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Love Triangle: How Trigonometry Shapes the World
An ode to trigonometry, the most important idea in mathematics and the key concept that enables our modern world, from the internationally bestselling author of Humble Pi
Contrary to what your friends may have grumbled in high school math class, trigonometry is perhaps the most essential concept humans have ever devised. The simple yet versatile triangle allows us to map the world, launch ships into space, and send cat gifs. Trigonometry also makes it possible to play the piano so that it sounds like a human voice and was crucial to prosecuting the balloon trip company that caused a pig stampede.
In Love Triangle , Matt Parker shares plenty of relevant and irreverent reasons we should all show a lot more love for the triangles in our lives. He tells extraordinary and entertaining stories of mathematicians, philosophers, and engineers—starting with Pythagoras—who dared to take triangles seriously. Humans have been using triangles for thousands of years to measure the earth and build structures. But trigonometry also underpins all modern data technology and is the essential component of GPS—without triangles, we’d still be at the gas station asking for directions.
Parker convincingly makes the case that trigonometry is vital, fun, and deeply useful. Its rules are the hidden pattern beneath the surface of just about everything we encounter, and we wouldn’t exist without them. Luckily, it’s never too late to learn!
Contrary to what your friends may have grumbled in high school math class, trigonometry is perhaps the most essential concept humans have ever devised. The simple yet versatile triangle allows us to map the world, launch ships into space, and send cat gifs. Trigonometry also makes it possible to play the piano so that it sounds like a human voice and was crucial to prosecuting the balloon trip company that caused a pig stampede.
In Love Triangle , Matt Parker shares plenty of relevant and irreverent reasons we should all show a lot more love for the triangles in our lives. He tells extraordinary and entertaining stories of mathematicians, philosophers, and engineers—starting with Pythagoras—who dared to take triangles seriously. Humans have been using triangles for thousands of years to measure the earth and build structures. But trigonometry also underpins all modern data technology and is the essential component of GPS—without triangles, we’d still be at the gas station asking for directions.
Parker convincingly makes the case that trigonometry is vital, fun, and deeply useful. Its rules are the hidden pattern beneath the surface of just about everything we encounter, and we wouldn’t exist without them. Luckily, it’s never too late to learn!
352 pages, Hardcover
First published August 20, 2024
393 people are currently reading
2999 people want to read
About the author
Matt Parker
5 books503 followersMatt Parker is a former maths teacher who communicates about mathematics via YouTube videos, stand-up comedy, and books.
Librarian note:
There is more than one author in the GoodReads database with this name
Librarian note:
There is more than one author in the GoodReads database with this name
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Displaying 1 - 30 of 167 reviews
May 18, 2025
Once again I have attempted to read a science book that really is over my paygrade mathematically. But this time, it isn’t because I cannot even begin to be able to understand any of the usually esoteric math that is described or mentioned in passing on those science subjects which rely heavily on maths. This time it’s because much of my memories of math I studied in college has rusted over with decaying proteins, or worse, the brain cells I used to pass my math exams have utterly died. I am in my seventh decade of life, and college was nearly half of a century ago in my past. There was a time when I did very well in Geometry and Algebra and Trigonometry, and even Calculus (shudder). However, it was truly a case of me following instructions without any understanding whatsoever.
Give me a break, people. I have difficulty remembering how to tie my shoes now.
Anyway.
‘Love Triangle’ by Matt Parker is fun to read, and very informative. He includes stories of how triangle area/angles/ratio formulas were discovered, and how they have been used throughout history to build our roads and buildings, and in solving extremely essential and important engineering and physics and art perspective convergence problems, and in history was the key to determining how big the Earth was. I was amazed! And mortified. My knowledge of trigonometry and geometry floated up, slowly, oh my god, so slowly, from the depths where my rusted over memories of past studies reside. I was able to follow most of Matt Parker’s fascinating stories about the practical uses of triangles only generally. And the sine/cosine chapters were very enlightening because of all the maths I took it was those classes where I most of all stared blankly at the work while I solved problems correctly without ANY understanding of it. The last couple of chapters on Fourier transforms and analysis lost me entirely, except that I could grok Fourier maths allowed scientists and musicians to see the different frequencies inside a sound or a molecule. Something like that.
I have copied the book blurb:
”An ode to triangles, the shape that makes our lives possible Trigonometry is perhaps the most essential concept humans have ever devised. The simple yet versatile triangle allows us to record music, map the world, launch rockets into space, and be slightly less bad at pool. Triangles underpin our day-to-day lives and civilization as we know it.
In Love Triangle, Matt Parker argues we should all show a lot more love for triangles, along with all the useful trigonometry and geometry they enable. To prove his point, he uses triangles to create his own digital avatar, survive a harrowing motorcycle ride, cut a sandwich, fall in love, measure tall buildings in a few awkward bounds, and make some unusual art. Along the way, he tells extraordinary and entertaining stories of the mathematicians, engineers, and philosophers—starting with Pythagoras—who dared to take triangles seriously.
This is the guide you should have had in high school—a lively and definitive answer to “Why do I need to learn about trigonometry?” Parker reveals triangles as the hidden pattern beneath the surface of the contemporary world. Like love, triangles actually are all around. And in the air. And they’re all you need.”
The above descriptions about trigonometry are not in the least hyperbolic. Math nerds have made civilization possible, not just space explorations. I don’t know how teachers teach maths today, but in my classes, it was all about memorizing formulas, not about how the formulas were discovered, or how architects, artists, scientists and engineers used them. This book fills in some of THOSE blanks, but unfortunately not the ones in my math education. My innate nerdiness begins and ends with the reading of books and seeing, no, feeling the atoms of meaning in the words. Maths for me are more a memorization exercise in which glimmers of light occasionally break through. I can tell the secrets of physical reality are there, somewhere, in those equations!
The book has lots and lots of pictures, most of which are very necessary. It also has a crazy index section to decipher.
Give me a break, people. I have difficulty remembering how to tie my shoes now.
Anyway.
‘Love Triangle’ by Matt Parker is fun to read, and very informative. He includes stories of how triangle area/angles/ratio formulas were discovered, and how they have been used throughout history to build our roads and buildings, and in solving extremely essential and important engineering and physics and art perspective convergence problems, and in history was the key to determining how big the Earth was. I was amazed! And mortified. My knowledge of trigonometry and geometry floated up, slowly, oh my god, so slowly, from the depths where my rusted over memories of past studies reside. I was able to follow most of Matt Parker’s fascinating stories about the practical uses of triangles only generally. And the sine/cosine chapters were very enlightening because of all the maths I took it was those classes where I most of all stared blankly at the work while I solved problems correctly without ANY understanding of it. The last couple of chapters on Fourier transforms and analysis lost me entirely, except that I could grok Fourier maths allowed scientists and musicians to see the different frequencies inside a sound or a molecule. Something like that.
I have copied the book blurb:
”An ode to triangles, the shape that makes our lives possible Trigonometry is perhaps the most essential concept humans have ever devised. The simple yet versatile triangle allows us to record music, map the world, launch rockets into space, and be slightly less bad at pool. Triangles underpin our day-to-day lives and civilization as we know it.
In Love Triangle, Matt Parker argues we should all show a lot more love for triangles, along with all the useful trigonometry and geometry they enable. To prove his point, he uses triangles to create his own digital avatar, survive a harrowing motorcycle ride, cut a sandwich, fall in love, measure tall buildings in a few awkward bounds, and make some unusual art. Along the way, he tells extraordinary and entertaining stories of the mathematicians, engineers, and philosophers—starting with Pythagoras—who dared to take triangles seriously.
This is the guide you should have had in high school—a lively and definitive answer to “Why do I need to learn about trigonometry?” Parker reveals triangles as the hidden pattern beneath the surface of the contemporary world. Like love, triangles actually are all around. And in the air. And they’re all you need.”
The above descriptions about trigonometry are not in the least hyperbolic. Math nerds have made civilization possible, not just space explorations. I don’t know how teachers teach maths today, but in my classes, it was all about memorizing formulas, not about how the formulas were discovered, or how architects, artists, scientists and engineers used them. This book fills in some of THOSE blanks, but unfortunately not the ones in my math education. My innate nerdiness begins and ends with the reading of books and seeing, no, feeling the atoms of meaning in the words. Maths for me are more a memorization exercise in which glimmers of light occasionally break through. I can tell the secrets of physical reality are there, somewhere, in those equations!
The book has lots and lots of pictures, most of which are very necessary. It also has a crazy index section to decipher.
June 6, 2024
He's done it again! This time with a book about triangles.
Matt Parker's books are always factual, fun and funny, this is no different.
His writing makes you feel you're chatting to a friend rather than reading and learning!
At some points I did feel like I was going through my A-Level physics/maths lessons but that didn't put me off due to the presentation of the examples.
Loved every page and cannot wait for more books by Matt.
Thank you to NetGalley, the publishers and Matt Parker for an eARC of Love Triangle.
Matt Parker's books are always factual, fun and funny, this is no different.
His writing makes you feel you're chatting to a friend rather than reading and learning!
At some points I did feel like I was going through my A-Level physics/maths lessons but that didn't put me off due to the presentation of the examples.
Loved every page and cannot wait for more books by Matt.
Thank you to NetGalley, the publishers and Matt Parker for an eARC of Love Triangle.
November 16, 2024
While I loved this book, I wouldn’t recommend it unless you have at least a moderate interest in math. Matt is a funny and easy to understand teacher, but it’s still a math-heavy book. If reading 300 pages about triangles feels a bit daunting, I’d highly recommend his YouTube channel which is more approachable for everyone.
October 20, 2024
I don’t know if I can express how much I appreciate and enjoyed this book. Mathematics is a topic that makes many cringe, and I’m know different. But over the last decade, because of my love of science and wanting to understand the universe, I have been learning. So I’m always looking out for good books on math, books that inspire. And believe me when I say, there are not many of them.
I’ve been struggling with trigonometry for some 3 years, so when I saw this book, the title grabbed me immediately. I’m so glad I bought it. This is not a textbook. It does not tell you how to solve problems, but what it does do is show the beauty of shapes, particularly triangle for how they shape our world. Engineering, science , computing and gaming, and even the natural world relies upon triangles.
The author’s passion comes through as he describes the many ways he uses math, and his love affair with triangles, shapes, and trigonometry. He covers uses from building strong bridges to predicting the orbits of asteroids. He also has a funny and sometimes cheesy sense of humor, corny turns of phrases that make the text all the more endearing.
I will reread this book in the future. It’s one that was hard to put down and will remain on my kitchen table so it’s easy to pick up again. This book is right up there with the Joy of X. I love it that more authors are writing fun books about mathematics and how it’s used everywhere.
I’ve been struggling with trigonometry for some 3 years, so when I saw this book, the title grabbed me immediately. I’m so glad I bought it. This is not a textbook. It does not tell you how to solve problems, but what it does do is show the beauty of shapes, particularly triangle for how they shape our world. Engineering, science , computing and gaming, and even the natural world relies upon triangles.
The author’s passion comes through as he describes the many ways he uses math, and his love affair with triangles, shapes, and trigonometry. He covers uses from building strong bridges to predicting the orbits of asteroids. He also has a funny and sometimes cheesy sense of humor, corny turns of phrases that make the text all the more endearing.
I will reread this book in the future. It’s one that was hard to put down and will remain on my kitchen table so it’s easy to pick up again. This book is right up there with the Joy of X. I love it that more authors are writing fun books about mathematics and how it’s used everywhere.
June 16, 2024
Matt Parker is a man who is very excited by triangles and he wants you to be excited by triangles too! The book skips along with humour and enthusiasm which is just so infectious.
I had fun and I learned stuff - what more could you want?
I had fun and I learned stuff - what more could you want?
July 11, 2024
Over all I enjoyed it. Matt Parker is funny and explains topics in a clear and interesting way, knowing how much detail to give so as to not bore the reader.
The topics he chose to speak about were a little applied for my tastes, but I think that issue lies with me.
The topics he chose to speak about were a little applied for my tastes, but I think that issue lies with me.
December 14, 2024
You can tell how much this guy loves maths and a good triangle. It kept the sections requiring more concentration engaging, especially listening to the audiobook. However, some parts felt like they were geared toward those with a very basic understanding of mathematical principles while others delved deeper without, in my opinion, adequate explanation or lead-in. Still a fun time though and would like to reread it eventually.
March 7, 2025
funny! mathy! liked it but not sure it would be up your alley if you don’t like math
May 23, 2025
One of the nerdiest books I've read to date - loved it :)
October 29, 2024
I don't think he's as funny as he thinks. Fun book. Kind of heavy at the last couple of chapters, but I survived. Triangles are fun. I have no use for his disrespect for God or those who believe. It's boring and low hanging fruit. No one ever wants to venture just where all this comes from...it just happened. Boring. But triangles are fun.
April 1, 2025
I really enjoyed this book! I listened to the audiobook read by the author and that was really fun. I’m already a triangle fan, so I really clicked with this book and enjoyed learning some new things along the way. Great book, would highly recommend!
October 12, 2024
One of my favorite things I've read in a while. He had me once he classified Pythagoras as "the Beyoncé of maths." I'm not even a math person. I just enjoyed learning about this and hearing someone so passionate speak about what they love (listen to this on audio btw)
April 17, 2025
Some subjects grabbed me more than others. I know the author avoided trig for a good chunk of the beginning of the book but I actually liked trig so I was thrilled when it finally came up. I'm one of those adults who remembers what soh cah toa stands for and that all those triangles come full circle.
I liked the chapter about reducing the number of edges using hexagons. Anytime astronomy concepts came up I was doubly interested and the author talked about how the James Webb Space Telescope was made up of 18 hexagons. Bees make hexagons as well, apparently quite by accident.
The chapter on art was also really good for me. I've been more interested in art lately. I learned about the vanishing point. Also enjoyed the discussion about how our brain's interpretation of what we are seeing can fool us.
Most of the goofy jokes were easy enough for me to get right away, like Hmmmmmmm = Hm^7
The only reason it's not 5 stars is because some of the chapters were really hard for me to get through. Most sports references were lost on me. Also, having a strong background in geometry and trig will definitely help. Some concepts were hard for me to understand. I will hold on to this book in hopes of reading it again!
I liked the chapter about reducing the number of edges using hexagons. Anytime astronomy concepts came up I was doubly interested and the author talked about how the James Webb Space Telescope was made up of 18 hexagons. Bees make hexagons as well, apparently quite by accident.
The chapter on art was also really good for me. I've been more interested in art lately. I learned about the vanishing point. Also enjoyed the discussion about how our brain's interpretation of what we are seeing can fool us.
Most of the goofy jokes were easy enough for me to get right away, like Hmmmmmmm = Hm^7
The only reason it's not 5 stars is because some of the chapters were really hard for me to get through. Most sports references were lost on me. Also, having a strong background in geometry and trig will definitely help. Some concepts were hard for me to understand. I will hold on to this book in hopes of reading it again!
July 22, 2025
This book does a disservice to Matt.
I enjoy his books and YouTube videos, which is why I'm reading this book.
First of all, what's with the page numbers? I understand that this is a book about math, specifically about triangles, so that angles will appear everywhere throughout the book. But what are the values in the place of page numbers??? Doesn't it warrant an explanation before the page number things? Perhaps in the intro?
I figured it out easily when I got to the value 0.500000 and counted 31. And value 1.000000 is on page 91.
Yes, everything seems to be off by 1, because it starts on 0.000000, so page 0. Where you would normally expect odd page numbers, there are now "even numbers", really the sine of that many degrees.
Page 0.707107 has a picture from a stock photo website. Matt invites readers to find 4 things wrong in the image of a double rainbow. The book is printed in low-resolution black and white. The only one of the 4 things you can see in the picture is about the shape. The other 3 things are about color and which part should be brighter. Needless to say, we are not afforded the opportunity to participate in this exercise.
I enjoy his books and YouTube videos, which is why I'm reading this book.
First of all, what's with the page numbers? I understand that this is a book about math, specifically about triangles, so that angles will appear everywhere throughout the book. But what are the values in the place of page numbers??? Doesn't it warrant an explanation before the page number things? Perhaps in the intro?
I figured it out easily when I got to the value 0.500000 and counted 31. And value 1.000000 is on page 91.
Yes, everything seems to be off by 1, because it starts on 0.000000, so page 0. Where you would normally expect odd page numbers, there are now "even numbers", really the sine of that many degrees.
Page 0.707107 has a picture from a stock photo website. Matt invites readers to find 4 things wrong in the image of a double rainbow. The book is printed in low-resolution black and white. The only one of the 4 things you can see in the picture is about the shape. The other 3 things are about color and which part should be brighter. Needless to say, we are not afforded the opportunity to participate in this exercise.
May 9, 2025
Delightful read on exactly what the title promises. I loved trigonometry in high school, but that was a thousand years ago. So, did I follow all the math in the book? Nope. But I happily skipped over most equations for the stories and the ooh-that's-fascinating moments which filled the book. Funny, too.
I chose this in my eternal struggle to find books my son might like reading, but I beat him to the end of this one.
I will absolutely read more books by this author.
I chose this in my eternal struggle to find books my son might like reading, but I beat him to the end of this one.
I will absolutely read more books by this author.
March 30, 2025
Always enjoy Matt Parker's books. He blends humor and mathematics into a great story. Most of his chapters discuss items that he covered in his YouTube channel but there was enough new stuff to keep it interesting.
December 31, 2024
Not as good as humble pi
January 3, 2025
Driehoeken zijn alles en alles is driehoeken
February 14, 2025
Fun romp through geometry and trigonometry with more puns than you can shake a stick at. If you follow his YouTube channel quite a lot has been covered there. Still, a fun read.
February 20, 2025
Listen to as audiobook. Which because of content would be much better as book. So likely would be higher rating of read.
June 20, 2024
As you should be able to tell from the title - but let me just confirm it - this really is a book all about triangles. Do not buy it if you are not remotely interested in these shapes with three sides! But if you like popular science books, are intrigued by the shapes that make up the universe, don't mind a little mathematics in your reading, or are actually a triangles-big-time-fan, then this book could be for you.
As for me, I like dabbling in a bit of popular science-type reading, but I'm not a mathematician or anywhere near it.
Essentially, the book explores all the ways triangles are used or can be found in the world, with plenty of illustrations and examples to go with it. The writing style is fun and engaging and you can learn a lot! The slight downside to me is that I didn't really find a narrative thread to follow, it was simply interesting a few pages at a time. So I'd read a section and then come back to it later, but there was no desperate urge to read more right away. Saying that, I think that can work well for a non-fiction book. I read it on Kindle, but I think it would be best in physical form where you can leave it somewhere handy and then pick it up for a short read now and then.
Plus, if you know anyone into maths/science/triangles, this would be an awesome gift.
As for me, I like dabbling in a bit of popular science-type reading, but I'm not a mathematician or anywhere near it.
Essentially, the book explores all the ways triangles are used or can be found in the world, with plenty of illustrations and examples to go with it. The writing style is fun and engaging and you can learn a lot! The slight downside to me is that I didn't really find a narrative thread to follow, it was simply interesting a few pages at a time. So I'd read a section and then come back to it later, but there was no desperate urge to read more right away. Saying that, I think that can work well for a non-fiction book. I read it on Kindle, but I think it would be best in physical form where you can leave it somewhere handy and then pick it up for a short read now and then.
Plus, if you know anyone into maths/science/triangles, this would be an awesome gift.
March 10, 2025
This is a book that talks about triangle and trigonometry and where they show up in the world. The author has a sense of humor and it shows up in the book. If you aren't a math person, I'm not sure that you'd appreciate this book as much as a math person.
February 4, 2025
I love Matt Parker, but the whole book is just recycled stories from his YouTube channel. Maybe there is original stuff at the end, but I wasn’t going to push through. His other books are better (and more mathematically rigorous).
I don’t feel bad about buying the book, because he does good stuff for math educators. I might gift the book to one of my students
I don’t feel bad about buying the book, because he does good stuff for math educators. I might gift the book to one of my students
September 4, 2024
In Love Triangle, Matt Parker gives a multitude of examples of triangles used in real life while skimming the surface of the core concepts of triangles, keeping it entertaining with his jokes and stories. The book also includes plenty of illustrations to complement what he is saying.
This was an enjoyable read that you can binge in one sitting or read a few pages at a time, as time allows. I enjoyed that the page numbers at the bottom were the sine of normal page numbers; it was a cute trigonometric quirk! I learned quite a bit about the world while reading this book. I highly suggest checking it out, even if you weren’t a fan of trigonometry or geometry in school!
I received an ARC of this book in return for my unbiased opinion.
This was an enjoyable read that you can binge in one sitting or read a few pages at a time, as time allows. I enjoyed that the page numbers at the bottom were the sine of normal page numbers; it was a cute trigonometric quirk! I learned quite a bit about the world while reading this book. I highly suggest checking it out, even if you weren’t a fan of trigonometry or geometry in school!
I received an ARC of this book in return for my unbiased opinion.
July 8, 2025
Enjoyed this way more than I'd like to admit. Loved the journey from triangle to geometry to trigonometry and all the way to physics. Prosthaphaeresis was something I thought I had invented. I'm sure it featured at some point in my education in the 90s. According to the book, it was 'a technique used between about 1590 and 1614.' I can confirm I'm not that old.
June 7, 2024
What an absolutely joyous book. I never expected to be so charmed by a book about triangles, but here we are. Matt Parker's writing style is easy to follow, comprehensive and comprehendible, and all-out fun to read.
Here are some of my absolute favourite stand-out facts from this wonderful book. I hope if you choose to read this book, you find more of your own. Trust me - you'll be repeating snippets from this book for weeks!
There were also moments in this book which made me laugh out loud - certainly not what I was expecting from this book on maths.
So all in all, this is acute book, and you should read it.
Thank you to Net Galley and Matt Parker for an advanced copy of this book in return for a fair and honest review.
Here are some of my absolute favourite stand-out facts from this wonderful book. I hope if you choose to read this book, you find more of your own. Trust me - you'll be repeating snippets from this book for weeks!
There were also moments in this book which made me laugh out loud - certainly not what I was expecting from this book on maths.
So all in all, this is acute book, and you should read it.
Thank you to Net Galley and Matt Parker for an advanced copy of this book in return for a fair and honest review.
June 12, 2025
This was an engaging read! As a STEM educator, I always scout for texts that break down complex subjects for the average reader. I recommend this book to anyone who wants a digestible and fun survey of the subject. Parker brought every day insight to concepts that I struggled with personally. I look forward to reading “Humble Pi”.
January 21, 2025
In classic Matt Parker fashion, it is filled with great math and great puns. It explains the maths in a fun way and doesn’t go too heavy with the technical details and instead focuses on fun stories and examples of how triangles help in the real world.
January 10, 2025
Buen libro de matemáticas y triángulos, y todo lo que viene de la mano. Me ha enganchado menos que Humble Pi. Capítulos muy interesantes pero otros que se me han hecho bola y han sido demasiado técnicos.
Aún así, muchos detalles muy interesantes que muestran como, efectivamente, el mundo son matemáticas.
The final wave kicker comes from quantum mechanics. We've already had a look at the sibling area of modern physics, General Relativity, which describes how things behave when they are very big or moving very fast. Quantum mechanics deals with the opposite end of the spectrum: the very, very small. If you zoom in far enough, the intuitive reality around us fades away and is replaced by pure mathematics. We live in a universe made of maths. What we think of as solid matter is actually wave functions.
You are literally made of sine waves. You are built of triangles.
Reality is triangles.
Triangles are everything and everything is triangles.
Loved the chapter about aperiodic tiles, einsteins and the discovery of The Hat:
There was something about the Hat that made it popular with the public and mathematicians alike: it was surprisingly simple. Given this shape had been eluding the entire mathematics community for over half a century, nobody expected i6 to be so straightforward. It's a 13-sided polygon, far fewer sides than I would have predicted. It is concave, as expected,but doesn't have any detached, fragmented bits or any holes.When I look at it, I see a slightly modified equilateral triangle.Even in the research paper announcing its discovery says,"The shape is almost mundane in its simplicity.'
I love triangles! We all rely on triangles to keep our modern world ticking along. I would argue (and have done, hence the book you're holding) that triangles unlock some of the most important bits of knowledge ever discovered by humans. Triangles are the gateway to the worids of geometry and trigonometry. Triangles help us out in our day-to-day lives and enable the civilization around us. Also, I just think they're neat.
Triangles are like a super-easy crossword where you get half the letters and the answer is always 'triangle'.
The Ahmes Papyrus being the first maths related document ever, and how it included triangles to calculate áreas of land.
Regular, non-maths humans often have a strange sense that maths would be capable of solving all of their problems if only they knew how. Like maths is a dark art understood only by a shadowy group of experts.
The granddaddy of triangle maths. First name: Pythagoras. Last name: who cares about his last name, this is Pythagoras we're talking about, the Beyoncé of maths. Pythagoras even sounds like it starts with π...because in Greek it does! The guy is as mathsy as it gets.
Engineers also triangulate surfaces, not because they want to build the structure out of triangles, but rather to double-check it's not going to fall over.
'Finite element analysis' is the process of splitting a structure or object up into composite parts that can be individually analyzed.Humans have had a pretty good understanding of forces for a few centuries, but engineers have been limited in how they can be applied to whole structures because of a lack of computing power. Thankfully, computing power is something computers are increasingly good at providing. It is no coincidence that the rise of non-cuboid architecture has coincided with the era of electronic computers.
In a perfect world some kind of supercomputer could calculate all of the forces as they vary across an entire structure(maybe going atom by atom), but for now we need to split a building into small-enough-is-good-enough chunks. Think of these finite elements as like the 'pixels' of a building, engine part, wing, or whatever object we need to calculate the forces of.
You would think the act of selecting a photo on your phone and printing it would be fairly straightforward. The phone just needs to transmit that image to the printer, which then does its printing thing. But the issue is that phones - indeed all digital cameras - speak a different color language than printers. And there is a device argument over whose problem the translation should be. An argument that is solved with a triangle mesh.
If you take a photo on your iPhone it will be saved in a color encoding called DCI-P3. When you send it to your computer and look at it on a screen it has probably been converted to sRGB. When the whim to print it takes over, the photo will be converted once more, to CMYK, before it's sent to the printer.Each of these different ways of representing colors is called a'color space', and converting from one color space to another- working out the equivalent color encoding for each color in the image - is a surprisingly complicated calculation.
The computer intermediary in the above hypothetical situation is the secret to success. It can listen to the iPhone talking in DCI-P3 and convert it over to CMYK so the printercan understand what it's saying.
The solution was twofold. First, it was agreed that sRGB would be the communal middle language. It is a good standard version of RGB insomuch as everyone assumes that the 's'stands for standard. But I directly contacted the Internationak Color Consortium, who are in charge of these things, and they confirmed the's' has no official definition. So in my headcanon'sRGB' stands for 'supercalibratinglogisticredgreenblueydocious' and you can't stop me.
This process has to happen fast and with the minimum possible calculations. We need those in-between values with minimal processor load otherwise I'm going to have yet another reason to want to throw my printer out the window.Peter needed a way to quickly calculate the volume of a tetrahedron. Actually, Peter needed to be able to quickly calculate the area of a triangle, the volume of a tetrahedron, or indeed the 'content size' of any simplex. He wanted this method to work for any data application in any number of dimensions.And what method did he use to find these volumes super fast?I kid you not. HERON'S AREA FORMULA.
I'm sorry. This is a very emotional chapter for me. First Eugénie's roller-coaster ride of reveals, and now Heron's Formula has slammed into the room with a practical applica tion.
Hexagons and bees:
Left to their own devices bees actually build circular cells, not hexagonal ones. That's right: a single bee cell would be a cylindrical tube of wax. But bees don't build just one, they build many cells all next to each other. It's the interaction ofadjacent cells thatcauses hexagons to form.
Wax is a flexible substance and bees take advantage of that by constantly pushing it around as they build, as well as scraping it back up and reforming it. While a bee building a cell will push all of the walls out to make a cylinder, a bee in the cell next door is pushing back. And it is that back-andforth pushing of the malleable wax that causes the cells to eventually settle into hexagonal shapes. Hexagons are a shape that can naturally result like this, which is why they pop up so often in nature.
GPS has been mentioned a few times in this book already, but'GPS', for 'Global Positioning System', is actually a brand name.There is more than one system of spacecraft beaming precise radio signals to form a global-navigation satellite system, giving us the generic name GNSS. The OS actually describe their 115 receivers as 'geodetic GNSS receivers', but this is a Kleenex/Hoover situation where the GPS brand-name has become a generic term in the eyes of the public. The OS have also called their sky-facing system OS Net that is a bit too Terminator-y for my liking.
The GNSS revolution has also exposed the mild inaccuracies of the old ways of measuring things. One of the touristy things to do in London is to go to the Greenwich observatory and stand on the o° line of longitude. The Earth's spin gives us a clear equator right in the middle of the planet, and the axis of rotation gives us precise north and south poles, so angles of latitude have an unambiguous reference frame. Longitude, however, which measures how far 'around' the planet a location is, has no such obvious starting point. For a while some countries just used their own starting points, but that was exactly as confusing as you would expect.
So in 1884 representatives from twenty-five countries got together in Washington, DC, and voted on it. Greenwich observatory in London won with twenty-two of the votes (the dissenting countries were Brazil, France, and Santo Domingo).This designated the starting point, or 'prime meridian', as the line that started at the Greenwich observatory and then went all the way around the planet, through both of the poles. The line is actually a massive circle going exactly around the very center of
Earth. Or it would have been, if they hadn't missed.
To draw a circle that goes around the center of the planet you need to know exactly which way is down. This was done very carefully using gravity to indicate the direction 'down', but the intrepid surveyors did not know that the density and deformities below the surface of the Earth could cause local gravity to misalign with the geometric center of the Earth. Thus the original'zeroth' circle around the Earth wasn't actually centered on the middle of the Earth. Nobody noticed at the time, though, and a big shiny metal line was put in the ground at Greenwich for tourists to jump back and forth over.
Satellites have no such problem locating the middle of the planet and so have a more accurate 'down'. With the global adoption of GNSS there was now a mismatch between the true longitudes and those based on the metal line in Greenwich. In 1984, exactly a century after the first gathering, countries from around the planet got together and decided on a new prime meridian, ever so slightly to the side of the old one.
I don't think it would take much arguing to persuade anyone that an understanding of waves makes our modern, digitally connected world possible. Signals fly at the speed of light between mobile phones and communication towers, as well as down fiber optic cables between continents. All impossible if we didn't have the knowledge of Fourier to disentangle multiple, combined sine waves.
Aún así, muchos detalles muy interesantes que muestran como, efectivamente, el mundo son matemáticas.
The final wave kicker comes from quantum mechanics. We've already had a look at the sibling area of modern physics, General Relativity, which describes how things behave when they are very big or moving very fast. Quantum mechanics deals with the opposite end of the spectrum: the very, very small. If you zoom in far enough, the intuitive reality around us fades away and is replaced by pure mathematics. We live in a universe made of maths. What we think of as solid matter is actually wave functions.
You are literally made of sine waves. You are built of triangles.
Reality is triangles.
Triangles are everything and everything is triangles.
Loved the chapter about aperiodic tiles, einsteins and the discovery of The Hat:
There was something about the Hat that made it popular with the public and mathematicians alike: it was surprisingly simple. Given this shape had been eluding the entire mathematics community for over half a century, nobody expected i6 to be so straightforward. It's a 13-sided polygon, far fewer sides than I would have predicted. It is concave, as expected,but doesn't have any detached, fragmented bits or any holes.When I look at it, I see a slightly modified equilateral triangle.Even in the research paper announcing its discovery says,"The shape is almost mundane in its simplicity.'
I love triangles! We all rely on triangles to keep our modern world ticking along. I would argue (and have done, hence the book you're holding) that triangles unlock some of the most important bits of knowledge ever discovered by humans. Triangles are the gateway to the worids of geometry and trigonometry. Triangles help us out in our day-to-day lives and enable the civilization around us. Also, I just think they're neat.
Triangles are like a super-easy crossword where you get half the letters and the answer is always 'triangle'.
The Ahmes Papyrus being the first maths related document ever, and how it included triangles to calculate áreas of land.
Regular, non-maths humans often have a strange sense that maths would be capable of solving all of their problems if only they knew how. Like maths is a dark art understood only by a shadowy group of experts.
The granddaddy of triangle maths. First name: Pythagoras. Last name: who cares about his last name, this is Pythagoras we're talking about, the Beyoncé of maths. Pythagoras even sounds like it starts with π...because in Greek it does! The guy is as mathsy as it gets.
Engineers also triangulate surfaces, not because they want to build the structure out of triangles, but rather to double-check it's not going to fall over.
'Finite element analysis' is the process of splitting a structure or object up into composite parts that can be individually analyzed.Humans have had a pretty good understanding of forces for a few centuries, but engineers have been limited in how they can be applied to whole structures because of a lack of computing power. Thankfully, computing power is something computers are increasingly good at providing. It is no coincidence that the rise of non-cuboid architecture has coincided with the era of electronic computers.
In a perfect world some kind of supercomputer could calculate all of the forces as they vary across an entire structure(maybe going atom by atom), but for now we need to split a building into small-enough-is-good-enough chunks. Think of these finite elements as like the 'pixels' of a building, engine part, wing, or whatever object we need to calculate the forces of.
You would think the act of selecting a photo on your phone and printing it would be fairly straightforward. The phone just needs to transmit that image to the printer, which then does its printing thing. But the issue is that phones - indeed all digital cameras - speak a different color language than printers. And there is a device argument over whose problem the translation should be. An argument that is solved with a triangle mesh.
If you take a photo on your iPhone it will be saved in a color encoding called DCI-P3. When you send it to your computer and look at it on a screen it has probably been converted to sRGB. When the whim to print it takes over, the photo will be converted once more, to CMYK, before it's sent to the printer.Each of these different ways of representing colors is called a'color space', and converting from one color space to another- working out the equivalent color encoding for each color in the image - is a surprisingly complicated calculation.
The computer intermediary in the above hypothetical situation is the secret to success. It can listen to the iPhone talking in DCI-P3 and convert it over to CMYK so the printercan understand what it's saying.
The solution was twofold. First, it was agreed that sRGB would be the communal middle language. It is a good standard version of RGB insomuch as everyone assumes that the 's'stands for standard. But I directly contacted the Internationak Color Consortium, who are in charge of these things, and they confirmed the's' has no official definition. So in my headcanon'sRGB' stands for 'supercalibratinglogisticredgreenblueydocious' and you can't stop me.
This process has to happen fast and with the minimum possible calculations. We need those in-between values with minimal processor load otherwise I'm going to have yet another reason to want to throw my printer out the window.Peter needed a way to quickly calculate the volume of a tetrahedron. Actually, Peter needed to be able to quickly calculate the area of a triangle, the volume of a tetrahedron, or indeed the 'content size' of any simplex. He wanted this method to work for any data application in any number of dimensions.And what method did he use to find these volumes super fast?I kid you not. HERON'S AREA FORMULA.
I'm sorry. This is a very emotional chapter for me. First Eugénie's roller-coaster ride of reveals, and now Heron's Formula has slammed into the room with a practical applica tion.
Hexagons and bees:
Left to their own devices bees actually build circular cells, not hexagonal ones. That's right: a single bee cell would be a cylindrical tube of wax. But bees don't build just one, they build many cells all next to each other. It's the interaction ofadjacent cells thatcauses hexagons to form.
Wax is a flexible substance and bees take advantage of that by constantly pushing it around as they build, as well as scraping it back up and reforming it. While a bee building a cell will push all of the walls out to make a cylinder, a bee in the cell next door is pushing back. And it is that back-andforth pushing of the malleable wax that causes the cells to eventually settle into hexagonal shapes. Hexagons are a shape that can naturally result like this, which is why they pop up so often in nature.
GPS has been mentioned a few times in this book already, but'GPS', for 'Global Positioning System', is actually a brand name.There is more than one system of spacecraft beaming precise radio signals to form a global-navigation satellite system, giving us the generic name GNSS. The OS actually describe their 115 receivers as 'geodetic GNSS receivers', but this is a Kleenex/Hoover situation where the GPS brand-name has become a generic term in the eyes of the public. The OS have also called their sky-facing system OS Net that is a bit too Terminator-y for my liking.
The GNSS revolution has also exposed the mild inaccuracies of the old ways of measuring things. One of the touristy things to do in London is to go to the Greenwich observatory and stand on the o° line of longitude. The Earth's spin gives us a clear equator right in the middle of the planet, and the axis of rotation gives us precise north and south poles, so angles of latitude have an unambiguous reference frame. Longitude, however, which measures how far 'around' the planet a location is, has no such obvious starting point. For a while some countries just used their own starting points, but that was exactly as confusing as you would expect.
So in 1884 representatives from twenty-five countries got together in Washington, DC, and voted on it. Greenwich observatory in London won with twenty-two of the votes (the dissenting countries were Brazil, France, and Santo Domingo).This designated the starting point, or 'prime meridian', as the line that started at the Greenwich observatory and then went all the way around the planet, through both of the poles. The line is actually a massive circle going exactly around the very center of
Earth. Or it would have been, if they hadn't missed.
To draw a circle that goes around the center of the planet you need to know exactly which way is down. This was done very carefully using gravity to indicate the direction 'down', but the intrepid surveyors did not know that the density and deformities below the surface of the Earth could cause local gravity to misalign with the geometric center of the Earth. Thus the original'zeroth' circle around the Earth wasn't actually centered on the middle of the Earth. Nobody noticed at the time, though, and a big shiny metal line was put in the ground at Greenwich for tourists to jump back and forth over.
Satellites have no such problem locating the middle of the planet and so have a more accurate 'down'. With the global adoption of GNSS there was now a mismatch between the true longitudes and those based on the metal line in Greenwich. In 1984, exactly a century after the first gathering, countries from around the planet got together and decided on a new prime meridian, ever so slightly to the side of the old one.
I don't think it would take much arguing to persuade anyone that an understanding of waves makes our modern, digitally connected world possible. Signals fly at the speed of light between mobile phones and communication towers, as well as down fiber optic cables between continents. All impossible if we didn't have the knowledge of Fourier to disentangle multiple, combined sine waves.
December 17, 2024
Because I am a popular math writer by profession, I have pretty high standards for popular math books. The main thing I'm looking for when I read a popular math book is: Does it tell me anything new, or is it a copycat that reproduces things you can find in many other books? By this test, I was DELIGHTED with Matt Parker's latest book, "Love Triangle," because he did such a good job of (a) following his own muse, and (b) talking with people who aren't professional mathematicians.
Without recapitulating the whole book, let me give a couple of examples. One reason that "trigonometry shapes the world," discussed in chapter 4, is that triangular meshes can be used to re-create smooth surfaces. This process is used a lot in engineering and in the visual effects industry, in computer games and films. Parker went out and interviewed VFX experts and found out some surprising things. For example, triangular meshes are used in computer games, but quadrilateral meshes are used in movies. He explains why, and the answers are good food for thought for anyone who might think that you can just model surfaces by connecting a bunch of dots. No, you really need to think about how you are doing it, and the right answer for a movie might be the wrong answer for an engineering model.
For another example, in the final chapter (Chapter 10) he finally gets to the topic of waves, which are indeed ubiquitous and are described mathematically by the sine and cosine functions of trigonometry fame. Here Parker does something almost no pop-math writer does: he actually gets into Fourier analysis, the process of breaking down complicated waves into their simple sine and cosine parts. This is the really good stuff that explains how "trigonometry shapes the world." Parker doesn't just give artificial math-class examples. He explains what a spectrogram is, and how you can actually see the difference between two species of monkey calls. He explains how Fourier analysis was a key to the elucidation of chemical structures by X-ray diffraction (e.g., the structure of DNA). And he shows the Fourier analysis of the first gravitational wave ever detected. This is some of the coolest science of the 20th and 21st centuries. Lots of credit and publicity (and even Nobel Prizes) go to the discoverers. But nobody thinks about the mathematics that made it all possible.
I also like the way that Parker follows his personal whims. When he finds out that there is a spot (*one* spot) in the UK where "map north" = "true north" = "magnetic north," he has to go there. When his personal friend, Alex Jones (bass guitarist of a band called Blur) wants him to come up with a math-inspired disco ball, he does some Internet sleuthing and discovers a chiral rhombification of a snub dodecahedron. You probably won't completely understand it (I can't say that I did), but it's fun to read about why this particular polyhedron was the perfect solution to Jones' request. It sure makes a pretty picture (see page 0.000000 for a computer rendering, and the following page for the actual party where Parker's mathematical disco ball was revealed).
Small cavils: 1) The page numbers in the book are the sine of the actual page numbers. Cute idea, but not explained anywhere and it actually gets in the way of finding things in the book later. 2) Parker says that he hates Heron's formula, then eats his words in the next chapter when he finds out that the formula is used for color interpolation. Kudos for honesty, but boo for underestimating Heron! When you see any formula this weird, there has to be some wonderful and mysterious math behind it. That math is the Cayley-Menger determinant. Read about it, Matt, and write about it in your next book! 3) In a book that is supposedly about trigonometry, we don't get to actual trig functions until chapter 7? 4) Chapter 10 was so fabulous that I just wanted it to keep going, and felt bummed that it seemed to end kind of abruptly.
To sum up: Parker knows people, he talks to people, he does crazy things and finds people who do crazy things. The result is a math book that doesn't feel the least bit like a math book. Recommended as a gift for any of your friends who love math but didn't love math class.
Without recapitulating the whole book, let me give a couple of examples. One reason that "trigonometry shapes the world," discussed in chapter 4, is that triangular meshes can be used to re-create smooth surfaces. This process is used a lot in engineering and in the visual effects industry, in computer games and films. Parker went out and interviewed VFX experts and found out some surprising things. For example, triangular meshes are used in computer games, but quadrilateral meshes are used in movies. He explains why, and the answers are good food for thought for anyone who might think that you can just model surfaces by connecting a bunch of dots. No, you really need to think about how you are doing it, and the right answer for a movie might be the wrong answer for an engineering model.
For another example, in the final chapter (Chapter 10) he finally gets to the topic of waves, which are indeed ubiquitous and are described mathematically by the sine and cosine functions of trigonometry fame. Here Parker does something almost no pop-math writer does: he actually gets into Fourier analysis, the process of breaking down complicated waves into their simple sine and cosine parts. This is the really good stuff that explains how "trigonometry shapes the world." Parker doesn't just give artificial math-class examples. He explains what a spectrogram is, and how you can actually see the difference between two species of monkey calls. He explains how Fourier analysis was a key to the elucidation of chemical structures by X-ray diffraction (e.g., the structure of DNA). And he shows the Fourier analysis of the first gravitational wave ever detected. This is some of the coolest science of the 20th and 21st centuries. Lots of credit and publicity (and even Nobel Prizes) go to the discoverers. But nobody thinks about the mathematics that made it all possible.
I also like the way that Parker follows his personal whims. When he finds out that there is a spot (*one* spot) in the UK where "map north" = "true north" = "magnetic north," he has to go there. When his personal friend, Alex Jones (bass guitarist of a band called Blur) wants him to come up with a math-inspired disco ball, he does some Internet sleuthing and discovers a chiral rhombification of a snub dodecahedron. You probably won't completely understand it (I can't say that I did), but it's fun to read about why this particular polyhedron was the perfect solution to Jones' request. It sure makes a pretty picture (see page 0.000000 for a computer rendering, and the following page for the actual party where Parker's mathematical disco ball was revealed).
Small cavils: 1) The page numbers in the book are the sine of the actual page numbers. Cute idea, but not explained anywhere and it actually gets in the way of finding things in the book later. 2) Parker says that he hates Heron's formula, then eats his words in the next chapter when he finds out that the formula is used for color interpolation. Kudos for honesty, but boo for underestimating Heron! When you see any formula this weird, there has to be some wonderful and mysterious math behind it. That math is the Cayley-Menger determinant. Read about it, Matt, and write about it in your next book! 3) In a book that is supposedly about trigonometry, we don't get to actual trig functions until chapter 7? 4) Chapter 10 was so fabulous that I just wanted it to keep going, and felt bummed that it seemed to end kind of abruptly.
To sum up: Parker knows people, he talks to people, he does crazy things and finds people who do crazy things. The result is a math book that doesn't feel the least bit like a math book. Recommended as a gift for any of your friends who love math but didn't love math class.
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