chaos, communication and consciousness module ph19510
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Chaos, Communication and Consciousness Module PH19510. Lecture 15 Fractals. Overview of Lecture. What are Fractals ? Fractal Dimensions How do fractals link to chaos ? Examples of fractal structures. Chaos – Making a New Science. James Gleick Vintage ISBN 0-749-38606-1 £8.99 - PowerPoint PPT PresentationTRANSCRIPT
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Chaos, Communication and ConsciousnessModule PH19510
Lecture 15
Fractals
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Overview of Lecture
What are Fractals ? Fractal Dimensions How do fractals link to chaos ? Examples of fractal structures
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Chaos – Making a New Science
James Gleick Vintage ISBN
0-749-38606-1
£8.99 http://www.around.com
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What are Fractals ?
"Clouds are not spheres, coastlines are not circles, bark is not smooth, nor does lightning travel in straight lines" - B.B. Mandelbrot
Fractals are rough or fragmented geometric shapes that can be subdivided into parts, each of which is exactly, or statistically a reduced-size copy of the whole : self-similarity
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The Koch curve
One of simplest fractals
Start with line Replace centre 1/3
with 2 sides of Repeat
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The Koch Snowflake
Start with equilateral triangle
Apply Koch curve to each edge
Perimeter increases by 4/3 at each iteration
Area bounded by circle
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Dimensions of Objects
Consider objects in 1,2,3 dimensions
Reduce length of ruler by factor, r
Quantity increases by N = rD
Take logs:
D is dimension
D = 1 D = 2 D = 3
r = 2
r = 3
N = 2
N = 3
N = 4
N = 9
N = 8
N = 27
rN
Dlog
log
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Fractal Dimensions "How long is the coast of Britain?" In Euclidian geometry, the dimension is
always an integer. For fractals, the dimension is usually a fraction.
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using a ruler of length L (green) - total length = 3L
using a ruler of length 3
L (red) - total length = 4L
using a ruler of length 9
L (blue) - total length =
3L16
To find the fractal dimension, either plot a graph of log(total length) against log(ruler length) - the gradient is (1-D)
Or 26134rND .)log()log(loglog
Fractal Dimension of Koch Snowflake
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Coastlines and Fractal Dimensions
Coastlines are irregular, so a measure with a straight ruler only provides an estimate.
The ruler on the right is half that used on the left, but the estimate of L on the right is longer.
If we halved the scale again, we would get a similar result, a longer estimate of L.
In general, as the ruler gets diminishingly small, the length gets infinitely large.
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Coastlines and Fractal Dimensions
Lewis Fry Richardson Relationship between length of national
boundary and scale size
• Linear on log-log plot
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Fractals and Chaos
System has boundary between stable and chaotic behaviour
Boundary is fractal in nature Strange attractor
Never repeatsFinite volume of phase space Infinite length Fractal in nature
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The Mandelbrot set
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The Mandelbrot Set
First Pictures 1978 Explored 1980s B.B.Mandelbrot Stability of iterated function
zn+1 zn2+c
z0 = 0
Stable if |z|<2
z
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Self Similarity of Mandelbrot set
Increasing magnification shows embedded ‘copies’ of main set
Similar but not identical
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The Mandelbrot Monk
Udo of Achen 1200-1270AD Nativity scene Discovered by Bob
Schpike 1999
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Fractals in Nature
Electrical Discharge from Tesla Coil
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Fractals in Nature
Lichtenberg Figure
Created by exposing plastic rod to electron beam & injecting chargeinto material. Discharged by touching earth connector to left hand end
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Fractals in Nature
Fern grown by nature Ferns grown in a computer
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Fractals in Nature
Romanesco
(a cross between broccoli and Cauliflower)
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Fractals in Nature
Blood vessels in lung
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Growth of mould
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Fractals in ArtMandalas
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Fractals in ArtVisage of War
Salvador Dali (1940)
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Fractals in Technology
Fractal antennae for radio comms
Many length scales broadband
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Review of Lecture
What are Fractals ? Fractal Dimensions How do fractals link to chaos ? Examples of fractal structures