Introducing Fractals Introducing Fractals
A Graphic Guide
Published on 5. June 2014
176 pages
978-1-84831-783-3 (ISBN)
Description
Fractals are the geometry of the natural world. They're about the broken, wrinkled, wiggly world- the uneven shapes of nature, unlike the idealised forms of Euclidean geometry. We see fractals everywhere; indeed, we are fractals ourselves.
Fractal geometry is an extension of classical geometry which can make precise models of physical structures, from ferns to galaxies. It can describe the shape of a cloud as precisely as an architect can describe a house.
Introducing Fractals traces the historical development of this mathematical discipline, explores its descriptive powers in the natural world, and then looks at the applications and the implications of the discoveries it has made.
As John Archibald Wheeler, protégé of Niels Bohr, friend of Albert Einstein and mentor of Richard Feynman has said, 'No one will be considered scientifically literate tomorrow, who is not familiar with fractals.'
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09/2009
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Persons
Nigel Lesmoir-Gordon is a producer of television documentaries.
Will Rood studied mathematics at Cambridge University. His fractal animations have graced many television documentaries and his artwork has featured on numerous magazines, posters and CD sleeves.
Ralph Edney trained as a mathematician, and has worked as a teacher, journalist, illustrator and political cartoonist.
Content
- Intro
- Title Page
- Copyright
- Contents
- Why Do Fractals Matter?
- A Smooth World or a Rough One?
- The Texture of Reality
- The Origins of Fractals
- Classical Geometry
- The Calculus
- The Paradox of Infinitesimals
- Effects of Calculus
- The First Fractal
- Explaining Numbers
- Firm Foundations and Sets
- What Are Sets?
- Cantor and the Continuum
- The Cantor Set
- Peano's Space-Filling Curve
- Topological and Fractal Dimension
- Self-Similarity
- The Koch Curve
- Similarity Dimension
- Similarity and Fractal Dimension
- Measuring Fractal Dimension
- Lewis Richardson
- How Long is a Coastline?
- The Sierpinski Gasket
- The Chaos Game
- Strange Attractor
- Pascal's Triangle
- Basins of Attraction
- Poincaré and Non-Linearity
- Malthus and Population Growth
- Negative Feedback
- The Logistic Equation
- Iteration
- Video Feedback
- Robert May and the Verhulst Model
- Bifurcation Points
- The Period-Doubling Cascade
- The Fig Tree
- Chaos Theory and Fractals
- The Feigenbaum Constant
- Real Numbers, Other Numbers
- Complex Numbers
- The Complex Plane
- Gaston Julia and Pierre Fatou
- Benoît Mandelbrot
- Uncle Szolem
- A Practical Education
- The Shape of Things
- The Wiles of Bourbaki
- On the Run Again
- The More the Merrier
- Emerging Patterns
- Fractals in Practice
- Bad Noise
- Fractal Errors
- A General Principle Emerges
- The Simplest Possible Transformation
- Two Types of Julia Sets
- A Map of Julia Sets
- A Whole New World
- Simple Rules, Complex Behaviour
- Newton's Law and Chaos
- Let's Admit - It is Complicated .
- Phase Transitions
- Hazy Areas of Calculation
- The Mathematics of Wrinkles
- Natural Templates
- Forest Fires: the Fractal Boundary
- More Phase Transitions
- Electro-plated Fractals
- Viscous Fingering
- Growth by Aggregation
- Brownian Motion and the Quantum World
- Dimensional Magic
- The Three-Quarter Power Law
- The Mysterious Brain
- Knowing the Mind of God?
- Fractals and Medical Research
- Viruses and Bacteria
- AIDS
- Detecting Cancer
- Women at Risk
- Bubbly Bones and Breaks
- Fractal Beats
- Practical Solutions
- Striking Oil
- The Spring is Sprung
- Stress Loadings
- Detection
- Fractal Ecology
- Orchid Fractals and Crowds
- Olbers' Paradox
- The Great Wall
- The Big Bang
- Cosmic Connections
- How Do We See?
- Fractal Image Compression
- Affine Transformations
- Morphogenesis
- Satellites
- The Money Markets
- What Are the Rules?
- Market Self-Affinity
- Generators
- Beyond Portfolio Theory
- Multifractals
- Fractals in Art: Mandalas
- Decorative Patterns: Self-Similarity
- Scaling and Repetition
- Cues in the Landscape
- Fractals in Music
- Dysfunction in Modern Architecture
- Organic Architecture
- Fractal Traditions
- Strength in the Details
- Fracology
- Quo Vadis?
- Not Everyone Agrees .
- The Special Talent of Mandelbrot
- The Order Is Out There
- Further Reading
- Acknowledgements
- Biographies
- Index
