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Vector: A Surprising Story of Space, Time, and Mathematical Transformation
A celebration of the seemingly simple idea that allowed us to imagine the world in new dimensions—sparking both controversy and discovery.ÌýÌýThe stars of this book, vectors and tensors, are unlikely celebrities. If you ever took a physics course, the word “vectorâ€� might remind you of the mathematics needed to determine forces on an amusement park ride, a turbine, or a projectile. You might also remember that a vector is a quantity that has magnitude and (this is the key) direction. In fact, vectors are examples of tensors, which can represent even more data. It sounds simple enough—and yet, as award-winning science writer Robyn Arianrhod shows in this riveting story, the idea of a single symbol expressing more than one thing at once was millennia in the making. And without that idea, we wouldn’t have such a deep understanding of our world.Vector and tensor calculus offers an elegant language for expressing the way things behave in space and time, and Arianrhod shows how this enabled physicists and mathematicians to think in a brand-new way. These include James Clerk Maxwell when he ushered in the wireless electromagnetic age; Einstein when he predicted the curving of space-time and the existence of gravitational waves; Paul Dirac, when he created quantum field theory; and Emmy Noether, when she connected mathematical symmetry and the conservation of energy. For it turned out that it’s not just physical quantities and dimensions that vectors and tensors can represent, but other dimensions and other kinds of information, too. This is why physicists and mathematicians can speak of four-dimensional space-time and other higher-dimensional “spaces,â€� and why you’re likely relying on vectors or tensors whenever you use digital applications such as search engines, GPS, or your mobile phone.In exploring the evolution of vectors and tensors—and introducing the fascinating people who gave them to us—Arianrhod takes readers on an extraordinary, five-thousand-year journey through the human imagination. SheÌýshows the genius required to reimagine the world—and how a clever mathematical construct can dramatically change discovery’s direction.
376 pages, Hardcover
First published July 1, 2024
101 people are currently reading
710 people want to read
About the author
Robyn Arianrhod
8Ìýbooks422ÌýfollowersRobyn Arianrhod is an Australian science writer historian of science known for her works on the predecessors to Albert Einstein, on Émilie du Châtelet and Mary Somerville, and on Thomas Harriot.
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Displaying 1 - 17 of 17 reviews
September 24, 2024
This is an outstanding book. It's non-fiction, and certainly not pop-sci, so only of interest to those with some knowledge of the field of mathematics, and specifically the fields of vector calculus, and real and complex analysis. And let's be honest, that's a narrow field.
Nevertheless, if your interest in mathematics has at least floated around the edge of these topics, then this book will blow your mind. You only need to follow the ideas, not the detail, though the ideas are mathematical not prose.
Most importantly there's a story here. It's the story of the development of vectors â€� and latterly tensors –Ìýtold via the critical path of physics: namely, Newton, Maxwell, Einstein, and then Quantum Mechanics via Dirac/Noether et al.
I've never seen mathematics told in this way, but I believe it should be more often. It shoes how the best minds, over many generations, struggled to uncover both the mathematical formulation, symbology, and physics that we pretty much take for granted today. It took a long time. Much longer than the popular myths suggests. And with far, far more effort.
And the story told here, if you can follow the maths, is easily an equal to Dickens, Shakespeare, Tolkien or whoever it is that floats your boat. It's a cracker.
Who should read it? If you are blown away by the fact that the stress tensor is included in Maxwell's Treatise, then this is the book for you. If you have no idea what that means, but you are curious, then you will enjoy it immensely, and perhaps be driven to greater things.
I think this is the best non-fiction book I've ever read. I also recognise that I'm the target audience, and we are a quite a small group.
Nevertheless, if your interest in mathematics has at least floated around the edge of these topics, then this book will blow your mind. You only need to follow the ideas, not the detail, though the ideas are mathematical not prose.
Most importantly there's a story here. It's the story of the development of vectors â€� and latterly tensors –Ìýtold via the critical path of physics: namely, Newton, Maxwell, Einstein, and then Quantum Mechanics via Dirac/Noether et al.
I've never seen mathematics told in this way, but I believe it should be more often. It shoes how the best minds, over many generations, struggled to uncover both the mathematical formulation, symbology, and physics that we pretty much take for granted today. It took a long time. Much longer than the popular myths suggests. And with far, far more effort.
And the story told here, if you can follow the maths, is easily an equal to Dickens, Shakespeare, Tolkien or whoever it is that floats your boat. It's a cracker.
Who should read it? If you are blown away by the fact that the stress tensor is included in Maxwell's Treatise, then this is the book for you. If you have no idea what that means, but you are curious, then you will enjoy it immensely, and perhaps be driven to greater things.
I think this is the best non-fiction book I've ever read. I also recognise that I'm the target audience, and we are a quite a small group.
November 10, 2024
By halfway through this, I will confess to having to admit the maths had the better of me, and calculus will need to remain the obscured art it has been for some time. I found it very difficult at school, so I suspect this is more me than the writing.
I did enjoy the history here, and I know understand much better how mathematics advances were essential to physics advancing, and also how they have laid the basis for so much modern technology. Maths is about modelling how things work, an approach I still wish I had had more embedded in my teaching. And by using abstractions, we can model things that we can't easily imagine (or measure - it was a revalation to me to realise how difficult calculating the length of a curve is when you can't just use string).
I did enjoy the history here, and I know understand much better how mathematics advances were essential to physics advancing, and also how they have laid the basis for so much modern technology. Maths is about modelling how things work, an approach I still wish I had had more embedded in my teaching. And by using abstractions, we can model things that we can't easily imagine (or measure - it was a revalation to me to realise how difficult calculating the length of a curve is when you can't just use string).
December 18, 2024
5/5
Fantastic overview of the develop of quaternions to vectors to tensors, culminating in Einstein's theory of relativity. Arianrhod goes quite deep into the math and while a layman should be able to follow along, some background knowledge of geometry and analysis will enhance the reading experiend. As someone with the background, I liked that the author was willing to dig further into the math and not just stick to surface level pop science. I picked this book up because quaternions are super neat and was disappointed to learn that vectors arose partially from quaternions due to an abuse (willing ignorance?) of quaternion notation.
Fantastic overview of the develop of quaternions to vectors to tensors, culminating in Einstein's theory of relativity. Arianrhod goes quite deep into the math and while a layman should be able to follow along, some background knowledge of geometry and analysis will enhance the reading experiend. As someone with the background, I liked that the author was willing to dig further into the math and not just stick to surface level pop science. I picked this book up because quaternions are super neat and was disappointed to learn that vectors arose partially from quaternions due to an abuse (willing ignorance?) of quaternion notation.
January 28, 2025
I recommend this book to anyone with interest in mathematics. However, I believe it would be difficult to follow without a university level understanding of mathematics. It is interesting to learn more about a general history of these concepts, something that I have not learnt much about in my undergraduate or masters. The authors passion for mathematics also reignites my interest, and reminds me of why I initially started studying mathematics! Looking forward to reading more from this author.
September 22, 2024
Disappointing. I’m not clear who the target audience is - early on she explains some quite basic maths but from the middle onwards the level of maths knowledge required is rather higher. Unlike Carlo Rovelli, for example, she is not able to distil the essence for the layman but instead awkwardly straddles the border between those with limited maths knowledge and those with maths degrees. The endnotes don’t work, for me at least. I’d rather the book had focussed on the ideas (of which there are many and they are intriguing) - their origin, development and interleaving, and make the personalities behind them secondary, maybe relegating their details to ‘end notes�.
November 28, 2024
Thoughts While Reading
* Algebraic symbolism appears so natural today that I rarely think of a time when it was not the rule of the land. Yet, it took a lot of work for human beings to abstract from a tangible way of thinking (geometry) to algebraic thinking. This seems to me one of the most important progress in human intelligence.
* Recalling from my own mathematics education, I remember a famous ancient Chinese math problem called (linked source in Chinese), which basically says that given 35 heads and 97 feet in a cage of chicken and rabbits, how many chicken and rabbits each. The way the problem and solution were conveyed was all text-based, and I remember having difficulty memorizing the "trick" to solve it. Yet later on when I learned about system of linear polynomial equations, the problem became trivial as the "trick" was no more than common operations on the algebraic symbols. This is but one simple example of the magic of algebraic symbolism.
Thoughts After Reading
I thought I was math-savvy enough to go through the entire book with full comprehension. I was very wrong. I was having a good time reading through the algebra, trigonometries, imaginary number, and calculus sections, until the quaternions. From there, the reading speed dropped but I was still able to keep up. Yet, when Maxwell's Equations hit, I had to skim through the proofs here and there, and not be able to fully comprehend the development of the theories. The final chapters of tensors and the general theory of relativity were where I gave up. While I understood what constituted a tensor and its significance in preserving invariance, grasping tensor notations and tensor calculus were beyond my capacity at a level of casual reading. Consequently, I could not fully appreciate the beauty of the gravitational field equation from the General Theory of Relativity.
In my opinion, there are two ways to enjoy this book. One is to fully soak in everything the book provides, but that would require prior knowledge of vector and tensor calculus. The other way is to stick to the stories, which are fascinating in their own rights without the mathematical details.
* Algebraic symbolism appears so natural today that I rarely think of a time when it was not the rule of the land. Yet, it took a lot of work for human beings to abstract from a tangible way of thinking (geometry) to algebraic thinking. This seems to me one of the most important progress in human intelligence.
* Recalling from my own mathematics education, I remember a famous ancient Chinese math problem called (linked source in Chinese), which basically says that given 35 heads and 97 feet in a cage of chicken and rabbits, how many chicken and rabbits each. The way the problem and solution were conveyed was all text-based, and I remember having difficulty memorizing the "trick" to solve it. Yet later on when I learned about system of linear polynomial equations, the problem became trivial as the "trick" was no more than common operations on the algebraic symbols. This is but one simple example of the magic of algebraic symbolism.
Thoughts After Reading
I thought I was math-savvy enough to go through the entire book with full comprehension. I was very wrong. I was having a good time reading through the algebra, trigonometries, imaginary number, and calculus sections, until the quaternions. From there, the reading speed dropped but I was still able to keep up. Yet, when Maxwell's Equations hit, I had to skim through the proofs here and there, and not be able to fully comprehend the development of the theories. The final chapters of tensors and the general theory of relativity were where I gave up. While I understood what constituted a tensor and its significance in preserving invariance, grasping tensor notations and tensor calculus were beyond my capacity at a level of casual reading. Consequently, I could not fully appreciate the beauty of the gravitational field equation from the General Theory of Relativity.
In my opinion, there are two ways to enjoy this book. One is to fully soak in everything the book provides, but that would require prior knowledge of vector and tensor calculus. The other way is to stick to the stories, which are fascinating in their own rights without the mathematical details.
August 12, 2024
10/10!
Coherent, flowing story able to captivate a wide range of audiences. With such a large number of (digestible and context driven) equations, this read was a truly joyous experience as a student of physics.
Coherent, flowing story able to captivate a wide range of audiences. With such a large number of (digestible and context driven) equations, this read was a truly joyous experience as a student of physics.
March 6, 2025
A fantastic book detailing the history of quaternions, vectors, and tensors. I definitely appreciated the author’s writing style, which was part historical and part mathematical, and enjoyed the further details and copious references in the end notes. This book was surely a labor of love for the author, as it was for me when I read it. It took me longer to read it than usual, mostly because I savored every moment I had with this book. There was a fair bit of actual math within these pages, which I very much enjoy! I cannot recommend this book highly enough to anyone intrigued by advanced mathematics and how it is used in our most robust physical theories to date. One of the best books I have read in a while!!
July 8, 2024
[Disclosure: reviewed for Tasmanian Times]
This one’s for the math-literate, and it’s terrific. If you have a general grasp, or reasonable memory, of higher math, you’ll be adequately equipped to read and enjoy Robyn Arianrhod’s latest work, Vector � A surprising story of space, time, and mathematical transformation.
If you take joy in thinking about how the world works, and how a few simple, tiny and beautiful mathematical tools, which express and do so much, evolve from the collective and individual consciousness of humans over time, you’ll love Arianrhod’s history of vectors, tensors and all the thought and struggle that birthed them.
Disclosure: I was punching above my weight reading this one, but it’s hard to turn away from the author’s warm, conversational and clear-sighted overview of the shared inner lives of the great mathematicians as they incrementally progress our understanding of what can be understood � or imagined � and the uses new math can be put to.
Arianrhod wrote the book to share “the magnificent intellectual creation that is mathematics and mathematical science� � a multi-cultural journey through five thousand years.
The curiosity that drove our species to learn our place in the stars is the same that drove the wonder of mathematicians to discover, or build (depending if you view math as either discovered or invented), field equations, tensor analysis, and quantum mechanics. Either way, every mathematician stands on the shoulders of others; mathematical progress is collaborative.
“Part of what I’ve been trying to show throughout this story is just how long it takes for ideas to develop and find their best form. I think that knowing something of this long journey can help students who are struggling to grasp an idea by showing earlier mathematicians struggled too.� Mathematicians do “the kind of deep thinking needed to make sense of even the simplest things we take for granted today� sometimes without ever realising they might apply to things other than just numbers.�
“[I]t is important to be able to appreciate this kind of intellectual beauty, just as it is important to appreciate beauty in music and the arts.�
Just as great artistic insight can be expressed in a short poem or a single photograph, so can an enormous amount of insight and information be expressed with a single symbol � a mathematical tool, or a short equation.
‘The great power of mathematics lies in its visual symbolism� [V]ectors and tensors made it possible to…discover new laws of nature, and new technological applications of these laws.� Simply by finding new ways to represent information via mathematical symbols, they, like words, can encompass whole worlds of meaning, and shortcut the way we think about the world and act in it.
The author’s relaxed writing style is superb, her erudition of her field and its history is genuinely impressive, and her understanding of intersectional issues provides nuance and context which all good histories need.
Despite the uphill battle for me (dysnumerate, on the left of the Bell Curve), Arianrhod’s writing is so good, I’ll be looking to read her other works for a lay audience.
This one’s for the math-literate, and it’s terrific. If you have a general grasp, or reasonable memory, of higher math, you’ll be adequately equipped to read and enjoy Robyn Arianrhod’s latest work, Vector � A surprising story of space, time, and mathematical transformation.
If you take joy in thinking about how the world works, and how a few simple, tiny and beautiful mathematical tools, which express and do so much, evolve from the collective and individual consciousness of humans over time, you’ll love Arianrhod’s history of vectors, tensors and all the thought and struggle that birthed them.
Disclosure: I was punching above my weight reading this one, but it’s hard to turn away from the author’s warm, conversational and clear-sighted overview of the shared inner lives of the great mathematicians as they incrementally progress our understanding of what can be understood � or imagined � and the uses new math can be put to.
Arianrhod wrote the book to share “the magnificent intellectual creation that is mathematics and mathematical science� � a multi-cultural journey through five thousand years.
The curiosity that drove our species to learn our place in the stars is the same that drove the wonder of mathematicians to discover, or build (depending if you view math as either discovered or invented), field equations, tensor analysis, and quantum mechanics. Either way, every mathematician stands on the shoulders of others; mathematical progress is collaborative.
“Part of what I’ve been trying to show throughout this story is just how long it takes for ideas to develop and find their best form. I think that knowing something of this long journey can help students who are struggling to grasp an idea by showing earlier mathematicians struggled too.� Mathematicians do “the kind of deep thinking needed to make sense of even the simplest things we take for granted today� sometimes without ever realising they might apply to things other than just numbers.�
“[I]t is important to be able to appreciate this kind of intellectual beauty, just as it is important to appreciate beauty in music and the arts.�
Just as great artistic insight can be expressed in a short poem or a single photograph, so can an enormous amount of insight and information be expressed with a single symbol � a mathematical tool, or a short equation.
‘The great power of mathematics lies in its visual symbolism� [V]ectors and tensors made it possible to…discover new laws of nature, and new technological applications of these laws.� Simply by finding new ways to represent information via mathematical symbols, they, like words, can encompass whole worlds of meaning, and shortcut the way we think about the world and act in it.
The author’s relaxed writing style is superb, her erudition of her field and its history is genuinely impressive, and her understanding of intersectional issues provides nuance and context which all good histories need.
Despite the uphill battle for me (dysnumerate, on the left of the Bell Curve), Arianrhod’s writing is so good, I’ll be looking to read her other works for a lay audience.
January 20, 2025
This is a very difficult book to read. I found it quite intense in the mathematical side, for a non expect like me I just skimmed through some parts hoping I might be able to get the big ideas.
I found this book to be challenging but thought provoking and filled with profound ideas, from how intricate and collaborative the history of math and science is, on how the idea of vectors (which is not old at all) has shaped the modern world, all the way to how Hamilton inspired Maxwell who later inspired Einstein, and much more.
I really recommend this book if you are mathematically inclined, a strong background in math would be very useful to make it much more enjoyable and powerful. The book requires patience for less mathematically experienced readers.
I found this book to be challenging but thought provoking and filled with profound ideas, from how intricate and collaborative the history of math and science is, on how the idea of vectors (which is not old at all) has shaped the modern world, all the way to how Hamilton inspired Maxwell who later inspired Einstein, and much more.
I really recommend this book if you are mathematically inclined, a strong background in math would be very useful to make it much more enjoyable and powerful. The book requires patience for less mathematically experienced readers.
March 29, 2025
An excellent treatment of a difficult subject
Comprehensive yet digestible, this book finally answered questions I have been pursuing for along time. I’ve read Wheeler and others, and while they are often definitive, they are (to me) difficult. I recommend this book unreservedly.
Comprehensive yet digestible, this book finally answered questions I have been pursuing for along time. I’ve read Wheeler and others, and while they are often definitive, they are (to me) difficult. I recommend this book unreservedly.
November 19, 2024
Utter garbage, if you want to understand why and what of vectors. If you want to enjoy politically-biased historical tales, then you might like this book. I have completed just 25% and it is painful to deal with the author's hand-wavy treatment of concepts.
December 27, 2024
Should be called “Tensor� instead based off the last 4-5 Chapters. Interesting history and math.
February 6, 2025
Interesting.
February 20, 2025
Beautiful history
It is written for someone with an undergraduate level of mathematics who wants to be inspired by what lies at the next level.
It is written for someone with an undergraduate level of mathematics who wants to be inspired by what lies at the next level.
March 4, 2025
She loses the plot quite a bit. Interesting and I liked the depth. It was just hard to go with her as she went as so many tangents.
February 4, 2025
Entertaining book - but the math was substantial. I don't have a calculus background; think it would have been useful to get the most out of the discussion. Still, it was interesting to learn the history and development of vectors and tensors thru the ages - and how they've become important to many of the advances in our lives.
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