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FRACTALS- CURVES

,

t , .

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(

)

Lorentz

Fractals

(Durer)

Escher

(Mandelbrot) -

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Fractals

Fractals

,

Weierstrass

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Weierstrass

1872

Durer

s,

, .

S

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Durer

Durer

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Durer

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Helge von Koch

1. 1/3

,

1/3.

1/3.

Helge Von Koch

Von Koch: 1 =4

r = 1/3. 2 4 (1) 1/3. . .

fractal

D= 1.2619

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Von Koch

Helge Von Koch

Von Koch:

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fractal

, .

Koch .

,

,

Koch . ,

,

Fractals

fractal

-

, Fractal, Fractal

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fractal

:

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Sierpinski

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Sierpinski

D= 1.584

Sierpinski

D=2

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Sierpinski

Sierpinski,

3,

1.

Sierpinski

: =43+41 : =32 -12

:

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Sierpinski

:

Sierpinski

:

!

, .

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Menger

Sierpinski

To Barnsley

.

,

.

,

.

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To Barnsley

(3) .

, Sierpinski!

Julia

:

z2 + c = (x2 - y2 + a) + i (2xy + b)

c = a +ib .

z0= x

0+ i y

0 z1 :

z1= f(x0) = z02 + c

(x1 , y1) = ({x02 - y0

2 + a}, {2 x0 y0 + b})

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Julia

c , (x0, y0) , , . Julia c. Julia (x0, y0)

(x0, y0), .

Julia

Julia fractal

, z0 (

Julia)

zn = zn-12 + c

zn-1=+ zn - c zn-1=- zn - c

c Julia

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Julia

.. c=-0.5+0.5i

Julia

.. c=i

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Julia

Julia f(z)=z8+3z4-4

Mandelbrot

Mandelbrot

(1980)

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Mandelbrot

Mandelbrot c Julia . Mandelbrot , Julia

Julia

Mandelbrot

Mandelbrot

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fractals

IFS - Iterated function system

fractals