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Mathematics in Nature: Modeling Patterns in the Natural World
From rainbows, river meanders, and shadows to spider webs, honeycombs, and the markings on animal coats, the visible world is full of patterns that can be described mathematically. Examining such readily observable phenomena, this book introduces readers to the beauty of nature as revealed by mathematics and the beauty of mathematics as revealed in nature.
Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks.
Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.
Generously illustrated, written in an informal style, and replete with examples from everyday life, Mathematics in Nature is an excellent and undaunting introduction to the ideas and methods of mathematical modeling. It illustrates how mathematics can be used to formulate and solve puzzles observed in nature and to interpret the solutions. In the process, it teaches such topics as the art of estimation and the effects of scale, particularly what happens as things get bigger. Readers will develop an understanding of the symbiosis that exists between basic scientific principles and their mathematical expressions as well as a deeper appreciation for such natural phenomena as cloud formations, halos and glories, tree heights and leaf patterns, butterfly and moth wings, and even puddles and mud cracks.
Developed out of a university course, this book makes an ideal supplemental text for courses in applied mathematics and mathematical modeling. It will also appeal to mathematics educators and enthusiasts at all levels, and is designed so that it can be dipped into at leisure.
392 pages, Paperback
First published January 1, 2003
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Displaying 1 - 7 of 7 reviews
January 8, 2025
My aunt gave this book to my dad, her brother-in-law, a college math professor, and I read it thinking of both of them. I could tell my dad read the book, or at least some of it, because of the notations he'd made in the margins.
My aunt was excited about the book, although I'd be surprised if she understood any of the math because her discipline took her in a different direction to a M.S. in special ed. I remember someone making the snide comment about her "finding God in a test tube," meaning, I think, in unlikely places not likely to prove His existence. I doubt that person would understand the math either.
This book is, actually, not about proving or disproving the existence of God at all. The author, a PhD in theoretical astrophysics, put Bible verses in a quote for each chapter's introduction, without really commenting on them. I think he liked the descriptive phrases used for the physics problem or phenomenon he was about to discuss.
He did use "By his stripes we are healed." - Isaiah 53:5a for his chapter on "Can the Leopard Change Its Spots?" about variation of skin pigmentation in animals, that is cheetahs, tigers, zebras, giraffes, etc. Most of the rest of the verses were more descriptive of the problem at hand. This one was more of a stretch, because the stripes of Christ were an injury and not skin pigmentation. I still happen to think it's a beautiful verse.
The author did not believe in a young-earth theory, and uses his last chapter to calculate the age of the earth from a heat diffusion/Fourier transform standpoint. It's been a long time since I looked at a Fourier transform and never in that context before.
Now that I've gotten that big topic out of the way, I'll review the rest of this like the mathematics book that it is.
One can tell that John A. Adam enjoyed writing this book. There's so much humor in it, and some of that I'll put in my favorite quotes section of my review at the bottom. I thought it was funny that he said that we could do some of this math as a party game. I don't see that as very likely, even in the event that it's a math dept party.
I found it engrossing, and I intend to put it on my shelf of physics books (I teach physics) in my classroom. I think part of what made it so engrossing were the vivid images that the mathematics and their related concepts conveyed to my mind.
Most of the time, I could just see it so clearly. Rainbows, mirages, iridescence, clouds including cloud streets, sand dunes, waves, wave bores (my husband wants to time a trip to see a bore), bees making hexagonal honeycombs, bubbles, foam, the goosenecks of a meandering river, trees, the position of leaves, and the shape of bird wings.
I found it captivating and didn't notice the passage of time, or when I should do something else. I got lost somewhere around the foam and wish there had been better pictures associated with that.
I didn't realize until I was several chapters into it that the photographs of the phenomena were in the center section of the book. Some of them were illuminating, others I was disappointed in, because I've seen it so much more clearly in real life, whether or not I actually have a photograph. Still, I feel like surely the author could've taken or borrowed better images than these, and there were no foam bubble photographs at all.
I recently discovered anew while teaching about levers to a very basic engineering class that not everyone who looks at math equations sees the images they portray. Some students just wanted to throw numbers at the equations and get out answers rather than trying to understand it, or being able to convert between the equations to words and back again. They complained that the answer to an exam question was not in the text they'd studied. I said that it was in their text in the equation, in understanding the relationships between the variables.
My point is that this "Mathematics in Nature" book is intended for people who can see images, or at least relationships, when they read math equations.
This book is written for people who have had math through at least Calculus 1 and there are some differential equations.
I liked the chapter "The Problem of Scale." It reminds me of the time(s) that I told students they'd miscalculated the amount of energy in a rolling ball across the floor, ending up with an answer that would've been more appropriate for a moving car.
I had not realized how the earth's elliptical orbit is so close to being circular, or even that there was an eccentricity measurement of orbits to evaluate how circular an orbit is.
I was glad to see that he used "arctan" rather than the more common phrase "inverse tan," because I use both terms with my students, so that they can understand the literature anywhere they go.
I had not realized that the second bow of a double rainbow was in reverse color order, with red on the bottom rather than on top or why they are brighter at the bottom.
I loved the ray diagrams of rainbows and double-rainbows, though. We did ray diagrams in the physics class I teach and I would add those as an interesting note, but the school wants me to teach a math review at the beginning of the class, and so ray diagrams have gone out the door.
I did not know that acoustic mirages can occur.
I liked the description of waves with gravity and surface tension components. It's interesting why waves usually slew around to being parallel to the shore. It's also interesting when a wave breaks (when the ratio is 7:1 in height (2A) to wavelength and the studies on the stability of waves (when they reach equilibrium and are no longer propagating.)
It's interesting how as the waves near the shore, a buoy goes from circular motion to elliptical to horizontal.
It was also interesting defining viscosity as a sort of friction between fluid layers and the idea of nonturbulent bores being undular bores.
I did not know that the sun had as much effect on the tides as it does, about half that of the moon, or that the tides are effected not by the gravitational force, per se, but the derivative of it as it changes over time.
At one point, I was convinced the author had an equation wrong, but when I looked it up, to my chagrin, I was the one misremembering it. Another time, I thought not all the variables had been defined, but in going back through it, I found they had. At another point, I had a hard time jumping from step to step with him in solving the equations, and he confessed later, "we shall wave our hands a little and just state the relevant formulas" again as he had with that section. So he didn't have all the steps there for me to follow along.
I have often thought of the bending of trees as a sort of spring, ever since I rode on one such bending tree as a child, jumping onto it from above and riding it as it bounced until it stopped.
I had always thought that the bird at the head of the v when migrating had the hardest position. At least they do in the animal stories where they are anthropomorphized, but this shows that mathematically that extra burden's not so. The v shape helps all the birds equally, and that the down draft of the air beneath the wings produces an updraft just beyond the wings that helps the neighboring birds.
I didn't know it takes about 3 months for sugar to diffuse into a cup of tea without stirring.
Favorite quotes:
"Like most others in my profession, I continue to be fascinated by the beauty, power, and applicability of mathematics ..." Well said. That is what I wish I could convey to my students, although I did overhear one student say to a classmate after I'd told them the physics class would feel like another math class, "I don't care. I LIKE math." It warmed my heart. So there are still students who appreciate the subject.
"Mathematics is to nature what Sherlock Holmes is to evidence." - Ian Stewart in "Nature's Numbers"
"... but the classic book by Goldstein should be consulted (and possibly tossed around)" about which way a falling plane is more stable.
"syzygy, a word that has been especially noted by Scrabble fanatics." I'll have to remember that for Scrabble or Bananagrams. It's when the sun, earth, and moon are collinear and tides are at their highest.
"It is easy to find all sorts of apparent relationships between numbers ... if one substitutes wishful thinking for a systematic search pattern and consistency of approximation." This was on misattributing things to the golden ratio. I had heard of the human body exhibiting the golden ratio. The source says that's a misconception.
"Just thinking of how much mathematics and physics is involved should prompt us all to offer to wash the dishes on a regular basis." On bubbles and foam.
"Man: the only animal to purposely wear plaid." - Reader's Digest, on animals' pigmentation variations
"The idea is that you are in a crowded room with perhaps fifty other people, all of whom are engaged in the kind of loud conversations that seem to be necessary at such a gathering." Sounds like an introvert talking.
"There is not necessarily a 'right' model, and obtaining results that are consistent with observations is only a first step; it does not imply that the model is the only one that applies or even that it is 'correct.' Furthermore, mathematical descriptions are not explanations, and never on their own can they provide a complete solution..."
I'd paraphrase that to mean that some mathematical models are just curve-fitting, and conceptually might be quite different from what is actually happening. I have a vague, sneaking suspicion that happens every now and then with cutting-edge physics. In engineering, we are just happy with a curve-fit that we can use with good results, but this is wanting something conceptual beyond that. Generally speaking, it doesn't explain where it came from or where it went or even what its exact role was in the mechanism.
My aunt was excited about the book, although I'd be surprised if she understood any of the math because her discipline took her in a different direction to a M.S. in special ed. I remember someone making the snide comment about her "finding God in a test tube," meaning, I think, in unlikely places not likely to prove His existence. I doubt that person would understand the math either.
This book is, actually, not about proving or disproving the existence of God at all. The author, a PhD in theoretical astrophysics, put Bible verses in a quote for each chapter's introduction, without really commenting on them. I think he liked the descriptive phrases used for the physics problem or phenomenon he was about to discuss.
He did use "By his stripes we are healed." - Isaiah 53:5a for his chapter on "Can the Leopard Change Its Spots?" about variation of skin pigmentation in animals, that is cheetahs, tigers, zebras, giraffes, etc. Most of the rest of the verses were more descriptive of the problem at hand. This one was more of a stretch, because the stripes of Christ were an injury and not skin pigmentation. I still happen to think it's a beautiful verse.
The author did not believe in a young-earth theory, and uses his last chapter to calculate the age of the earth from a heat diffusion/Fourier transform standpoint. It's been a long time since I looked at a Fourier transform and never in that context before.
Now that I've gotten that big topic out of the way, I'll review the rest of this like the mathematics book that it is.
One can tell that John A. Adam enjoyed writing this book. There's so much humor in it, and some of that I'll put in my favorite quotes section of my review at the bottom. I thought it was funny that he said that we could do some of this math as a party game. I don't see that as very likely, even in the event that it's a math dept party.
I found it engrossing, and I intend to put it on my shelf of physics books (I teach physics) in my classroom. I think part of what made it so engrossing were the vivid images that the mathematics and their related concepts conveyed to my mind.
Most of the time, I could just see it so clearly. Rainbows, mirages, iridescence, clouds including cloud streets, sand dunes, waves, wave bores (my husband wants to time a trip to see a bore), bees making hexagonal honeycombs, bubbles, foam, the goosenecks of a meandering river, trees, the position of leaves, and the shape of bird wings.
I found it captivating and didn't notice the passage of time, or when I should do something else. I got lost somewhere around the foam and wish there had been better pictures associated with that.
I didn't realize until I was several chapters into it that the photographs of the phenomena were in the center section of the book. Some of them were illuminating, others I was disappointed in, because I've seen it so much more clearly in real life, whether or not I actually have a photograph. Still, I feel like surely the author could've taken or borrowed better images than these, and there were no foam bubble photographs at all.
I recently discovered anew while teaching about levers to a very basic engineering class that not everyone who looks at math equations sees the images they portray. Some students just wanted to throw numbers at the equations and get out answers rather than trying to understand it, or being able to convert between the equations to words and back again. They complained that the answer to an exam question was not in the text they'd studied. I said that it was in their text in the equation, in understanding the relationships between the variables.
My point is that this "Mathematics in Nature" book is intended for people who can see images, or at least relationships, when they read math equations.
This book is written for people who have had math through at least Calculus 1 and there are some differential equations.
I liked the chapter "The Problem of Scale." It reminds me of the time(s) that I told students they'd miscalculated the amount of energy in a rolling ball across the floor, ending up with an answer that would've been more appropriate for a moving car.
I had not realized how the earth's elliptical orbit is so close to being circular, or even that there was an eccentricity measurement of orbits to evaluate how circular an orbit is.
I was glad to see that he used "arctan" rather than the more common phrase "inverse tan," because I use both terms with my students, so that they can understand the literature anywhere they go.
I had not realized that the second bow of a double rainbow was in reverse color order, with red on the bottom rather than on top or why they are brighter at the bottom.
I loved the ray diagrams of rainbows and double-rainbows, though. We did ray diagrams in the physics class I teach and I would add those as an interesting note, but the school wants me to teach a math review at the beginning of the class, and so ray diagrams have gone out the door.
I did not know that acoustic mirages can occur.
I liked the description of waves with gravity and surface tension components. It's interesting why waves usually slew around to being parallel to the shore. It's also interesting when a wave breaks (when the ratio is 7:1 in height (2A) to wavelength and the studies on the stability of waves (when they reach equilibrium and are no longer propagating.)
It's interesting how as the waves near the shore, a buoy goes from circular motion to elliptical to horizontal.
It was also interesting defining viscosity as a sort of friction between fluid layers and the idea of nonturbulent bores being undular bores.
I did not know that the sun had as much effect on the tides as it does, about half that of the moon, or that the tides are effected not by the gravitational force, per se, but the derivative of it as it changes over time.
At one point, I was convinced the author had an equation wrong, but when I looked it up, to my chagrin, I was the one misremembering it. Another time, I thought not all the variables had been defined, but in going back through it, I found they had. At another point, I had a hard time jumping from step to step with him in solving the equations, and he confessed later, "we shall wave our hands a little and just state the relevant formulas" again as he had with that section. So he didn't have all the steps there for me to follow along.
I have often thought of the bending of trees as a sort of spring, ever since I rode on one such bending tree as a child, jumping onto it from above and riding it as it bounced until it stopped.
I had always thought that the bird at the head of the v when migrating had the hardest position. At least they do in the animal stories where they are anthropomorphized, but this shows that mathematically that extra burden's not so. The v shape helps all the birds equally, and that the down draft of the air beneath the wings produces an updraft just beyond the wings that helps the neighboring birds.
I didn't know it takes about 3 months for sugar to diffuse into a cup of tea without stirring.
Favorite quotes:
"Like most others in my profession, I continue to be fascinated by the beauty, power, and applicability of mathematics ..." Well said. That is what I wish I could convey to my students, although I did overhear one student say to a classmate after I'd told them the physics class would feel like another math class, "I don't care. I LIKE math." It warmed my heart. So there are still students who appreciate the subject.
"Mathematics is to nature what Sherlock Holmes is to evidence." - Ian Stewart in "Nature's Numbers"
"... but the classic book by Goldstein should be consulted (and possibly tossed around)" about which way a falling plane is more stable.
"syzygy, a word that has been especially noted by Scrabble fanatics." I'll have to remember that for Scrabble or Bananagrams. It's when the sun, earth, and moon are collinear and tides are at their highest.
"It is easy to find all sorts of apparent relationships between numbers ... if one substitutes wishful thinking for a systematic search pattern and consistency of approximation." This was on misattributing things to the golden ratio. I had heard of the human body exhibiting the golden ratio. The source says that's a misconception.
"Just thinking of how much mathematics and physics is involved should prompt us all to offer to wash the dishes on a regular basis." On bubbles and foam.
"Man: the only animal to purposely wear plaid." - Reader's Digest, on animals' pigmentation variations
"The idea is that you are in a crowded room with perhaps fifty other people, all of whom are engaged in the kind of loud conversations that seem to be necessary at such a gathering." Sounds like an introvert talking.
"There is not necessarily a 'right' model, and obtaining results that are consistent with observations is only a first step; it does not imply that the model is the only one that applies or even that it is 'correct.' Furthermore, mathematical descriptions are not explanations, and never on their own can they provide a complete solution..."
I'd paraphrase that to mean that some mathematical models are just curve-fitting, and conceptually might be quite different from what is actually happening. I have a vague, sneaking suspicion that happens every now and then with cutting-edge physics. In engineering, we are just happy with a curve-fit that we can use with good results, but this is wanting something conceptual beyond that. Generally speaking, it doesn't explain where it came from or where it went or even what its exact role was in the mechanism.
April 29, 2010
Really complicated math; presented in an astute manner. Reads well with Mario Livio...but you might need a graphing calculator to actually comprehend what he's talking about.
August 3, 2013
Fantastic book with beautiful topics ... rainbows, dunes, trees, etc ... However, the mathematics is not easy, you need to be fit in algebra and (for some parts) calculus
December 14, 2018
A lovely pot pourri of maths applied to natural phenomena. The level is slightly beyond high school maths, I imagine, as I'm out of touch. You need to be at ease with differential equations to enjoy this book. That being said, it is a great way to understand the maths underlying things we see every day: clouds, waves, eroded rivers, etc. The author casts his not wide and strikes the right balance for me between oversimplifying and delving too deep into the intricacies.
July 2, 2024
I barely understood half of this book (nature written in math formulas!!) but the author is brilliant and has done remarkable work in investigating nature and the awe-inspiring patterns that make them the phenomenon that they are.
January 29, 2020
This book is interesting and there's a lot of learnings
This entire review has been hidden because of spoilers.
November 15, 2015
great book...complicated but amazing
Displaying 1 - 7 of 7 reviews