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Very Short Introductions #367
Fractals: A Very Short Introduction
From the contours of coastlines to the outlines of clouds, and the branching of trees, fractal shapes can be found everywhere in nature. In this Very Short Introduction , Kenneth Falconer explains the basic concepts of fractal geometry, which produced a revolution in our mathematical understanding of patterns in the twentieth century, and explores the wide range of applications in science, and in aspects of economics.
About the
Oxford's Very Short Introductions series offers concise and original introductions to a wide range of subjects--from Islam to Sociology, Politics to Classics, Literary Theory to History, and Archaeology to the Bible. Not simply a textbook of definitions, each volume in this series provides trenchant and provocative--yet always balanced and complete--discussions of the central issues in a given discipline or field. Every Very Short Introduction gives a readable evolution of the subject in question, demonstrating how the subject has developed and how it has influenced society. Eventually, the series will encompass every major academic discipline, offering all students an accessible and abundant reference library. Whatever the area of study that one deems important or appealing, whatever the topic that fascinates the general reader, the Very Short Introductions series has a handy and affordable guide that will likely prove indispensable.
About the
Oxford's Very Short Introductions series offers concise and original introductions to a wide range of subjects--from Islam to Sociology, Politics to Classics, Literary Theory to History, and Archaeology to the Bible. Not simply a textbook of definitions, each volume in this series provides trenchant and provocative--yet always balanced and complete--discussions of the central issues in a given discipline or field. Every Very Short Introduction gives a readable evolution of the subject in question, demonstrating how the subject has developed and how it has influenced society. Eventually, the series will encompass every major academic discipline, offering all students an accessible and abundant reference library. Whatever the area of study that one deems important or appealing, whatever the topic that fascinates the general reader, the Very Short Introductions series has a handy and affordable guide that will likely prove indispensable.
132 pages, Paperback
First published August 6, 2013
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Displaying 1 - 30 of 32 reviews
December 30, 2013
We’ve all come across images of fractals: almost infinitely intricate and complex visual patterns that challenge almost all of our intuitions about geometry. Fractal lines are oftentimes infinitely long, yet they are contained within very well defined areas. The same goes for other measures of fractals in higher dimensions: area, volume, etc., In fact, the very notion of dimension as we normally understand it loses meaning when applied to fractals.
This short book tries to give a very intuitive and easy-to-follow introduction to fractals. It starts by examining some prototypical fractal sets that are relatively easy to construct, at least in principle. Fractals and fractal-related notions actually have a pretty long history, but they had only become popular in the last few decades. This is largely thanks to the advent of modern computers, and the ability to visualize many of the more interesting fractals for the first time.
Fractals are not just pretty pictures. They are based on some really profound and intricate mathematical concepts. What makes fractals from the mathematical viewpoint particularly fascinating is that the rules that are required for describing a fractal are seemingly very simple, and yet in order to understand the full intricacy of a fractal requires some exceedingly complex higher mathematics. To this book’s credit it tries to explain some of the richness of this mathematics, without, of course, going into any detail. To fully appreciate this material the reader should be able to understand at least some more abstract mathematical concepts – such as imaginary and complex numbers – but other than that a curious mind and a willingness to be intellectually engaged should be sufficient.
The book also covers several applications of fractals – in nature, science and finance to name a few. These examples illustrate that fractals, far from being just an idle abstract curiosity, are actually a very useful and powerful tool for the understanding of many aspects of the world around us.
The book is very elegantly written, and it is very accessible and a pleasure to read. This is perhaps one of the best examples of popular math book that I’ve ever come across.
This short book tries to give a very intuitive and easy-to-follow introduction to fractals. It starts by examining some prototypical fractal sets that are relatively easy to construct, at least in principle. Fractals and fractal-related notions actually have a pretty long history, but they had only become popular in the last few decades. This is largely thanks to the advent of modern computers, and the ability to visualize many of the more interesting fractals for the first time.
Fractals are not just pretty pictures. They are based on some really profound and intricate mathematical concepts. What makes fractals from the mathematical viewpoint particularly fascinating is that the rules that are required for describing a fractal are seemingly very simple, and yet in order to understand the full intricacy of a fractal requires some exceedingly complex higher mathematics. To this book’s credit it tries to explain some of the richness of this mathematics, without, of course, going into any detail. To fully appreciate this material the reader should be able to understand at least some more abstract mathematical concepts – such as imaginary and complex numbers – but other than that a curious mind and a willingness to be intellectually engaged should be sufficient.
The book also covers several applications of fractals – in nature, science and finance to name a few. These examples illustrate that fractals, far from being just an idle abstract curiosity, are actually a very useful and powerful tool for the understanding of many aspects of the world around us.
The book is very elegantly written, and it is very accessible and a pleasure to read. This is perhaps one of the best examples of popular math book that I’ve ever come across.
August 11, 2023
The word geometry might conjure up images or circles, squares, cylinders, and cubes, but many phenomena in nature and science are anything but regular or smooth. Think coastlines, mountain ranges, clouds, and trees. Fractals allow these complex and irregular objects to be described in simple terms.
This book outlines how fractals may be constructed and analysed as geometrical objects and how mathematical fractals relate to fractals out there in nature.
Fractals became 'popular' in the 1980s thanks to Benoit Mandelbrot who died in 2010 and is referred to as 'the Father of Fractals'. Fractals are increasingly used in sophisticated scientific research.
Brownian Motion was first noticed by the Scottish botanist Robert Brown in 1827 when he observed that particles of pollen moved on irregular paths in water. This motion is a very good example of a natural fractal phenomenon.
This book outlines how fractals may be constructed and analysed as geometrical objects and how mathematical fractals relate to fractals out there in nature.
Fractals became 'popular' in the 1980s thanks to Benoit Mandelbrot who died in 2010 and is referred to as 'the Father of Fractals'. Fractals are increasingly used in sophisticated scientific research.
Brownian Motion was first noticed by the Scottish botanist Robert Brown in 1827 when he observed that particles of pollen moved on irregular paths in water. This motion is a very good example of a natural fractal phenomenon.
May 7, 2020
The field of fractals is undoubtedly replete with rich mathematical and artistic ideas. But the book provides just a glimpse of the concepts of fractals like self-similarity, fractal dimensions and Julia sets that do not do justice to the beauty of these mathematical beasts. Maybe reading a popular-sci (math?) book for fractals was not a good idea and I should have chosen to read the standard textbook by Benoit Mandelbrot.
January 15, 2023
comprehensive👍
October 11, 2024
Got to the topic of fractals via the works of Nassim Taleb, mentioning the work of Mendelbrot; from there it was a short link to the magisterial work by Geoffrey West and his unsurpassed Scale: The Universal Laws of Growth, Innovation, Sustainability, and the Pace of Life in Organisms, Cities, Economies, and Companies; this is a welcome primer; recommended.
May 20, 2022
Written in 2013. I give this book 05 stars. As true to its name its really a very good introduction to the topic FRACTALS.
The topics of Mandelbrot set & Julia set and their intricate relation is very well explained
Like all other Euclid geometry figures, In real world we only encounter approximate fractals like tree, leaf, our lungs, our blood vessels, cloud etc. All defined through Fractal Dimension and self similarity ..
A good read if you like mathematics
The topics of Mandelbrot set & Julia set and their intricate relation is very well explained
Like all other Euclid geometry figures, In real world we only encounter approximate fractals like tree, leaf, our lungs, our blood vessels, cloud etc. All defined through Fractal Dimension and self similarity ..
A good read if you like mathematics
February 22, 2015
I'd describe this as an adequate explanation of fractals. I feel like I have a grasp of some of the basic concepts, but I was left feeling that most of what I was being told was very abstract. For such a graphically rich field, I would have liked to have seen more figures and equations, and to be given some more information about the real-world applications. We are told that fractals can be used to generate artificial landscapes or compress data, but not how.
September 6, 2015
This is thrilling because you can plot examples yourself. Meaning that the whole thing is simple but the result will spray a complex beauty at your face. Behold! You are about to fall in love :-) Very entertaining book.
January 22, 2017
This book was a very good introduction for someone who wants to know the fundamental math fractals are build upon and their applications. Mathematical explanations are very clear for the most part with several examples that the author walks through with the reader.
January 4, 2017
Fascinating subject. Clear explanations plus illuminating examples.
July 31, 2025
awesome sauce
August 6, 2020
Fractals, A Very Short Introduction
is a short book about fractal geometry. It was written by Kenneth Falconer and published in 2009.
It is very small in size, runs just 132 pages including the index, and has 44 illustrations. Despite its small size, it took me a few days to read it, albeit piecemeal and in my spare time.
As someone just starting to learn about fractals — and wanting to experiment with drawing them myself — I found it to be the perfect primer. I earned a degree in mathematics from Virginia Commonwealth University in the 1980s, but haven't thought about exponents and logarithms in quite awhile.
I am giving Fractals, A Very Short Introduction 5 of 5 stars because it is an excellent introduction to the topic. Its focus is on explaining the basics of fractal geometry, which relies on some advanced mathematics, but it does not have more math than it needs to have. And for the benefit of those who could use a gentle reminder of some of the basics of topics like logarithms and complex numbers, the author includes an Appendix with precisely just enough information to revive those dormant brain cells.
Fractal geometry is a very complicated topic, yet Fractals, A Very Short Introduction does an excellent job of starting from the beginning and sticking to the basics.
In Chapter 1, The fractal concept, Falconer uses the very simple von Koch curve to introduce some of the core ideas underlying fractal geometry. And in Chapter 2, Self-similarity, he dives into one of the most important of these concepts, by introducing the mesmerizing Sierpinski triangle and showing readers some of the self-similarity found in nature, such as that found in snowflakes and plants.
Chapter 3, Fractal dimension, is where things start to get complicated. Mr. Falconer, however, clearly knows how to explain the not-very-intuitive idea of fractional dimensions in a way that makes it as intuitive as possible. Having read this I think that I, despite still being a bit new to all this, could — with a bit of effort and a willing listener — actually explain this to someone. At worst, if I failed, I could return to this chapter, figure out what I got wrong, and learn how I could do better next time.
Chapter 4, Julia sets and the Mandlebrot set, is as deep as Chapter 3, if not deeper. The author remains true to form, however, explaining in a step-by-step manner how mathemeticians found and construct these forms — which are far more complicated than the von Koch curves.
Fractals, A Very Short Introduction is now the third book I have read about fractal geometry. I found it in the Denver Public Library, but when I got to Illustration 31 in Chapter 4, Julia sets corresponding to various points of the Mandelbrot set, I decided to buy my own copy. It was just under $10, and in my opinion — especially in light of this incredibly simple yet profound illustration — will be worth every penny as a reference.
Chapter 5, Random walks and Brownian motion, is much less complicated than its two predecessors and introduces a fractal topic that is surprisingly quite relevant to many things we see everyday. The essence of a random or "drunkard's" walk is that each step is randomly either forward or backward. Kenneth explains how this can be seen as a place to begin mathematically modeling things such as how molecules, prices, and even the stock markets move, which often seems to be in a rather random fashion.
Chapter 6, Fractals in the real world, includes brief descriptions of fractal geometry's relevance to things ranging from coastlines and landscapes, to the fractal patterns in our bodies, to clouds and galaxies, and all the way to cell phone antenna design. Finally Chapter 7, A little history, wraps up the book with a few pages about the key players in this research during the past few centuries.
I highly recommend Introducing Fractals, A Very Short Introduction to anyone wanting a solid, substantive introduction to fractal geometry. Even though the book is small in size, it is extremely well-organized and concise, and every one of its small pages is packed with information.
As someone who is anxious to try creating some of these designs I feel Fractals, A Very Short Introduction has given me the background I need to read some of the more advanced and comprehensive books on the subject. Hopefully it has not made me overconfident, because I have also ordered copies of Falconer's Fractal Geometry: Mathematical Foundations and Applications and Fractals: Form, Chance and Dimension by Benoit Mandelbrot!
It is very small in size, runs just 132 pages including the index, and has 44 illustrations. Despite its small size, it took me a few days to read it, albeit piecemeal and in my spare time.
As someone just starting to learn about fractals — and wanting to experiment with drawing them myself — I found it to be the perfect primer. I earned a degree in mathematics from Virginia Commonwealth University in the 1980s, but haven't thought about exponents and logarithms in quite awhile.
I am giving Fractals, A Very Short Introduction 5 of 5 stars because it is an excellent introduction to the topic. Its focus is on explaining the basics of fractal geometry, which relies on some advanced mathematics, but it does not have more math than it needs to have. And for the benefit of those who could use a gentle reminder of some of the basics of topics like logarithms and complex numbers, the author includes an Appendix with precisely just enough information to revive those dormant brain cells.
Fractal geometry is a very complicated topic, yet Fractals, A Very Short Introduction does an excellent job of starting from the beginning and sticking to the basics.
In Chapter 1, The fractal concept, Falconer uses the very simple von Koch curve to introduce some of the core ideas underlying fractal geometry. And in Chapter 2, Self-similarity, he dives into one of the most important of these concepts, by introducing the mesmerizing Sierpinski triangle and showing readers some of the self-similarity found in nature, such as that found in snowflakes and plants.
Chapter 3, Fractal dimension, is where things start to get complicated. Mr. Falconer, however, clearly knows how to explain the not-very-intuitive idea of fractional dimensions in a way that makes it as intuitive as possible. Having read this I think that I, despite still being a bit new to all this, could — with a bit of effort and a willing listener — actually explain this to someone. At worst, if I failed, I could return to this chapter, figure out what I got wrong, and learn how I could do better next time.
Chapter 4, Julia sets and the Mandlebrot set, is as deep as Chapter 3, if not deeper. The author remains true to form, however, explaining in a step-by-step manner how mathemeticians found and construct these forms — which are far more complicated than the von Koch curves.
Fractals, A Very Short Introduction is now the third book I have read about fractal geometry. I found it in the Denver Public Library, but when I got to Illustration 31 in Chapter 4, Julia sets corresponding to various points of the Mandelbrot set, I decided to buy my own copy. It was just under $10, and in my opinion — especially in light of this incredibly simple yet profound illustration — will be worth every penny as a reference.
Chapter 5, Random walks and Brownian motion, is much less complicated than its two predecessors and introduces a fractal topic that is surprisingly quite relevant to many things we see everyday. The essence of a random or "drunkard's" walk is that each step is randomly either forward or backward. Kenneth explains how this can be seen as a place to begin mathematically modeling things such as how molecules, prices, and even the stock markets move, which often seems to be in a rather random fashion.
Chapter 6, Fractals in the real world, includes brief descriptions of fractal geometry's relevance to things ranging from coastlines and landscapes, to the fractal patterns in our bodies, to clouds and galaxies, and all the way to cell phone antenna design. Finally Chapter 7, A little history, wraps up the book with a few pages about the key players in this research during the past few centuries.
I highly recommend Introducing Fractals, A Very Short Introduction to anyone wanting a solid, substantive introduction to fractal geometry. Even though the book is small in size, it is extremely well-organized and concise, and every one of its small pages is packed with information.
As someone who is anxious to try creating some of these designs I feel Fractals, A Very Short Introduction has given me the background I need to read some of the more advanced and comprehensive books on the subject. Hopefully it has not made me overconfident, because I have also ordered copies of Falconer's Fractal Geometry: Mathematical Foundations and Applications and Fractals: Form, Chance and Dimension by Benoit Mandelbrot!
September 4, 2021
Fractals are …. What are fractals? That alone is a problem and part of the fun of fractals. Unlike, say, triangles, they are not defined in the usual rigorous way of mathematics. But mathematics they most certainly involve with some surprising and spectacular theorems relating, by proof, observations of remote effects. But again, what are fractals? They are known to everybody, if not by name, as recurring multilevel shapes that you see every day in trees and bushes and in coastlines and in maps of blood vessels and in stock market charts. A little more precisely, they are objects where the pattern repeats at different scales and where the pattern has a nearly trivial generator. Most people find fractals cool.
For the mathematically inclined, this coolness goes to a different level. They are impossible to characterize using familiar geometry. For example, even though the construction may be based on a 1D line, it can fill a plane. But yet leave infinite space. A different, but natural, notion of dimension emerges. Fractals invariably have dimensionality not equal counting number, like 1.47…. When you read popular accounts about fractals or other pure math you invariably read about the “beauty” – I don’t know if I see beauty, but there is a basic joy and satisfaction and amazement.
This very short introduction gives you everything that you need to get started. There are examples and a little history. There is definition. There are explanations of the key mathematical concepts, especially fractal dimension. There are basic extensions, most prominently Julia sets. There are connections into incredibly deep and important ideas like probability distributions, random variables, and Brownian motion. Really amazing. I need to read this again soon!
For the mathematically inclined, this coolness goes to a different level. They are impossible to characterize using familiar geometry. For example, even though the construction may be based on a 1D line, it can fill a plane. But yet leave infinite space. A different, but natural, notion of dimension emerges. Fractals invariably have dimensionality not equal counting number, like 1.47…. When you read popular accounts about fractals or other pure math you invariably read about the “beauty” – I don’t know if I see beauty, but there is a basic joy and satisfaction and amazement.
This very short introduction gives you everything that you need to get started. There are examples and a little history. There is definition. There are explanations of the key mathematical concepts, especially fractal dimension. There are basic extensions, most prominently Julia sets. There are connections into incredibly deep and important ideas like probability distributions, random variables, and Brownian motion. Really amazing. I need to read this again soon!
February 14, 2022
Fractals entered my reading space via Robert G. Hagstrom and his book, Investing the Last Liberal Art and it was so enticing i stopped reading and added it to my kindle immediately. The book is interesting from a theoretical viewpoint and the concepts are easy enough to comprehend and understand, but the maths was not for me.
There is no shame to say that a boring and or dull part of a book has been skipped, and with Fractals i did skip some parts. These though were the pure mathematics, you know the type Y x P + L = something or other to the square of a hippopotamus. These were skipped through with genuine remorse, not because i have no interest in learning or understanding, but due to not needing to know and not willing to waste my time attempting to digest this information.
A strong recommendation is advised for those who do enjoy there maths and want to have a better understanding of Fractals because this is a great guide. I would also commend this book to anyone who would have an interest in a fascinating interest in theoretical concepts that can be taken out of from their original field and adapted to others.
As an example one field that had no mention was education, but what i do see is that with modern education and people development there would a huge opportunity to apply fractals and the theoretical construct to educational and pedagogical structures.
Pick it up, have a read and try something different; even if you have to skip a page or two.
There is no shame to say that a boring and or dull part of a book has been skipped, and with Fractals i did skip some parts. These though were the pure mathematics, you know the type Y x P + L = something or other to the square of a hippopotamus. These were skipped through with genuine remorse, not because i have no interest in learning or understanding, but due to not needing to know and not willing to waste my time attempting to digest this information.
A strong recommendation is advised for those who do enjoy there maths and want to have a better understanding of Fractals because this is a great guide. I would also commend this book to anyone who would have an interest in a fascinating interest in theoretical concepts that can be taken out of from their original field and adapted to others.
As an example one field that had no mention was education, but what i do see is that with modern education and people development there would a huge opportunity to apply fractals and the theoretical construct to educational and pedagogical structures.
Pick it up, have a read and try something different; even if you have to skip a page or two.
May 2, 2019
Many years ago, a program called CorelPhoto Paint introduced me to fractals. Back then, I thought of them as stunningly beautiful, cosmic-looking things that you could change and "mutate" by pressing various controls on the program. Only later, in college, did I see a course offering on the topic, which was, however, reserved for the few "chosen ones" who could do advanced math without adversely affecting their GPA. This book filled some, but not all, of the gap in what I've long wanted to know about fractals, but as I took the audiobook version, it was difficult to listen to. Perhaps a hardcopy would perform better. Still, it is a topic worth exploring, and while I felt the book was a little dry and technical on a topic that I've always seen as poetic and beautiful (if you use a fractal explorer software and all the math turns into art), it is still a step in the right direction. Especially considering that the topic is highly technical, regardless of how beautiful its applications may be.
September 7, 2025
The book is easy to read, though I found some sections, especially at the beginning, overly simplistic. Its treatment of the material feels somewhat unbalanced: certain topics, like the Julia and Mandelbrot sets, receive extensive coverage, while others, such as physical applications or random walks, are given only brief attention. As a physicist, I was more interested in the latter, so I found myself less engaged when the book did not explore those aspects in depth.
The mathematics presented is very elementary, so readers without a mathematical background need not be concerned. On the other hand, those comfortable with mathematics may find the book lacking, as it does not offer any real mathematical insight into fractals.
Overall, I would recommend it to complete newcomers to fractals without a technical background in mathematics, particularly those not seeking detailed discussions of physical applications and explanations.
The mathematics presented is very elementary, so readers without a mathematical background need not be concerned. On the other hand, those comfortable with mathematics may find the book lacking, as it does not offer any real mathematical insight into fractals.
Overall, I would recommend it to complete newcomers to fractals without a technical background in mathematics, particularly those not seeking detailed discussions of physical applications and explanations.
April 18, 2020
2015.01.16–2015.01.17
Contents (with formatting slightly broken in List of illustrations)
Falconer K (2013) Fractals - A Very Short Introduction
Preface
List of illustrations
• 01. The basic step in constructing the von Koch curve: (a) divide a line into three equal parts, (b) remove the middle part, (c) replace the missing part by the other two sides of an equilateral triangle
• 02. (a) The first few stages E0, E1, E2, ... of construction of the von Koch curve F, (b) three such curves joined together to form the von Koch snowflake
• 03. Repeated magnification reveals more irregularity
• 04. A squig curve (above) and a fractal grass (below) with their generators: the line segments in the generators are repeatedly replaced by scaled down copies to form the fractals
• 05. Points in the plane with their coordinates
• 06. Two itineraries, one starting at (2,5) under the function (x,y) → (x + 1 , y), the other starting at (8,4) under the function (x,y) → (½x, ½y)
• 07. The Hénon attractor with a portion enlarged to display its banded structure
• 08. The figures shown are all similar to each other
• 09. The von Koch curve together with its defining template
• 10. Reconstruction of the von Koch curve by repeated substitution of the template in itself
• 11. The Sierpiński triangle with its template consisting of a large square of side 1 and three squares of side ½
• 12. Self-similar fractals with their templates: (a) a snowflake, (b) a spiral
• 13. The eight symmetries of the square indicated by the positions of the face
• 14. Self-similar fractals with the same template but with different orientations of the similar copies
• 15. A function moves points to points and so transforms sets of points to sets of points
• 16. A self-affine fractal with its template
• 17. A self-affine fern and tree
• 18. Construction of a random von Koch curve—when replacing each line segment a coin is tossed and the ‘V’ is drawn pointing upwards for a head and downwards for a tail
• 19. Box-counting with grids of side ¼ and ⅛ for (a) a line segment of length 1, (b) a square of side 1
• 20. Box-counting with grids of side ¼ and ⅛ for the Sierpiński triangle
• 21. Variants on the von Koch curve with the dimensions depending on the angles used in the constructions
• 22. (a) Box-counting to estimate the dimension of the coastline of Britain, (b) a ‘log-log’ plot of the box counts—the slope v / h gives an estimate of the dimension
• 23. The effect on length, area, and d-dimensional measurement of enlargement by a factor 2
• 24. Complex numbers in the complex plane, with the magnitude and angle of 4 + 2i indicated
• 25. (a) Addition of two complex numbers shifts one parallel to the other, (b) squaring a complex number squares the magnitude and doubles the angle
• 26. Itineraries under (a) z → z2 and (b) z → z2 + 0.1 + 0.1i, showing the curves that separate the two types of itineraries
• 27. The filled-in Julia set of z → z2 – 0.9 with its exterior shaded according to the escape time
• 28. Julia sets for a range of complex numbers c, given by (a) 0.2–0.2i, (b) –0.6–0.3i, (c) –1, (d) –0.1 + 0.75i, (e) 0.25 + 0.52i, (f) –0.5 + 0.55i, (g) 0.66i, (h) –i
• 29. The Mandelbrot set comprises those complex numbers a + bi in the blackened portion of the complex plane
• 30. Successive zooms on the Mandelbrot set
• 31. Julia sets corresponding to various points of the Mandelbrot set
• 32. An exotic Julia set with c = –0.6772 + 0.3245i, a point very close to the boundary of the Mandelbrot set
• 33. Progression between the loops of the Julia set for z → z2 – 1 under iteration
• 34. Progress of a typical random walk
• 35. Several independent random walks plotted together showing the spread of the walkers widening as time progresses
• 36. The distribution of positions of 130 random walkers after 20 steps
• 37. Graph of a Brownian process
• 38. A random walk in the plane
• 39. A Brownian path in the plane
• 40. Computer simulation of fractal fingering: successive particles are released from random points A on the circle and perform random walks until they reach a square next to one already shaded, B. This square is shaded and the process repeated, so the shaded squares grow out from the initial ‘seed’ square S
• 41. A computer simulation of fractal fingering
• 42. Processes with different Hurst indices
• 43. Multifractal spectrum of the distribution of the proportion of 0s in numbers
• 44. The middle-third Cantor set F. Each stage of the construction, E1, E2, … is obtained by removing the middle third of each interval in the preceding stage
1. The fractal concept
• The rise of fractals
• A first fractal construction–the von Koch curve
• Some more examples
• Coordinates, functions, and itineraries
• Fractals by iteration
• What can be done with fractals?
2. Self-similarity
• Self-similar fractals and their templates
• Orientation
• Templates and functions
• Drawing fractals
• Self-affine fractals
• Statistically self-similar fractals
• Fractal image compression
3. Fractal dimension
• Box-counting dimension
• The role of logarithms
• Practical box-counting
• Dimension of self-similar fractals
• Measurement with dimension
• Properties of dimension
• Limitations of dimension
4. Julia sets and the Mandelbrot set
• Complex numbers
• Iteration and Julia sets
• The zoo of Julia sets
• The Mandelbrot set
• Back to Julia sets
• A historical note
5. Random walks and Brownian motion
• Random walks and Brownian motion in the plane or space
• Fractal fingering
• Fractal time records
6. Fractals in the real world
• Coastlines and landscapes
• Turbulent fluids
• Fractals in our bodies
• Medical diagnosis
• Clouds
• Galaxies
• Fractal antennae
• Multifractals
7. A little history
Appendix
• Powers and logarithms
• Squaring complex numbers
Further reading
Index
Contents (with formatting slightly broken in List of illustrations)
Falconer K (2013) Fractals - A Very Short Introduction
Preface
List of illustrations
• 01. The basic step in constructing the von Koch curve: (a) divide a line into three equal parts, (b) remove the middle part, (c) replace the missing part by the other two sides of an equilateral triangle
• 02. (a) The first few stages E0, E1, E2, ... of construction of the von Koch curve F, (b) three such curves joined together to form the von Koch snowflake
• 03. Repeated magnification reveals more irregularity
• 04. A squig curve (above) and a fractal grass (below) with their generators: the line segments in the generators are repeatedly replaced by scaled down copies to form the fractals
• 05. Points in the plane with their coordinates
• 06. Two itineraries, one starting at (2,5) under the function (x,y) → (x + 1 , y), the other starting at (8,4) under the function (x,y) → (½x, ½y)
• 07. The Hénon attractor with a portion enlarged to display its banded structure
• 08. The figures shown are all similar to each other
• 09. The von Koch curve together with its defining template
• 10. Reconstruction of the von Koch curve by repeated substitution of the template in itself
• 11. The Sierpiński triangle with its template consisting of a large square of side 1 and three squares of side ½
• 12. Self-similar fractals with their templates: (a) a snowflake, (b) a spiral
• 13. The eight symmetries of the square indicated by the positions of the face
• 14. Self-similar fractals with the same template but with different orientations of the similar copies
• 15. A function moves points to points and so transforms sets of points to sets of points
• 16. A self-affine fractal with its template
• 17. A self-affine fern and tree
• 18. Construction of a random von Koch curve—when replacing each line segment a coin is tossed and the ‘V’ is drawn pointing upwards for a head and downwards for a tail
• 19. Box-counting with grids of side ¼ and ⅛ for (a) a line segment of length 1, (b) a square of side 1
• 20. Box-counting with grids of side ¼ and ⅛ for the Sierpiński triangle
• 21. Variants on the von Koch curve with the dimensions depending on the angles used in the constructions
• 22. (a) Box-counting to estimate the dimension of the coastline of Britain, (b) a ‘log-log’ plot of the box counts—the slope v / h gives an estimate of the dimension
• 23. The effect on length, area, and d-dimensional measurement of enlargement by a factor 2
• 24. Complex numbers in the complex plane, with the magnitude and angle of 4 + 2i indicated
• 25. (a) Addition of two complex numbers shifts one parallel to the other, (b) squaring a complex number squares the magnitude and doubles the angle
• 26. Itineraries under (a) z → z2 and (b) z → z2 + 0.1 + 0.1i, showing the curves that separate the two types of itineraries
• 27. The filled-in Julia set of z → z2 – 0.9 with its exterior shaded according to the escape time
• 28. Julia sets for a range of complex numbers c, given by (a) 0.2–0.2i, (b) –0.6–0.3i, (c) –1, (d) –0.1 + 0.75i, (e) 0.25 + 0.52i, (f) –0.5 + 0.55i, (g) 0.66i, (h) –i
• 29. The Mandelbrot set comprises those complex numbers a + bi in the blackened portion of the complex plane
• 30. Successive zooms on the Mandelbrot set
• 31. Julia sets corresponding to various points of the Mandelbrot set
• 32. An exotic Julia set with c = –0.6772 + 0.3245i, a point very close to the boundary of the Mandelbrot set
• 33. Progression between the loops of the Julia set for z → z2 – 1 under iteration
• 34. Progress of a typical random walk
• 35. Several independent random walks plotted together showing the spread of the walkers widening as time progresses
• 36. The distribution of positions of 130 random walkers after 20 steps
• 37. Graph of a Brownian process
• 38. A random walk in the plane
• 39. A Brownian path in the plane
• 40. Computer simulation of fractal fingering: successive particles are released from random points A on the circle and perform random walks until they reach a square next to one already shaded, B. This square is shaded and the process repeated, so the shaded squares grow out from the initial ‘seed’ square S
• 41. A computer simulation of fractal fingering
• 42. Processes with different Hurst indices
• 43. Multifractal spectrum of the distribution of the proportion of 0s in numbers
• 44. The middle-third Cantor set F. Each stage of the construction, E1, E2, … is obtained by removing the middle third of each interval in the preceding stage
1. The fractal concept
• The rise of fractals
• A first fractal construction–the von Koch curve
• Some more examples
• Coordinates, functions, and itineraries
• Fractals by iteration
• What can be done with fractals?
2. Self-similarity
• Self-similar fractals and their templates
• Orientation
• Templates and functions
• Drawing fractals
• Self-affine fractals
• Statistically self-similar fractals
• Fractal image compression
3. Fractal dimension
• Box-counting dimension
• The role of logarithms
• Practical box-counting
• Dimension of self-similar fractals
• Measurement with dimension
• Properties of dimension
• Limitations of dimension
4. Julia sets and the Mandelbrot set
• Complex numbers
• Iteration and Julia sets
• The zoo of Julia sets
• The Mandelbrot set
• Back to Julia sets
• A historical note
5. Random walks and Brownian motion
• Random walks and Brownian motion in the plane or space
• Fractal fingering
• Fractal time records
6. Fractals in the real world
• Coastlines and landscapes
• Turbulent fluids
• Fractals in our bodies
• Medical diagnosis
• Clouds
• Galaxies
• Fractal antennae
• Multifractals
7. A little history
Appendix
• Powers and logarithms
• Squaring complex numbers
Further reading
Index
February 8, 2025
Frustrated with this as an audiobook. Yes there is a pdf included but only very rarely does the author tell you what figure he is talking about and as I am usually driving or doing something while listening it is super frustrating to try and find the figure he is talking about. Several times I found myself lost in the weeds. Initial and later chapters are the best. He explains the origins of fractals as a name and its origins as a study topic in the waning seconds of the book. I feel like that maybe should have led the way. Do NOT recommend as an audiobook unless you are on a computer and can open the pdf and study it along with the listening. On my old and small phone it was a bear and not worth it.
November 7, 2019
As a thorough introduction to the world of fractals, this book was both interesting and overwhelming. I'll have to read it another two or three times before I fully understand half of what is going on, but this was a good start.
May 30, 2020
A fascinating introduction to fractals that provides the right amount of detail for newcomers, from von Koch curves to Julia sets and the Mandelbrot set, and even complex numbers. The most amazing section of the book was the section on Julia sets and how they comprise the Mandelbrot set.
February 23, 2021
Highly recommend for a thorough and elementary introduction
This book provided a comprehensive and intuitive progress of ideas into the pervasive geometry of fractals. I’d recommend this to anyone interested in using math as tool to understand reality.
This book provided a comprehensive and intuitive progress of ideas into the pervasive geometry of fractals. I’d recommend this to anyone interested in using math as tool to understand reality.
June 4, 2018
Just what it claims to be - a lucid and (relatively) nontechnical glimpse at an interesting branch of mathematics.
July 3, 2020
This is a very easy introduction into the topic. Math is simple, good explanations, enough references. Also, a good list of what to read next.
August 17, 2020
I see this book has not a very high rating and I wonder why. Althought is not as accesible as Gleick's "Chaos", it delivers its promise, being a lot less anecdotical, if a little more technical.
June 14, 2021
A good introduction to the fascinating world of fractals.
July 1, 2021
a very interesting read but the section on applications of fractals falls short of the rest of the book
July 6, 2023
Too many typographical errors in my edition, also a glaring lack of illustrations.
June 11, 2024
rrly interesting stuff. good intro before looking into it with a more mathematical approach
June 27, 2025
Apt name because it was a very general overview, but neat nonetheless
September 10, 2015
I'm not sure what possessed me to read a book about fractals. Probably I did it because fractals seem like an interesting subject and the book was short. And probably because I figured there'd be no math. Well, fractals remain interesting and the book is still short. I struck out on the math part, and that's where the book strikes out.
The first chapter, which introduces the concept of fractals and gives basic background info on them, is fine. Then the book goes to math hell. There is a lot of abstruse math in the next three chapters, which from my perspective was indistinguishable from mumbo jumbo. Things like self-similarity, box counting, Julia sets, were Greek to me. Logarithms show up at one point. There's even a discussion of the square root of -1. Everyone knows negative numbers don't have square roots! Apparently mathematicians simply invented this one by fiat to help their calculations. What help it provided I can't say, since I was adrift on a sea of gobbledygook. I have no doubt this was important. But whatever the explanation for why it was important was in the form of numbers and not in English.
The last couple chapters improve. The discussion of how fractals can represent Brownian motion was quite lucid. The penultimate chapter, about fractals in the real world, is also good. I was expecting more of this kind of thing, and less of the mathematical theory stuff. I wanted more about how knowledge of fractals is used in the real world – in things like making cancer diagnoses and the determining the size of the lungs. Even galaxies and the universe itself exhibit fractal behavior. That stuff is interesting. Box-counting? Not interesting.
The last chapter gives a brief account of the history of fractal geometry. Some of its principles have been around for a long time, but it's only in the last century, and really the last half-century, that it's become a field in its own right. More of this would have been good, too. Or less math. That would be even better.
Added Wednesday, September 9, 2015
The first chapter, which introduces the concept of fractals and gives basic background info on them, is fine. Then the book goes to math hell. There is a lot of abstruse math in the next three chapters, which from my perspective was indistinguishable from mumbo jumbo. Things like self-similarity, box counting, Julia sets, were Greek to me. Logarithms show up at one point. There's even a discussion of the square root of -1. Everyone knows negative numbers don't have square roots! Apparently mathematicians simply invented this one by fiat to help their calculations. What help it provided I can't say, since I was adrift on a sea of gobbledygook. I have no doubt this was important. But whatever the explanation for why it was important was in the form of numbers and not in English.
The last couple chapters improve. The discussion of how fractals can represent Brownian motion was quite lucid. The penultimate chapter, about fractals in the real world, is also good. I was expecting more of this kind of thing, and less of the mathematical theory stuff. I wanted more about how knowledge of fractals is used in the real world – in things like making cancer diagnoses and the determining the size of the lungs. Even galaxies and the universe itself exhibit fractal behavior. That stuff is interesting. Box-counting? Not interesting.
The last chapter gives a brief account of the history of fractal geometry. Some of its principles have been around for a long time, but it's only in the last century, and really the last half-century, that it's become a field in its own right. More of this would have been good, too. Or less math. That would be even better.
Added Wednesday, September 9, 2015
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