Fang Qi Pu Wan Xue Rui

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Fang QiPu WanXue Rui
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A fractal is a geometric object which is rough or
irregular on all scales of length, and therefore
appears to be 'broken up' in a radical way.
Fractals can be most simply defined as images
that can be divided into parts, each of which is
similar to the original object. Fractals are
said to possess infinite detail, and some of them
have a self-similar structure that occurs at
different scales, or levels of magnification
What is Fractal
Fractal Dragon
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The Dimension of a Fractal
If each initial set had b subsets, each identical
to the initial one. And each subset is 1/a the
size of the initial one, then we define the
dimension of the fractal
For example the Kohn snowflake, here a3, b4,
so the dimension Dclog4/log31.26
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Julia Set
Let R(z) be a rational function, R(z)P(z)/Q(z),
where z is a complex number, and P(z) and Q(z)
are polynomials without common divisors. The
Julia set J is the boundary of the filled-in
set after R(z) is repeatedly applied. There are
two types of Julia sets connected sets
(Mandelbrot set) and unconnected set (Cantor set)
Quadratic Julia sets are generated by the
quadratic mapping
For almost every c, this transformation
generates a fractal. Examples are shown next
for various values of c
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dendrite fractal ci
Douady's rabbit fractal c-0.1230.745 i
San Marco fractal c-0.75
Siegel disk fractal c-0.391-0.587 i
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Newton Fractal
The Newton fractal is a boundary set in the
complex plane which is characterized by Newtons
method applied to a fixed complex polynomial
p(z), where the coefficients are complex number.
It divides the complex plane into regions Gk,
each of which is associated with a root of the
polynomial p(z) Each point of the complex plane
is associated with one of the deg(p) roots of the
polynomial in the following way the point is
used as starting value zo for Newton's iteration
yielding a sequence of points z1, z2, .... If the
sequence converges to the kth root, then zo was
an element of the region Gk. By color Gk in
different colors, we get the Newton Fractal.
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Newton fractal for p(z)z³-1 coloured by root
reached
Newton fractal for p(z) z5 -1 coloured by root
reached shaded by number of iterations required
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Applications of Fractal

• Fractal landscapes
• chemical industry
• Meteorology
• Filters
• Aerosol Deposition Studies
• Stock market
• computer graphics