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The Fractal Geometry of Nature
Clouds are not spheres, mountains are not cones, and lightning does not travel in a straight line. The complexity of nature's shapes differs in kind, not merely degree, from that of the shapes of ordinary geometry, the geometry of fractal shapes.
Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.
Now that the field has expanded greatly with many active researchers, Mandelbrot presents the definitive overview of the origins of his ideas and their new applications. The Fractal Geometry of Nature is based on his highly acclaimed earlier work, but has much broader and deeper coverage and more extensive illustrations.
468 pages, Hardcover
First published January 1, 1977
About the author
Benoît B. Mandelbrot
25 books288 followersBenoît B. Mandelbrot, O.L.H., Ph.D. (Mathematical Sciences, University of Paris, 1952; M.S., Aeronautics, California Institute of Technology, 1949) was a mathematician best known as the father of fractal geometry. He was Sterling Professor Emeritus of Mathematical Sciences at Yale University; IBM Fellow Emeritus at the Thomas J. Watson Research Center; and Battelle Fellow at the Pacific Northwest National Laboratory.
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
Mandelbrot was born in Poland, but his family moved to France when he was a child; he was a dual French and American citizen and was educated in France. He has been awarded with numerous honors, including induction into the Legion d'honneur, as well as the 1986 Franklin Medal for Physics, the 1993 Wolf Prize for Physics, the 2000 Lewis Fry Richardson Medal of the European Geophysical Society, and the 2003 Japan Prize "for the creation of universal concepts in complex systems."
Ratings & Reviews
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Displaying 1 - 30 of 53 reviews
August 13, 2018
We are in 1982, Mandelbrot got a computer and is showing us the geometry nature has been using. Have a good look through that window. Euclid’s geometry works for human made stuff (before 3D printers) but fails with clouds, trees, your hair (in that reflection) and so on. Nature is ingenious using a blend of chance, scale and repetition creating regular irregularities here called fractals. Being a maverik, Mandelbrot introduced the fractal dimension assuming that “to see is to believe”. Lucky us. There are plenty of puzzling plots in this book. Some even defined as nondifferentiable monsters, say a snowflake (Koch’s monster). What a show! But Madelbrot’s erudition and eccentricity makes this book a fractal itself. Often diverging and sharing bits of history and biographical notes. But always discussing typical anomalies within our aberrant universe.
January 1, 2009
More a reference than a read, Mandelbrot was the first mathematician and naturalist to embrace fractals, second citizen equations that turned out to be elegant expressions of much of the patterns, sequences, arrangements we can see in molecules, simple life forms, organ systems, complex life forms, environments, ecosystems.... Some of it is user friendly, some needs to be skimmed by those who don't care about the math, but Mandelbrot is supportive of both kinds of readers.
December 4, 2011
I believe this is where I first encountered such oddities of geometry and math, some of which were once thought to be monsters, as Cantor dust, Sierpinski triangles, Koch snowflakes, Peano curves, and the like. More important, it's where I first found explicated the idea of fractional dimensionality, from which I believe Mandelbrot derived the term "fractal." (The concept itself comes from a German mathematician named Hausdorff, but Mandelbrot generalized it). A solid understanding of the book is likely to be beyond anyone who lacks some training in higher mathematics, but the intrepid reader can still learn much. And nowadays, the Internet is at hand to assist, which wasn't the case when I read this in the 80s, not very long after its publication.
December 17, 2014
I have to admit it: I simply love this book. However, this book is quite out of the ordinary. If you are a linear reader, this book is definitely not for you. The book is about fractals, but the book is fractal. Its structure, the way topics are introduced and developed, everything in it resembles a fractal. But the book is useful, very useful: it has thousands of ideas. Most of them still unexplored. It asks many, many questions and let you find the answers. If you are a researcher, you will find plenty of inspiration in this book. If you are a non-linear, casual fractal lover you will enjoy this book. If you are a classical, linear reader you will stop reading not even one-fifth-way through.
January 11, 2023
Went to reread this classic, but our local libraries don't have it. Still in print? Yes, for $30:
https://www.amazon.com/Fractal-Geomet...
Order at ZIP in due time. Date read is just a guess, as is the rating: 1982 book.
https://www.amazon.com/Fractal-Geomet...
Order at ZIP in due time. Date read is just a guess, as is the rating: 1982 book.
April 2, 2012
Mandlebrot does a good job of writing for both mathematicians and those who simply have a passing interest. He makes it easy to skip over the in-depth mathematical sections whilst still providing solid explanations of the concepts being discussed. I will admit that some of it was way over my head and there were sections that could be quite dry. However, it was thrilling to gain a greater understanding of fractals like the Menger sponge, Koch snowflakes and Cantor Dust along with how they can be applied to things like star distribution in the galaxy or modeling ecosystems. Definitely worth reading.
Read
October 20, 2009I have to admit that this book has been sitting on my shelves half-read and gathering dust. Mandelbrot's a fairly good speaker (from what I've seen of him on documentaries), but a trying and tedious writer.
If you're looking for pretty pictures of fractals, there are some. If you're looking for the formula for the Mandelbrot set, it's here. If you want to understand the theory behind fractals, this is not the best introduction.
If you're looking for pretty pictures of fractals, there are some. If you're looking for the formula for the Mandelbrot set, it's here. If you want to understand the theory behind fractals, this is not the best introduction.
January 13, 2015
A modern classic. The style of writing lurks between the literary and the scientific, very much like the topic itself.
November 21, 2015
the essential reading for chaos and fractal
October 10, 2023
THE ‘FATHER OF FRACTALS’ EXPLAINS THEM IN THIS WELL-ILLUSTRATED BOOK
Author Benoit Mandelbrot wrote in the Foreword to this revised edition (1982) of his book, “This work follows and largely replaces my 1977 Essay, ‘Fractals: Form, Chance and Dimension,’ which had followed and largely replaced my 1975 Essay in French… Each stage involved new art, a few deletions, extensive rewriting that affected nearly every section, additions devoted to my older work, and---most important---extensive additions devoted to new developments… My first scientific publication came out on April 30, 1951. Over the years, it had seemed to many that each of my investigations was aimed in a different direction. But this apparent disorder was misleading: it hid a strong unity of purpose, which the present Essay, like its two predecessors, is intended to reveal. Against odds, most of my works turn out to have been the birth pangs of a new scientific discipline.”
He states in the Introduction, “Why is geometry often described as ‘cold’ and ‘dry’? One reason is its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared with … standard geometry---Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite. The existence of these patterns challenges us to … investigate the morphology of the ‘amorphous.’… Responding to this challenge, in conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call ‘fractals.’ The most useful fractals involve CHANCE and both their regularities and their irregularities are statistical… Somme fractal sets are curves of surfaces, others are disconnected ‘dusts,’ and yet others are so oddly shaped that there are no good terms for them in either the sciences or the arts. The reader is urged to sample them now, by browsing through the book’s illustrations.” (Pg. 1)
He continues, “Despite its length, I describe [this book] as a scientific Essay because it is written from a personal point of view and without attempting completeness… This Essay brings together a number of analyses in diverse sciences, and it promotes a new mathematical and philosophical synthesis. Thus, it serves as both a ‘casebook’ and a ‘manifesto.’ Furthermore, it reveals a totally new world of plastic beauty.” (Pg. 2)
He goes on, “fractal geometry reveals that some of the most austerely formal chapters of mathematics had a hidden face: a world of pure plastic beauty unsuspected until now… I coined ‘fractal’ from the Latin adjective ‘fractus.’ The corresponding Latin verb … means … to create irregular fragments… how appropriate for our needs!... in addition to ‘fragmented’ … ‘fractus’ should also mean ‘irregular.’ … the present Essay describes the solutions I propose to a host of concrete problems, including very old ones, with the help of mathematics that… had never been used in this fashion. The cases this mathematics allows us to tackle, and the extensions these cases require, lay the foundation of a new discipline.” (Pg. 5)
He clarifies, “This Essay was designed to help make its contents accessible in various degrees to help make its contents accessible in various degrees to a wide range of readers, and to try to convince even the purest among mathematicians that the understanding of known concepts and the search for new concepts and conjectures are both helped by fine graphics. Rarely does contemporary scientific literature show such trust in the usefulness of graphics. However, showing pretty pictures is NOT the main purpose in this Essay; they are an essential tool, but only a tool.” (Pg. 21-22)
He notes, ‘The distribution of the stars, the galaxies, the clusters of galaxies, and so on, fascinates the amateur as well as the specialist, yet clustering remains peripheral to astronomy and to astrophysics as a whole. The basic reason is that no one has yet explained why the distribution of matter falls into an irregular hierarchy, at least within a certain range of scales. While there are allusions to clustering in most works on the subject, serious theoretical developments hasten to sweep it under the rug, claiming that on scales beyond some large but unspecified threshold galaxies are uniformly distributed. Less fundamentally, the hesitation in dealing with the irregular arises from the absence of tools to describe it mathematically. Statistics is asked to decide between two assumptions, only one of which is thoroughly explored (asymptotic uniformity). Is it surprising that the results are inconclusive?” (Pg. 84)
He states, “Whether or not all fractal attractors are strange is a matter of semantics. Increasing numbers of authors agree with me that ‘for most purposes an attractors is strange when it is a fractal.’ This is a healthy attitude, if ‘strange’ is taken to be a synonym to ‘monstrous,’ ‘pathological,’ and other epithets once applied to individual fractals. But ‘strange’ is sometimes given a technical sense… With this definition of ‘strange,’ the argument in the preceding section ceases to be compelling. But it becomes compelling again if strangeness is modified from being a topological to being a fractal notion. Thus, I think that those who define ‘strange’ as ‘fractal’ deserve to win. Since indeed they are winning, there is little reason to preserve a term whose motivation vanished when I showed that fractals are no stranger than coastlines or mountains. Anyhow, I cannot conceal a personal dislike for the term ‘strange.’” (Pg. 197-198)
He explains, “The Fractal Geometry of Nature was first set forth by this author. This geometry combines the mathematics and the science necessary to tackle a certain broad and widespread class of natural shapes… We perceive three different kinds of form in this newly created world: circles, waves, and ‘wiggles.’ The studies of circles and waves benefited from colossal investments of effort by man, and they form the very foundation of science. In comparison, ‘wiggles’ have been left almost totally untouched… To apply dividers to circles and waves had long proven an easy task. But what if we apply dividers to …coastlines on Earth? The result is unexpected…” (Pg. C2)
He says, “I am reluctant to reply (heavy-handedly) by criticizing the concrete ‘mainstream’ theories of relief for failure to come forth with fake landscapes anywhere close in realism to those due to my ‘abstract’ theories. It seems better to point out that many among the finest theories of science did start with exquisite combinations of pistons, strings, and pulleys, only to end (several generations later) with bare-bones invariance principles.” (Pg. 100)
He states, “Irregularity is easy to inject---through randomness. As to invariance by translation, our hoped-for substitute for the Creator dust will only be required to match up with its translation in a statistical sense. In probabilistic terminology, this means that a set has to be stationary, or at least satisfy a suitably weakened condition of stationarity.” (Pg. 281)
Of course, everyone expects a book about fractals to have lots of ‘pretty pictures’ of them---and this book certainly satisfies this desire.
Author Benoit Mandelbrot wrote in the Foreword to this revised edition (1982) of his book, “This work follows and largely replaces my 1977 Essay, ‘Fractals: Form, Chance and Dimension,’ which had followed and largely replaced my 1975 Essay in French… Each stage involved new art, a few deletions, extensive rewriting that affected nearly every section, additions devoted to my older work, and---most important---extensive additions devoted to new developments… My first scientific publication came out on April 30, 1951. Over the years, it had seemed to many that each of my investigations was aimed in a different direction. But this apparent disorder was misleading: it hid a strong unity of purpose, which the present Essay, like its two predecessors, is intended to reveal. Against odds, most of my works turn out to have been the birth pangs of a new scientific discipline.”
He states in the Introduction, “Why is geometry often described as ‘cold’ and ‘dry’? One reason is its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared with … standard geometry---Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite. The existence of these patterns challenges us to … investigate the morphology of the ‘amorphous.’… Responding to this challenge, in conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call ‘fractals.’ The most useful fractals involve CHANCE and both their regularities and their irregularities are statistical… Somme fractal sets are curves of surfaces, others are disconnected ‘dusts,’ and yet others are so oddly shaped that there are no good terms for them in either the sciences or the arts. The reader is urged to sample them now, by browsing through the book’s illustrations.” (Pg. 1)
He continues, “Despite its length, I describe [this book] as a scientific Essay because it is written from a personal point of view and without attempting completeness… This Essay brings together a number of analyses in diverse sciences, and it promotes a new mathematical and philosophical synthesis. Thus, it serves as both a ‘casebook’ and a ‘manifesto.’ Furthermore, it reveals a totally new world of plastic beauty.” (Pg. 2)
He goes on, “fractal geometry reveals that some of the most austerely formal chapters of mathematics had a hidden face: a world of pure plastic beauty unsuspected until now… I coined ‘fractal’ from the Latin adjective ‘fractus.’ The corresponding Latin verb … means … to create irregular fragments… how appropriate for our needs!... in addition to ‘fragmented’ … ‘fractus’ should also mean ‘irregular.’ … the present Essay describes the solutions I propose to a host of concrete problems, including very old ones, with the help of mathematics that… had never been used in this fashion. The cases this mathematics allows us to tackle, and the extensions these cases require, lay the foundation of a new discipline.” (Pg. 5)
He clarifies, “This Essay was designed to help make its contents accessible in various degrees to help make its contents accessible in various degrees to a wide range of readers, and to try to convince even the purest among mathematicians that the understanding of known concepts and the search for new concepts and conjectures are both helped by fine graphics. Rarely does contemporary scientific literature show such trust in the usefulness of graphics. However, showing pretty pictures is NOT the main purpose in this Essay; they are an essential tool, but only a tool.” (Pg. 21-22)
He notes, ‘The distribution of the stars, the galaxies, the clusters of galaxies, and so on, fascinates the amateur as well as the specialist, yet clustering remains peripheral to astronomy and to astrophysics as a whole. The basic reason is that no one has yet explained why the distribution of matter falls into an irregular hierarchy, at least within a certain range of scales. While there are allusions to clustering in most works on the subject, serious theoretical developments hasten to sweep it under the rug, claiming that on scales beyond some large but unspecified threshold galaxies are uniformly distributed. Less fundamentally, the hesitation in dealing with the irregular arises from the absence of tools to describe it mathematically. Statistics is asked to decide between two assumptions, only one of which is thoroughly explored (asymptotic uniformity). Is it surprising that the results are inconclusive?” (Pg. 84)
He states, “Whether or not all fractal attractors are strange is a matter of semantics. Increasing numbers of authors agree with me that ‘for most purposes an attractors is strange when it is a fractal.’ This is a healthy attitude, if ‘strange’ is taken to be a synonym to ‘monstrous,’ ‘pathological,’ and other epithets once applied to individual fractals. But ‘strange’ is sometimes given a technical sense… With this definition of ‘strange,’ the argument in the preceding section ceases to be compelling. But it becomes compelling again if strangeness is modified from being a topological to being a fractal notion. Thus, I think that those who define ‘strange’ as ‘fractal’ deserve to win. Since indeed they are winning, there is little reason to preserve a term whose motivation vanished when I showed that fractals are no stranger than coastlines or mountains. Anyhow, I cannot conceal a personal dislike for the term ‘strange.’” (Pg. 197-198)
He explains, “The Fractal Geometry of Nature was first set forth by this author. This geometry combines the mathematics and the science necessary to tackle a certain broad and widespread class of natural shapes… We perceive three different kinds of form in this newly created world: circles, waves, and ‘wiggles.’ The studies of circles and waves benefited from colossal investments of effort by man, and they form the very foundation of science. In comparison, ‘wiggles’ have been left almost totally untouched… To apply dividers to circles and waves had long proven an easy task. But what if we apply dividers to …coastlines on Earth? The result is unexpected…” (Pg. C2)
He says, “I am reluctant to reply (heavy-handedly) by criticizing the concrete ‘mainstream’ theories of relief for failure to come forth with fake landscapes anywhere close in realism to those due to my ‘abstract’ theories. It seems better to point out that many among the finest theories of science did start with exquisite combinations of pistons, strings, and pulleys, only to end (several generations later) with bare-bones invariance principles.” (Pg. 100)
He states, “Irregularity is easy to inject---through randomness. As to invariance by translation, our hoped-for substitute for the Creator dust will only be required to match up with its translation in a statistical sense. In probabilistic terminology, this means that a set has to be stationary, or at least satisfy a suitably weakened condition of stationarity.” (Pg. 281)
Of course, everyone expects a book about fractals to have lots of ‘pretty pictures’ of them---and this book certainly satisfies this desire.
September 17, 2015
This beautiful book is about Mandelbrot's love of science, mathematics and all forms of knowing. He is humorous at times, dense, and waxing on about fractals and changes that are self similar. It is through the figure of a difference that something is known. Fractals are unique in that they are that difference regardless of scale, that is, as Mandelbrot said of Leibniz, that Leibniz first recognized a straight line as being a curve whose arbitrary measure was universally applicable by itself by any other arbitrary measure.
Fractals are thus, a balance of form and measure and thus perfectly applicable to describing self similiarities that occur throughout various scales. By necessity these fractals are thus found in areas of maximal distribution where it be biological, informational, materially, socially or otherwise. Each bit of aggregate from each context can be traced through out some of the other contexts so that a distribution of their differences can be expressed mathematically as a formalization of unithood -- difference -- itself. Fractals can thus be understood as the limit of scaleless models of difference. Mandelbrot goes over a variety of contexts in which we can understand their expressedly different dimensions, as structured rules through time, or a static interface that modifies itself as scale is adjusted.
At times, Mandelbrot can become overwhelming as he notices as particular "cut" in an equation, be it a variable or an expressed tendency, and in vocalizing it, circulates around that textual point to arrange chapters, whorls on whorls, in which sections and sections of sections let us know when one thing was described and another thing began.
And so, as this is a book about the fractal geometry of nature, Mandelbrot shows us his love by talking admiringly of other mathematicians, many not celebrated, or fully acknowledged in their time. These technicians and their stories become the backdrop of those who developed this metric enough to let us see, and explore these subtle differences and their odd refinements. Indeed, it is really to those quiet, anonymous men, who established the halls of science that Mandelbrot writes this book to, for he would have liked for them to experience the joy he feels at being able to explore these monsters, while many of them did not, due to the contemporaneous level of mathematical understanding not yet understanding how to recognize (and thus, explore) the fractal nature of geometry in all its glories.
Fractals are thus, a balance of form and measure and thus perfectly applicable to describing self similiarities that occur throughout various scales. By necessity these fractals are thus found in areas of maximal distribution where it be biological, informational, materially, socially or otherwise. Each bit of aggregate from each context can be traced through out some of the other contexts so that a distribution of their differences can be expressed mathematically as a formalization of unithood -- difference -- itself. Fractals can thus be understood as the limit of scaleless models of difference. Mandelbrot goes over a variety of contexts in which we can understand their expressedly different dimensions, as structured rules through time, or a static interface that modifies itself as scale is adjusted.
At times, Mandelbrot can become overwhelming as he notices as particular "cut" in an equation, be it a variable or an expressed tendency, and in vocalizing it, circulates around that textual point to arrange chapters, whorls on whorls, in which sections and sections of sections let us know when one thing was described and another thing began.
And so, as this is a book about the fractal geometry of nature, Mandelbrot shows us his love by talking admiringly of other mathematicians, many not celebrated, or fully acknowledged in their time. These technicians and their stories become the backdrop of those who developed this metric enough to let us see, and explore these subtle differences and their odd refinements. Indeed, it is really to those quiet, anonymous men, who established the halls of science that Mandelbrot writes this book to, for he would have liked for them to experience the joy he feels at being able to explore these monsters, while many of them did not, due to the contemporaneous level of mathematical understanding not yet understanding how to recognize (and thus, explore) the fractal nature of geometry in all its glories.
August 11, 2019
I admit, I bought this on a whim while at university because I'd suddenly got into mathematics of fractals and their beauty (mainly due to a heavy dose psychedelic trip that opened my eyes up to the mathematics in nature all around us.. Fibonacci is literally everywhere - in petals, sunflower seeds, branches etc. Fractals are the shape they take.. clouds, coastlines, trees etc..)
I understood about 10% of the mathematics, but this book deserves high praise because this is THE man himself who discovered the absolute simplicity of a formula
Zn+1 = Zn2 + C
That produces the infinitely zoomable fractals we all know and love and that explains more about geometry that the numbers that make up natures patterns perfectly. Until Benoit B Mandelbrot.. we didn't know how to describe any shape that wasn't connected by two dots!
Also, with this review I get to write my only fractal joke..
Q. What does the B in Benoit B Mandelbrot stand for ?
A. Benoit B Mandelbrot
Its worth having on your shelf to look smart and in the know about all things fractal. There are plenty of stunning colour glossy prints in it, the text is readable, the mathematics is beyond me.
Thankfully, its a great mix of all these things, so it will appeal to all levels.
I understood about 10% of the mathematics, but this book deserves high praise because this is THE man himself who discovered the absolute simplicity of a formula
Zn+1 = Zn2 + C
That produces the infinitely zoomable fractals we all know and love and that explains more about geometry that the numbers that make up natures patterns perfectly. Until Benoit B Mandelbrot.. we didn't know how to describe any shape that wasn't connected by two dots!
Also, with this review I get to write my only fractal joke..
Q. What does the B in Benoit B Mandelbrot stand for ?
A. Benoit B Mandelbrot
Its worth having on your shelf to look smart and in the know about all things fractal. There are plenty of stunning colour glossy prints in it, the text is readable, the mathematics is beyond me.
Thankfully, its a great mix of all these things, so it will appeal to all levels.
January 10, 2023
A fractal is by definition
A set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.
The definition of a fractal pretty much sets the tone for the book. There are mostly definitions and monochrome diagrams to explain the more classical fractals. The book does show some practical geometric uses for fractals but I would not let it get anywhere near my Koch Curve.
I am not being kind to this book as there is a color section in the center. That shows “The Great Wave” by Katsushika Hokusai (1760-12849.) And an extensive reference section.
The book itself could easily be used as a textbook for school.
A set for which the Hausdorff Besicovitch dimension strictly exceeds the topological dimension.
The definition of a fractal pretty much sets the tone for the book. There are mostly definitions and monochrome diagrams to explain the more classical fractals. The book does show some practical geometric uses for fractals but I would not let it get anywhere near my Koch Curve.
I am not being kind to this book as there is a color section in the center. That shows “The Great Wave” by Katsushika Hokusai (1760-12849.) And an extensive reference section.
The book itself could easily be used as a textbook for school.
July 23, 2015
"The Fractal Geometry of Nature" is a book to be skimmed by many but read closely by few because its excellences are so uneven. On the one hand Mandelbrot creates truly wonderful images that allow most people to see how nature can be described by fractals. On the other hand he freely engages in formulas that any non-mathematician reads but fails to grasp. In all honesty I probably understood all of his images, 4/5 of his prose, and perhaps a 1/4 of his formulae. Still, I am far richer intellectually for reading his work and you will be too - but you might just want to look at the pictures.
January 15, 2018
The writing style is difficult, but soldier on. It is worth the effort. If you are interested in Chaos theory, Spinoza, the beauty of maths, this will give you goosebumps.
July 9, 2014
Amazing images and a very cool concept, well explained. Fractal make sense out of nature.
October 10, 2023
"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
A book that mainly delves into the concept of fractals, which are complex, self-replicating geometric patterns found in nature.
Fractals offer a fascinating intersection of mathematics, geometry, and art, with countless other equations and variations to explore and create stunning visual patterns. Some of the most famous ones include:
1. Mandelbrot Set
2. Julia Set
3. Koch Snowflake
4. Sierpinski Triangle
5. Dragon Curve
What makes this book an interesting read is its ability to bridge the gap between mathematics and the natural world. It challenges conventional geometric thinking and offers a new perspective on understanding the complexity and beauty of nature.
My favourite part of the book is where Mandelbrot explores various natural phenomena and demonstrates how fractal geometry can be used to describe them, for example:
Clouds and Trees: Mandelbrot shows how the branching patterns in trees and the irregular shapes of clouds can be described using fractal mathematics. For example, the branching of tree limbs follows a self-replicating pattern that can be expressed through fractal equations.
Coastlines: Mandelbrot discusses how coastlines are not smooth curves but exhibit intricate, jagged patterns. He introduces the concept of coastline paradox, where the measured length of a coastline depends on the scale of measurement.
Overall this book will be an insightful read for anyone interested in the intersection of mathematics and nature.
A book that mainly delves into the concept of fractals, which are complex, self-replicating geometric patterns found in nature.
Fractals offer a fascinating intersection of mathematics, geometry, and art, with countless other equations and variations to explore and create stunning visual patterns. Some of the most famous ones include:
1. Mandelbrot Set
2. Julia Set
3. Koch Snowflake
4. Sierpinski Triangle
5. Dragon Curve
What makes this book an interesting read is its ability to bridge the gap between mathematics and the natural world. It challenges conventional geometric thinking and offers a new perspective on understanding the complexity and beauty of nature.
My favourite part of the book is where Mandelbrot explores various natural phenomena and demonstrates how fractal geometry can be used to describe them, for example:
Clouds and Trees: Mandelbrot shows how the branching patterns in trees and the irregular shapes of clouds can be described using fractal mathematics. For example, the branching of tree limbs follows a self-replicating pattern that can be expressed through fractal equations.
Coastlines: Mandelbrot discusses how coastlines are not smooth curves but exhibit intricate, jagged patterns. He introduces the concept of coastline paradox, where the measured length of a coastline depends on the scale of measurement.
Overall this book will be an insightful read for anyone interested in the intersection of mathematics and nature.
Read
January 27, 2023The Fractal Geometry of Nature by Benoit Mandelbrot is a fascinating look into the world of fractals and their application in various fields of study. Mandelbrot's writing is clear and easy to understand, making the complex concepts of fractal geometry accessible to readers with little to no background in mathematics.
One of the key concepts in the book is the idea that fractals can be found in many natural phenomena, such as coastlines, rivers, and trees. Mandelbrot uses a plethora of examples to illustrate this concept, including photographs of coastlines and detailed drawings of fractal patterns in plants. He also explores the mathematical properties of fractals, such as their infinite complexity and self-similarity.
Another important concept in the book is the idea that fractals can be used to model and understand complex systems in fields such as finance and weather forecasting. Mandelbrot discusses the ways in which fractal geometry can be used to analyze stock market fluctuations and predict weather patterns.
Overall, The Fractal Geometry of Nature is a comprehensive and engaging introduction to the world of fractals. It provides a deep understanding of the concepts and applications of fractal geometry, making it a great read for anyone interested in mathematics, physics, and natural phenomena.
Overall, I highly recommend this book to anyone interested in mathematics, physics, and natural phenomena. It's a great read for anyone who wants to understand the concepts and applications of fractal geometry.
One of the key concepts in the book is the idea that fractals can be found in many natural phenomena, such as coastlines, rivers, and trees. Mandelbrot uses a plethora of examples to illustrate this concept, including photographs of coastlines and detailed drawings of fractal patterns in plants. He also explores the mathematical properties of fractals, such as their infinite complexity and self-similarity.
Another important concept in the book is the idea that fractals can be used to model and understand complex systems in fields such as finance and weather forecasting. Mandelbrot discusses the ways in which fractal geometry can be used to analyze stock market fluctuations and predict weather patterns.
Overall, The Fractal Geometry of Nature is a comprehensive and engaging introduction to the world of fractals. It provides a deep understanding of the concepts and applications of fractal geometry, making it a great read for anyone interested in mathematics, physics, and natural phenomena.
Overall, I highly recommend this book to anyone interested in mathematics, physics, and natural phenomena. It's a great read for anyone who wants to understand the concepts and applications of fractal geometry.
November 27, 2017
The fractal geometry of nature is about fractals and how it will never end. Each chapter is all different but related to fractals. It talks about things that will help you in your life if you are an engineer,mathematician, etc. This book gives you so much knowledge if you want to be a mathematician.Fractals are a genomic figure each part if it is that same as the hole.
The thing I like about this book is that it teaches you so many things. You'r know more things in the book than others.It's teaching you about different things in nature and in other things like models for clothes, mountains, etc. The things I didn't like about was that I didn't understand some of the words because they were so complex to say. The words were to small in the book for me to read.
I would recommend this book to a person who loves to read about math.If they like to learn about knew things I would recommended this book to them.I would give this to a person who is 10 or older.
The thing I like about this book is that it teaches you so many things. You'r know more things in the book than others.It's teaching you about different things in nature and in other things like models for clothes, mountains, etc. The things I didn't like about was that I didn't understand some of the words because they were so complex to say. The words were to small in the book for me to read.
I would recommend this book to a person who loves to read about math.If they like to learn about knew things I would recommended this book to them.I would give this to a person who is 10 or older.
This entire review has been hidden because of spoilers.
April 2, 2018
This is an absolute classic!
Mandelbrot claims that patterns of Nature are so irregular that, compared with Euclid, Nature exhibits not simply a higher degree but an altogether different level of complexity. This is why he developed fractal geometry to describe the "formless" irregular and fragmented patterns around us. He called this family of shapes fractals.
Mandelbrot provides a mathematical definition of fractal (p.15), but believes that one would do better without a definition (p.361). A common misunderstanding is that the notion of fractal is wedded to self-similarity. This is emphatically not the case (p.166). There are non-self-similar fractals. It all depends on how they are generated.
Mandelbrot claims that patterns of Nature are so irregular that, compared with Euclid, Nature exhibits not simply a higher degree but an altogether different level of complexity. This is why he developed fractal geometry to describe the "formless" irregular and fragmented patterns around us. He called this family of shapes fractals.
Mandelbrot provides a mathematical definition of fractal (p.15), but believes that one would do better without a definition (p.361). A common misunderstanding is that the notion of fractal is wedded to self-similarity. This is emphatically not the case (p.166). There are non-self-similar fractals. It all depends on how they are generated.
June 3, 2019
Amazing book, only getting better over time, even good read for non-mathematicians, like me. Fascinating process - the birth of new math allowing to model nature shapes - trees, clouds, rocks, mountains etc (including monstrous curves and snowflakes), direct from a father of computer graphics.
Want to read
January 25, 2020Got to know while reading Black Swan book. You can see it in the page 259
October 14, 2021
Beautiful pictures and excellent introduction to fractals. Perhaps a bit too technical for a casual reader.
Read
February 8, 2022Mandelbrot er mer enn ett geni! Hans fractals og scaling er fascinerende, men det blir for matematisk og teoretisk for min forståelse, for nå…
Read
March 20, 2022Too difficult.
July 25, 2022
Not really a book for simple readers or hobbyists. Better to look at pictures and read about the basics online.
March 26, 2024
math is gay
November 5, 2020
Wow, just wow.
This book is simply a masterpiece; mind-blowing content & breathtaking graphics.
Within this title Benoit Mandelbrot outlines his findings in the science of fractal geometry, a science I previously had not explicitly encountered prior to this reading experience, but from working my way through the book found myself seeing the world & in particular my perception of it, completely different.
A strong motivation for this personal enlightenment was the confrontation that what is observed is merely so in the context of the viewpoint of the observer, I choose to believe this to be a powerful insight as to how we, human beings, perceive the world around us, more so than necessarily true insight of the world around us.
I absolutely fell in love with the science of fractals, however found myself still lacking complete ownership of the ideas, if there is anyone else that has sought a personal/ professional pursuit of fractal science, I would be most grateful if a path to best grasp the ideas was shared - which complimentary disciplines & associated titles.
I have a mechanical engineering background, but clearly lack applied mathematical skills to truly grasp the literature as a whole.
This book is simply a masterpiece; mind-blowing content & breathtaking graphics.
Within this title Benoit Mandelbrot outlines his findings in the science of fractal geometry, a science I previously had not explicitly encountered prior to this reading experience, but from working my way through the book found myself seeing the world & in particular my perception of it, completely different.
A strong motivation for this personal enlightenment was the confrontation that what is observed is merely so in the context of the viewpoint of the observer, I choose to believe this to be a powerful insight as to how we, human beings, perceive the world around us, more so than necessarily true insight of the world around us.
I absolutely fell in love with the science of fractals, however found myself still lacking complete ownership of the ideas, if there is anyone else that has sought a personal/ professional pursuit of fractal science, I would be most grateful if a path to best grasp the ideas was shared - which complimentary disciplines & associated titles.
I have a mechanical engineering background, but clearly lack applied mathematical skills to truly grasp the literature as a whole.
May 23, 2008
This book may just provide a hunger in a student to learn more about what fractals are in math. It is something I have studied for quite sometime and I hope to incorporate in my future math classes as a teacher. The images and mathematical properties of fractals are fascinating! Even if someone doesn't understand the mathematics behind fractals they can use technology to create some interesting fractals.
December 28, 2011
The complexity of nature as explained by the genius behind the study of fractals. I can never look at a fern leaf the same way again. The illustrations alone in this gem of a book are worth its purchase price. Gaussian curves, the Julia Set, all explained - albeit a little technical for me in many chapters.
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