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Fractals and iterated algoritms

An overview of fractals and iterated algorithms in order to understand key concepts of Tom Stoppard's Arcadia

Sue Doodle

on 30 November 2012

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Transcript of Fractals and iterated algoritms

Fractals and iterated algorithms To understand Tom Stoppard's Arcadia Timeline Generating fractals What is a fractal? A small definition “A rough or fragmented geometric shape that can be subdivided in parts, each of which is approximately a reduced size copy of the whole” Benoît Mandelbrot In other words... Fractals Graphically represent the behavior of an equation Look similar at any scale Are also present in nature Romanesco broccoli Fern The term was coined in 1975 by the French mathematician who also developed the chaos theory. In Latin, "fractus" means "broken" A few noteworthy fractals... The Mandelbrot set Sierpinski carpet Space-filling curve The Julia set Apollonian gasket Sierpinski triangle Using an iterated algorithm Complex fractals: many iterations Issue of time 2D and 3D fractals Helps to predict changes Koch snowflake Population Biology What is population biology? A study of population organisms So what is population ecology? It is the study of how the population sizes of species living together in groups change over time and space. In Arcadia A parallel between Thomasina and Valentine "She started with an equation and turned it into a graph" "I've got a graph and I'm trying to find the equation." Valentine's Goldfish ? X Y Formulas Fk(x) = kx(1-x) X is the population
K is a constant “A population will grow (or decline) exponentially as long as the environment experienced by all individuals in the population remains constant.”
Thomas Malthus Fermat's last theorem Pierre de Fermat A lawyer
Amateur mathematician
Now considered a great theorist The Conjecture I have a truly marvelous demonstration of this proposition... ...Which this margin is too narrow to contain. Proving Fermat's last theorem Not proved until 1995
Academics encouraged the research greatly There were many steps to proving this theorem from 1665 to 1995 Sophie Germain Andrew Wiles' general proof
in 1994... ...358 years after it was conjectured. Is Fermat's theorem useful? By itself no
A very important specific case : Pythagore's theorem
Influence on the study of elliptic curves So, did Fermat have the proof? Probably not. Reference in Arcadia Act 1 scene 1 Thank you!
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