Fractals ChinSung Lin Eleanor Roosevelt High School Fractals

Fractals Chin-Sung Lin Eleanor Roosevelt High School

Fractals • What is Fractals? • Mandelbrot Set

Mandelbrot Set

Mandelbrot Set • A set of values of c in the complex plane. • Plug into the complex quadratic polynomial Zn+1 = Zn 2 + c where Z 0 = 0 • Generate a sequence Z 0, Z 1, Z 2, …… Zn, …… • Z 0 = 0 Z 1 = Z 0 2 + c Z 2 = Z 12 + c = (Z 02 + c)2 + c Z 3 = Z 22 + c = (Z 12 + c)2 + c = ((Z 02 + c)2 + c …………. Zn+1 = Zn 2 + c = ……………………. y (Im) (cx , cy ) c = c x + i cy x (Re) Complex Plane

Mandelbrot Set • Zn = Znx + i Zny • c = c x + i cy • Zn+1 = Zn 2 + c y (Im) = (Znx + i Zny)2 + (cx + i cy) = (Znx 2 – Zny 2) + i (2 Znx. Zny) + (cx + i cy) = (Znx – Zny + cx) + i (2 Znx. Zny + cy) 2 2 • Z(n+1)x = Znx 2 – Zny 2 + cx • Z(n+1)y = 2 Znx. Zny + cy (cx , cy ) c = c x + i cy x (Re) Complex Plane

Mandelbrot Set • A set of values of c in the complex plane. • Plug into the complex quadratic polynomial Zn+1 = Zn 2 + c where Z 0 = 0 • c = 1 gives the sequence 0, 1, 2, 5, 26, …, which tends to infinity. c = − 1 gives the sequence 0, − 1, 0, …, which is bounded. y (Im) (Zx , Zy ) Z 2 Zo Z 3 Z 1 Zn = Znx + i Zny x (Re) Complex Plane

Mandelbrot Set • Under iterations repeatedly, Zn remains bounded however large n gets. • “Bounded means | Z | ≤ 2. • | Zn | = sqrt ( Znx 2 + Zny 2) or y (Im) (Zx , Zy ) 2 Znx 2 + Zny 2 ≤ 4 Zn = Znx + i Zny Z 2 Zo Z 3 Z 1 2 x (Re) Complex Plane

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