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  £8.99

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 = r D Take logs: D is dimension D = 1D = 2D = 3 r = 2 r = 3 N = 2 N = 3 N = 4 N = 9 N = 8 N = 27

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.

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 repeats  Finite 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  z n+1  z n 2 +c  z 0 = 0  Stable if |z|<2

Self Similarity of Mandelbrot set Increasing magnification shows embedded ‘copies’ of main set Similar but not identical

The Mandelbrot Monk Udo of Achen AD 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 charge into 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 Art Mandalas

Fractals in Art Visage 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