Fractals in nature

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Fractals in nature
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A fractal fern
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A fractal tree
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How to grow a digital tree?
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A fractal is an object with a fractional
dimension!
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0.6039
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Other example of fractal Kochs snowflake
Dlog4/log31.261
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Self-similarity in Kochs curve
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Two classic examples of fractal the Julia set
and the Mandelbrot set
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How to create a Julia set? Consider the
map f z --gt z2 c where z x iy (x, y)
and c a ib (a, b) is a parameter in the
mapping. It is equivalent to the two-dimensional
map (Polar coordinate) r ei?--gt r2 e2i? c
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This map of the complex numbers is equivalent to
3 successive transformations on the complex
plane.
Stretch points inside the unit circle towards the
origin. Stretch points outside towards infinity
Cut along the positive x-axis. Wrap the plane
around itself once by doubling every angle.
Shift the plane over so the origin lies on (a, b).
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Despite all this stretching, twisting, and
shifting there is always a set of points that
transforms into itself. Such sets are called the
Julia sets (after the French mathematician Gaston
Julia who discovered them in the 1910s.) The
Julia set for c (0, 0) is easy to find the set
is the unit circle. For other values of c we
need a computer to find out the fixed points
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Examples of the Julia set on z plane

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A Julia set is either totally connected
or totally disconnected!
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Self-similarity of the Julia set
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An artistic visualization of the Julia set
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Whether a Julia set is connected or not depends
on the parameter c. Plot the Julia sets for all
parameter values c. If the value of c makes the
Julia set connected, then we say this c belongs
to the Mandelbrot set. We can plot the Mandelbrot
set on the c plane. (Note the Julia set is
defined on the z plane) Examine the Julia set to
determine whether it is connected or not takes a
long time. Luckily, we need to study only one
point in the z plane the origin If the origin
never escapes to infinity then it is either a
part of the Julia set or is trapped inside it. In
both cases, the Julia set is connected.
(Mandelbrot) (Note If the origin is part of the
set, the set is dendritic (branch-like). If it is
trapped inside the set, the set is topologically
equivalent to a circle.)
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Mandelbrot set on the c plane
(x,y)(-2,0)
(x,y)(1/4,0)
(x,y)(-3/4,0)
(x,y)(0,0)
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Mandelbrot set and the bifurcation diagram!
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8
3
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The first computer print-out of the Mandelbrot set
All the islands in the set are connected!!
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The fascinating universe of the Mandelbrot set
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The end
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Bulbs with different periods
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Period 3
3
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Period 4
4
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Period 5
5
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Period 7
7
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You can find thousands of artistic fractals on
the web, for example...
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etc...