Fractal Worlds
Grown, Built, and Imagined
1st Edition
Published on 21. June 2016
512 pages
978-0-300-22070-4 (ISBN)
Description
Fractal geometry is a uniquely fascinating area of mathematics, exhibited in a range of shapes that exist in the natural world, from a simple broccoli floret to a majestic mountain range. In this essential primer, mathematician Michael Frame-a close collaborator with Benoit Mandelbrot, the founder of fractal geometry-and poet Amelia Urry explore the amazing world of fractals as they appear in nature, art, medicine, and technology. Frame and Urry offer new insights into such familiar topics as measuring fractal complexity by dimension and the life and work of Mandelbrot. In addition, they delve into less-known areas: fractals with memory, the Mandelbrot set in four dimensions, fractals in literature, and more. An inviting introduction to an enthralling subject, this comprehensive volume is ideal for learning and teaching.
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06/2016
Yale University Press
€24.75
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Persons
Michael Frame taught mathematics at Yale University for twenty years. Amelia Urry is a journalist and poet.
Content
- Cover
- Half Title
- Title
- Copyright
- Contents
- Foreword
- Introduction
- 1 What is the shape of the world aroundus?
- 1.1 Types of symmetries
- 1.2 Symmetry under magnification
- 1.3 The Sierpinski gasket
- 1.4 Fractals with memory
- 1.5 Self-affine fractals
- 1.6 Nonlinear fractals
- 1.7 Random fractals
- 1.8 Some non-fractals
- 1.9 Physical world examples
- 2 Self-similarity in geometry
- 2.1 A simple way to grow fractals
- 2.2 Some classical fractals
- 2.3 Fractal trees and ferns
- 2.4 Fractal spirals
- 2.5 IFS with memory
- 2.6 Random rendering of fractal images
- 2.7 Circle inversion fractals
- 2.8 Random fractals
- 2.9 And flavors stranger still
- 3 Self-similarity in the wild
- 3.1 MathWorld vs. PhysicalWorld
- 3.2 A foreshadowing of fractal dimension
- 3.3 Coastlines, mountains, and rivers
- 3.4 How (and why) the lungs are fractal
- 3.5 Power laws
- 3.6 Forests and trees
- 3.7 Our complex hearts
- 3.8 Metabolic rates
- 3.9 Fractals and DNA
- 3.10 How planets grow
- 3.11 Reversals of the Earth's magnetic field
- 3.12 The distribution of galaxies
- 3.13 Is this all?
- 4 Manufactured fractals
- 4.1 Chemical mixers
- 4.2 Capacitors
- 4.3 Wide-band antennas
- 4.4 Invisibility cloaks
- 4.5 Fractal metamaterials
- 4.6 Internet infrastructure
- 4.7 Music
- 4.8 Literature
- 4.9 Visual art
- 4.10 Building fractals
- 5 The Mandelbrot set: infinite complexity from a simple formula
- 5.1 Some pictures
- 5.2 The algorithm
- 5.3 Julia sets
- 5.4 The Mandelbrot set
- 5.5 Other Mandelbrot sets
- 5.6 The universality of the Mandelbrot set
- 5.7 The Mandelbrot set in four dimensions
- 5.8 Unanswered questions
- 6 Quantifying fractals: What is fractal dimension?
- 6.1 Similarity dimension
- 6.2 Box-counting dimension
- 6.3 Mass dimension
- 6.4 Random, with memory, and nonlinear
- 6.5 Dimension rules
- 6.6 Are unearthly dimensions of earthly use?
- 6.7 A speculation about dimensions
- 7 Further developments
- 7.1 Driven IFS
- 7.2 Driven IFS and synchronization
- 7.3 Multifractals from IFS
- 7.4 Applications of multifractals
- 7.5 Fractals and stories, again
- 8 Valediction
- A: A look under the hood:Some technical notes
- B: Solutions ti the problems
- References
- Acknowledgments
- Figure credits
- Index
- A
- B
- C
- D
- E
- F
- G
- H
- I
- J
- K
- L
- M
- N
- O
- P
- Q
- R
- S
- T
- U
- V
- W
- X
- Y
- Z
