Fractals

1
Fractals Grammarsin GraphicsComp 575 - Fall
2008
2
Texturing with Images

• Texture maps with images are good because images
  are (sometimes) easy to create.
• In some cases, just taking a picture is easier
  than writing a program to generate the picture.
• Images are hard to get for things that arent
  real.
• Images consume space.
• Images have finite precision.

3
Procedural Methods.

• Write a program to generate the texture.
• Save space.
• In some cases, can zoom in forever and retain
  precision
• Hard to write the right procedure.
• However, ideas based on fractals go a long way.
• Especially for objects in nature.

4
Euclidean vs. Fractal

• Standard objects are represented in Euclidean
  geometry
• Described by equations
• This gives smooth, regular objects
• Spheres
• Polygons
• B-spline surfaces
• How do we model natural objects?
• Clouds
• Mountains
• Plants
• These kinds of objects are better represented
  with fractal geometry

5
What is a Fractal?

• Benoit Mandelbrot, 1982,
• clouds are not spheres, mountains are not
  cones, coastlines are not circles, and bark is
  not smooth, nor does lightning travel in a
  straight line.
• Objects are represented as procedures rather than
  equations
• Repeating the fractal procedure produces more and
  more detail

6
Properties of Fractals

• Self-similarity between object parts and the
  overall object features
• Fractional dimension
• No specific size or scaling
• Infinite detail looks good at every resolution
• Generated by a function, so no data to store.
• Some of them look like things you find in nature.
  

7
Fractal Mountain
The closer you get, the more detail you see
8
Fractal Generation

• A fractal object is generated by repeatedly
  applying a specified transformation
• Can apply the transformation to a point set, set
  of primitives (lines, curves, colors ,etc.), or
  anything else
• Theoretically, the procedure is applied an
  infinite number of times
• Practically, we only need to iterate a few times.

9
Fractal Dimension (Hausdorff variant)
Split into ¼ the size and multiply by
4 Build original by splitting into 4 parts
and multiplying by 4
Split into ¼ the size And multiply by 16
42 Build original by splitting into 4 parts
and multiplying by 42
Build original by splitting into 4 parts and
multiplying by 43
10
Fractal Dimension (Hausdorff variant)
Split into ¼ the size and multiply by
4 Build original by splitting into 4 parts
and multiplying by 4
Split into ¼ the size And multiply by 16
42 Build original by splitting into 4 parts
and multiplying by 42
Build original by splitting into 4 parts and
multiplying by 43
11
Fractal Dimension (Hausdorff variant)
12
Sierpinski Triangle

• Take a triangle.
• Reduce by ½
• Make 3 copies and arrange them into a triangle
• Repeat forever.

13
Sierpinski Triangle
14
Cantor Set

• Lots of interesting and counter-intuitive
  properties
• Uncountably infinite number of points.
• Lebesgue measure of 0.
• What is its fractal dimension?
• Reduce by 1/3
• Make 2 copies

15
A Few Examples
Penrose tiling 1.974 (a non-periodic tiling)
Cauliflower 2.33
Menger sponge 2.726
http//en.wikipedia.org/wiki/List_of_fractals_by_H
ausdorff_dimension
16
c. 13 branches each 1/3 smaller, D
log(13)/log(3) 2.33
17
Fractals in Nature
http//www.funpike.com/images/plant-bushes/2fracta
l_nature_1.jpg
http//www.treklens.com/gallery/Oceania/New_Zealan
d/photo22113.htm
18
Fractal Classification

• Self-similar fractals
• Parts are scaled-down versions of the initial
  object
• Deterministic self-similar
• Non-random
• Statistically self-similar
• Some randomness introduced
• Self-affine
• Different scaling parameters in each coordinate
  direction
• Invariant Fractal Sets
• Formed with nonlinear transformations
• Sets in the complex plane (usually)

19
Self-Similar Fractals

• Parts are scaled-down versions of the entire
  object
• Start with the initial shape
• Construct sub-parts by duplicating the initial
  shape and scaling it
• Can apply different scaling factors for different
  parts
• Example
• The von Koch snowflake
• Can also introduce random variations to the
  sub-parts
• These fractals are statistically self-similar
• Typically used to model trees, shrubs, and other
  plants

20
Von Koch Snowflake

• Start with initiator
• And generator
• At each iteration, replace each piece of the
  initiator with the generator

21
Von Koch Snowflake

• Iteration 0

22
Von Koch Snowflake

• Iteration 1

23
Von Koch Snowflake

• Iteration 2

24
Von Koch Snowflake

• Iteration 3

25
Other Self-Similar Fractals
26
Peano Curve

• Space-Filling Curve

1st Iteration
2nd Iteration
3rd Iteration
27
Statistically Self-Similar Fractals

• Self-similar fractals in which random variations
  are made to the sub-parts
• Typically used to model trees, shrubs, and other
  plants

Marco Bubkewww.xfrogdownloads.com
Bernd Lintermann www.xfrogdownloads.com
28
L-systems

• Lindenmayer systems
• Invented by a biologist
• Used to describe plant growth
• Formal Grammar
• Rewrite a bunch of symbols in parallel.

29
Fibonacci Numbers as an L-system

• variables  A B
• start   A
• rules   A ? B
• B ? AB
• This L-system produces the following sequence of
  strings
• n 0  A
• n 1  B
• n 2  AB
• n 3  BAB
• n 4  ABBAB
• n 5  BABABBAB
• n 6  ABBABBABABBAB
• n 7  BABABBABABBABBABABBAB
• Count the length of each string 1,1, 2, 3, 5, 8,
  13, 21, 34,

30
Cube-Free

• Is it possible to create a string of 0s and 1s
  where no substring repeats three times?
• No substrings like 000, 010101, or 101101101
• This is called the cube-free property.
• Yes, the Thue-Morse Sequence.
• Under some rules of chess, you can have an
  infinitely long game without a stalemate.

31
Thue-Morse Sequence

• variables  0 1
• start   0
• rules   0 ? 01
• 1 ? 10
• n 0  0
• n 1  01
• n 2  0110
• n 3  01101001
• n 4  0110100110010110
• n 5  0110100110010110 1001011001101001
• n 6  0110100110010110 1001011001101001
• 1001011001101001 0110100110010110

32
Von Koch Snowflake

• Start with initiator
• And generator
• At each iteration, replace each piece of the
  initiator with the generator

33
L-systems Used to Model Plant Growth
34
Fractal Trees
35
Trees using L-systems
36
Modeling Venation Patterns
37
Split Grammars for Buildings
38
Results using Split Grammars
39
Midpoint Displacement
40
Midpoint Displacement
41
Midpoint Displacement
42
Invariant Fractal Sets

• Formed with non-linear transformations
• Includes the self-squaring fractals and the
  self-inverse fractals

43
Mandelbrot Set

• Iterate over a complex function
• Color the point in space based on how quickly the
  function diverges at that point
• Points that do not diverge are in the set
• The set is typically colored black, points that
  diverge at different speeds are colored
  accordingly

44
The Mandelbrot Set
45
Zooming into the Mandelbrot Set
http//www.softlab.ntua.gr/miscellaneous/mandel/ma
ndel.html
46
Computing the Mandelbrot Set
For each pixel on the screen do x x0
// x co-ordinate of pixel y y0 // y
co-ordinate of pixel x2 xx y2 yy
iteration 0 maxIteration 1000 while (
x2 y2 lt (22) iteration lt maxIteration )
y 2xy y0 // Complex
Multiplication x x2 - y2 x0 x2 xx
y2 yy iteration iteration 1
if ( iteration maxIteration ) color
black else color iteration
47
Fractal Mountains

• Start with the basic shape of the mountain
• Subdivide edges of the shape
• Randomly perturb the new vertices
• Repeat recursively
• 2D for coastlines
• 3D for mountains

48
Fractal Terrain Results
49
Terrain
50
Terrain
Other effects like erosion can be added.
51
Terrain
52
Other Uses of Fractals

• Sierpinski Tetrahedron

53
Other Uses of Fractals

• Fractal Brociflower

54
Other Uses of Fractals

• Fractal Chandelier

55
Procedural Textures - Perlin Noise

• Instead of perturbing the geometry perturb
  texture values.
• http//www.noisemachine.com/talk1/20.html
• http//blip.tv/file/760578/