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MANDELBROT

&

JULIASETS

PRESENTED BY:

JONATHAN ROSS

JACOB MAUCK

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S

IMPLY

C

OMPLEX

zn+1

= zn

2 + c

Mandelbrot and Julia sets are complex iterations of z2 + c wremaining a fixed constant

Mandelbrot sets: iterate with z0

= 0

For Julia sets: z0

varies

- The starting value of z not only determines the limit but happroaches it

Jonathan RossJacob Mauck

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GASTON JULIA

February 3, 1893 March 19, 1978

Born in Sidi Bel Abbes, Algeria

While serving as a conscript in the French Army during WWhe sustained an injury that resulted in the loss of his nose

His first article on the iteration of rational functions was pub1918 and was widely celebrated within the mathematical co

He received the Grand Prix of the Academy of Sciences awawork

Although his work was widely celebrated, it went largely oveuntil the discovery of Mandelbrot Sets

Jonathan RossJacob Mauck

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BENOIT MANDELBROT

November 20, 1924 October 14, 2010

Born in Warsaw, Poland and emigrated to France in 1936

Studied under Gaston Julia at cole Polytechnique from 1945-1947 Employed by IBM in 1958, he remained for 35 years and was one of th

use computer graphics to create fractal images.

In 1975 he coined the term fractal as a means to describe repeatingsimilar mathematical patterns.

In 1979 IBM gave him the task of solving signal interferences occurringcomputer data transfers over phone lines

The Mandelbrot Sets were the result of his research

Jonathan RossJacob Mauck

A cloud is made of billows upon billows upon billows that look like clouds. As you

closer to a cloud you don't get something smooth, but irregularities at a smaller sc

-Benoit Ma

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FIXED-POINT BOUNDARIES

Jonathan RossJacob Mauck

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ATTRACTIVE, REPELLING INDIFFERENT PO

Jonathan RossJacob Mauck

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THE MAIN CARDIOID

The boundary between attractive fixed points and repelling points is the set of indifferent fixed points and is represente

|(

)(z2 + c)| = |2z| = 1

Jonathan RossJacob Mauck

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COMBINATORICS PRINCIPLE SERI

Each n-cycle is represented by a disc around the main cardio

Upper principle series these are located above the top halmain cardioid

Lower principle series these are located below the bottomthe main cardioid

Each cycle is represented by

a number (2, 3, 4)

Jonathan RossJacob Mauck

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Jonathan RossJacob Mauck

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Jonathan RossJacob Mauck

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Jonathan RossJacob Mauck

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Jonathan Ross Jacob Mauck CMDA 3605 Matthias Chung

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