GVC-432
Lecture –13
Fractals
Ref: Donald Hearn & M. Pauline Baker ,
Computer Graphics (Chapter – 10.18 )
Foley, van Dam, Feiner & Hughes,
Computer Graphics Principles & Practice
Dr Pavan Chakraborty
IIIT-Allahabad
Indian Institute of Information Technology - Allahabad
What are fractals?
• Iterative patterns
• Infinitely complex
• Appear in nature
– Leaves
– Snowflakes
– DNA molecules
Examples of fractals in nature
The Ship Fractal
3-D Fractals
Non-Fractal
Fractal
Non - Fractal
Size of Features
1 cm
1 characteristic
scale
Fractal
Size of Features
2 cm
1 cm
1/2 cm
1/4 cm
many different scales
Fractals
Self-Similarity
Self-Similarity
Pieces resemble the whole.
Water Water Water
Land Land
Land
Sierpinski Triangle
Branching Patterns
blood vessels air ways
in the retina in the lungs
Family, Masters, and Platt 1989 West and Goldberger 1987
Physica D38:98-103 Am. Sci. 75:354-365
Mainster 1990 Eye 4:235-241
Blood Vessels in the Retina
Modeling
• Geometric
–Meshes
–Hierarchical
–Curves and Surfaces
• Procedural
–Particle Systems
–Fractal
Box Fractal
The box fractal is a fractal also called the anticross-stitch
curve which can be constructed using string rewriting
beginning with a cell [1] and iterating the rules
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Box Fractal
An outline of the box fractal can encoded as a Lindenmayer
system with initial string "F-F-F-F", string rewriting rule "F" ->
"F-F+F+F-F", and angle (J. Updike, pers. comm.,
Oct. 26, 2004).
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Let be the number of black boxes, the length of a side of a
white box, and the fractional area of black boxes after the th
iteration.
(2)
(3)
(4)
(5)
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The sequence is then 1, 5, 25, 125, 625, 3125, 15625, ...
(Sloane's A000351). The capacity dimension is therefore
(6)
(7)
(8)
(9)
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Sierpinski Gasket
Rule based:
Repeat n times. As n →∞
Area→0
Perimeter →∞
Not a normal geometric object
Coastline Problem
What is the length of the coastline of England?
Answer: There is no single answer
Depends on length of ruler (units)
If we do experiment with maps at various
scales we also notice self-similarity
each part looks a whole
Fractal Geometry
• Created by Mandelbrot
– Self similarity
– Dependence on scale
• Leads to idea of fractional dimension
• Graftals: graphical fractal objects
Koch Curve/Snowflake
Fractal Dimension
• Start with unit line, square, cube which
we should agree are 1, 2, 3 dimensional
respectively under any reasonable
dimension
• Consider scaling each one by a h = 1/n
How Many New Objects?
• Line: n
• Square: n2
• Cube: n3
• The whole is the sum of its parts implies
=1 d =
Examples
• Koch Curve
– Scale by 3 each time
– Create 4 new objects
– d = ln 4 / ln 3 = 1.26186
• Sierpinski Gasket
– Scale by
– Create 3 new objects
– d = ln 3 / ln 4 = 1.58496
Volumetric Examples
d = ln 4/ ln 2 = 2
D = ln 20 / ln 3 = 2.72683
Midpoint subdivision
Randomize displacement using a Gaussian
random number generator. Reduce
displacement each iteration by reducing
variance of generator.
Fractal Brownian Motion
-(2-d)
variance ~ length
Brownian motion d = 1.5
Fractal Mountains
Iteration in the Complex
Plane
Mandelbrot Set
iterate on zk+1=zk2+c
with z0 = 0 + j0
Two cases as k →∞
|zk |→∞
|zk | remains finite
If for a given c, |zk | remains finite, then c belongs to
the Mandelbrot set
Mandelbrot Set
Mandelbrot Set
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Indian Institute of Information Technology - Allahabad
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An Escheresque fractal by Peter Raedschelders.
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Recursion Trees
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Sierpinski Fractals
Julia and Mandelbrot Sets znew = zold² + c
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Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
Indian Institute of Information Technology - Allahabad
