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Chaos, Communication and Consciousness Module PH19510

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

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Chaos, Communication and Consciousness Module PH19510

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  1. Chaos, Communication and ConsciousnessModule PH19510 Lecture 15 Fractals

  2. Overview of Lecture • What are Fractals ? • Fractal Dimensions • How do fractals link to chaos ? • Examples of fractal structures

  3. Chaos – Making a New Science • James Gleick • Vintage • ISBN • 0-749-38606-1 • £8.99 • http://www.around.com

  4. 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

  5. The Koch curve • One of simplest fractals • Start with line • Replace centre 1/3 with 2 sides of  • Repeat

  6. 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

  7. D = 1 D = 2 D = 3 r = 2 N = 2 N = 4 N = 8 r = 3 N = 3 N = 9 N = 27 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

  8. 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.

  9. Fractal Dimension of Koch Snowflake

  10. 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.

  11. Coastlines and Fractal Dimensions • Lewis Fry Richardson • Relationship between length of national boundary and scale size • Linear on log-log plot

  12. 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

  13. The Mandelbrot set

  14. 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

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

  16. The Mandelbrot Monk • Udo of Achen • 1200-1270AD • Nativity scene • Discovered by Bob Schpike 1999

  17. Fractals in Nature Electrical Discharge from Tesla Coil

  18. 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

  19. Fractals in Nature Fern grown by nature Ferns grown in a computer

  20. Fractals in Nature Romanesco (a cross between broccoli and Cauliflower)

  21. Fractals in Nature Blood vessels in lung

  22. Growth of mould

  23. Fractals in ArtMandalas

  24. Fractals in Art Visage of War Salvador Dali (1940)

  25. Fractals in Technology • Fractal antennae for radio comms • Many length scales  broadband

  26. Review of Lecture • What are Fractals ? • Fractal Dimensions • How do fractals link to chaos ? • Examples of fractal structures

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