains unchanged when changing scale

dynamic quantity=f(x,t)x/tz=cdynamic\ quantity = f(x, t) \\ x / {t^z} = c
dynamic\ quantity = f(x, t) \\ x / {t^z} = c

Zooming in on a Kock curve

-SIMILAR

SELF-

RECURSIVE STRUCTURE

var angle = 0;
var slider;

function setup() {
  createCanvas(400, 400);
  slider = createSlider(0, TWO_PI, PI / 4, 0.01);
}

function draw() {
  background(51);
  angle = slider.value();
  stroke(255);
  translate(200, height);
  branch(100);
}

function branch(len) {
  line(0, 0, 0, -len);
  translate(0, -len);
  if (len > 4) {
    push();
    rotate(angle);
    branch(len * 0.67);  // left branch
    pop();
    push();
    rotate(-angle);
    branch(len * 0.67);  // right branch 
    pop();
  }
}

The SUB-STRUCTURE Can be recursive generated. In the fractal tree, each branch is a fractal tree itself

LET's UNDERSTAND DIMENSION AGAIn

NON-INTEGER DIMenSION?!

LENGTH = 1
MASS = 1
SCALE = 1

LENGTH CHANGES WITH SCALE

MASS CHANGES WITH SCALE BUT WHAT IS THE AMOUNT OF CHANGE?

LENGTH = 1/21/2
MASS = 1/21/2
SCALE = 2

LENGTH = 1 MASS = 1
SCALE =1

LENGTH = 1/21/2 MASS = 1/41/4
SCALE = 2

LENGTH = 1
MASS = 1
SCALe = 1

LENGTH = 1/21/2
MASS = 1/31/3 ??
SCALe = 1/21/2

SCALE∗LENGTH=SCALE1∗MASSSCALE * LENGTH = SCALE ^ 1 * MASS

SCALE∗LENGTH=SCALE1.58∗MASSSCALE * LENGTH = SCALE ^ {1.58} * MASS

SCALE∗LENGTH=SCALE2∗MASSSCALE * LENGTH = SCALE ^ 2 * MASS

fORMULA

SCALE∗LENGTH=SCALEz∗MASSSCALE * LENGTH = SCALE ^ {z} * MASS
WHERE Z IS THE DIMENSION OF THE SHAPE

FRACTALS FOR LANDSCAPE GENERATION

XN+1=f(XN)X_{N+1} = f(X_N)
XN+2=f(XN+1)X_{N+2} = f(X_{N+1})

THE LANDSCAPE CAN BE RECURSIVELY DIVIDED BY PROCESSING ITSELF

ADD COMPLEXITY BY SUMMING TOGETHER DIFFERENT LAYERS

FILTER OUT OUTLIERS FOR A NATURAL TRANSITION

THANK YOU

REFERENCES

  1. Recursive fractal trees

  2. Generating landscapes with fractals

  3. Fractal dimension

Fractals - Weird dimensions and Procedural Landscapes

By Ishan Bhanuka

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Fractals - Weird dimensions and Procedural Landscapes

Presentation for Computer Graphics course assignment

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