Chaos, Communication and ConsciousnessModule 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 http://www.around.com

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

The Koch curve

One of simplest fractals

Start with line Replace centre 1/3

with 2 sides of Repeat

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

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

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.

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

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.

Coastlines and Fractal Dimensions

Lewis Fry Richardson Relationship between length of national

boundary and scale size

• Linear on log-log plot

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

The Mandelbrot set

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

Self Similarity of Mandelbrot set

Increasing magnification shows embedded ‘copies’ of main set

Similar but not identical

The Mandelbrot Monk

Udo of Achen 1200-1270AD Nativity scene Discovered by Bob

Schpike 1999

Fractals in Nature

Electrical Discharge from Tesla Coil

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

Fractals in Nature

Fern grown by nature Ferns grown in a computer

Fractals in Nature

Romanesco

(a cross between broccoli and Cauliflower)

Fractals in Nature

Blood vessels in lung

Growth of mould

Fractals in ArtMandalas

Fractals in ArtVisage of War

Salvador Dali (1940)

Fractals in Technology

Fractal antennae for radio comms

Many length scales broadband

Review of Lecture

What are Fractals ? Fractal Dimensions How do fractals link to chaos ? Examples of fractal structures