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

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 James Gleick
 Vintage
 ISBN
 £8.99
 http://www.around.com

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 "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 reducedsize
copy of the whole : selfsimilarity

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 One of simplest fractals
 Start with line
 Replace centre 1/3 with 2 sides of D
 Repeat

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 Start with equilateral triangle
 Apply Koch curve to each edge
 Perimeter increases by 4/3 at each iteration à ¥
 Area bounded by circle

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 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

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 In Euclidian geometry, the dimension is always an integer.
 For fractals, the dimension is usually a fraction.

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 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.

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 Lewis Fry Richardson
 Relationship between length of national boundary and scale size

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 System has boundary between stable and chaotic behaviour
 Boundary is fractal in nature
 Strange attractor
 Never repeats
 Finite volume of phase space
 Infinite length
 è Fractal in nature

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 First Pictures 1978
 Explored 1980s B.B.Mandelbrot
 Stability of iterated function
 z_{n+1 }ß z_{n}^{2}+c
 z_{0} = 0
 Stable if z<2

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

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 Udo of Achen
 12001270AD
 Nativity scene
 Discovered by Bob Schpike 1999

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 Fractal antennae for radio comms
 Many length scales à
broadband

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