Quaternions are Amazing and so is William Rowan Hamilton!

Quaternions are amazing, they made modern vector mathematics. But how were they created and how do they work? In order to explain all that I am providing both a fun biography of Hamilton AND a biography of quaternions and the couplets that led to them.
My website: www.KathyLovesPhysics.com
My book "The Lightning Tamers" on Amazon: amzn.to/3I7N4mq
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Anne van Weerden's Website:
annevanweerden.nl/Entrance.html
"A Victorian Marriage"
annevanweerden.nl/VictorianMa...
Full song of "William Rowan Hamilton": • William Rowan Hamilton...
Cover art uses (with permission) the amazing sand sculpture by Daniel Doyle.
I highly recommend glancing at his blog about it, just beautiful!
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0:00 Why I made this video
2:08 Hamilton's Career (History)
6:46 Couple/Couplet Theory (Math)
10:28 Hamilton's Personal Life (History)
15:34 Quaternions (Math)
31:55 Hamilton's Legacy (History)
39:20 What's Next

[@korolev23]

If you like quaternions you’re going to love geometric (Clifford) algebra, which finally situates the quaternion concept in its rightful setting.

[@davidhand9721]

Bingo. Complex, Quaternion, and Octonions are all hacks in comparison. Heck, most linear algebra feels like a hack once you know GA.

[@bradhayes8294]

This is an incredible video! As a mechanical engineer, I've always believed the most enjoyable and intellectually fulfilling way of teaching and learning any mathematically based subject is to include both the mathematics and the history.

[@exsollertan7366]

I very much agree. Sometimes the best way to absorb new concepts is to learn how mathematicians and physicists developed their ideas and how they overcame the pitfalls on the way. John Fannon

[@djredrover]

I am literally using Quaternions for my hovercraft UAV project. I use Quaternions to obtain my vehicle's attitude (roll, pitch, yaw which are Euler angles phrasing but used in Aerospace) to avoid the Gimbal Lock that is inherent in regular Euler Angles method of obtaining attitude from the Inertial Measurement Unit (IMU). In fact, all smartphones also use Quaternions to obtain phone attitude and pose.

[@mekkler]

Quaternions are everywhere. They are used millions of times a second in any modern video game.

[@djredrover]

@@mekkler facts!

[@djredrover]

Also, I am an Electrical/computer Engineer from UBC Canada, and we did NOT learn quaternions in school. Not even the concept. I had to teach this to my self thanks to 3b1b and others.

[@labibbidabibbadum]

Well, then you're obviously a drunkard.

[@TheEvertw]

Not to mention that there are many different variants of Euler angles, all incompatible. Roll-Pitch-Yaw is a different system of coordinates than Pitch-Roll-Yaw, for instance. To make testing nicer, the differences between them are only visible in compound angles, with multiple non-zero values, and are small for small values.

[@sebastian-azcarate]

I really think math/physics should be teached with history alongside. So much can be appreciated this way! Thanks Kathy for these videos and your book.

[@DJF1947]

Perhaps they should teach English as well.

[@emjizone]

Good teachers put their students in the situation of the previous discoverers, so they don't only deliver the results but also the *joy* of the discovery with the results.

[@larrywildman4381]

Too often schools teach you things like they had fallen from they sky, without the understanding of why scientists got there and how. It's a pity.

[@PanglossDr]

Hamilton was an ancestor of mine. When the plaque to him was unveiled at Broom Bridge my Dad was there as one of his closest living relations. I think the genes are still there in the family, I never had any problems with maths and my daughter got a two Es offer to Cambridge at the age of 16 on the strength of her maths.

[@JohnVKaravitis]

What are "E"s?

[@Kathy_Loves_Physics]

They stand for excellent it’s equivalent to an A.

[@PanglossDr]

@@Kathy_Loves_Physics Not exactly. An E is a very low grade, A, B, C, D, E, just above an F for Fail. What a 2 Es offer means is they don't care what grades you get, you are in.

[@PhilFogle]

@@PanglossDr Congratulations on your talented Daughter! May the light of mathematics glow brightly for her...hmmm this is probably historic? Oh well, congrats anyway :))

​@@PanglossDr this, but to clarify it's something that's given to someone who in practice wouldn't have problems at getting to the university/college. It's a "we know you're skilled enough to join, no need to stress on finals" kind of a thing.

[@tomkerruish2982]

They should make a Broadway musical about this guy!

[@Kathy_Loves_Physics]

I would watch TF out of that 🤣

[@leyasep5919]

@@Kathy_Loves_Physics only watch ? not script ?...

[@frankrizzo7454]

Only if they morph him into a black transvestite.

[@b43xoit]

LOL

[@le-m0nke]

@@frankrizzo7454 LOL

[@enricolucarelli816]

I love physics, mathematics, its history... I am 64, and yet, when I listen to your wonderful videos, I feel like a little kid listening mesmerized to the most beautiful fairy tales. Thank you very much ❤

[@Kathy_Loves_Physics]

Aw that was lovely, thank you.

[@pikiwiki]

but are they fairy tales, or, reality?

[@TheKosmicCraftsman]

Great video! As some commenters have already mentioned....Geometric Algebra / Clifford Algebra is the *actual* powerhouse that gives rise to complex numbers and quaternions naturally. As W. K. Clifford's life ended too soon, his work remained almost unrecognized for a long time. Luckily, it was rediscovered and built upon in recent times by David Hestenes. GA has been a real eye opener for me and others. Maybe it will have the same effect on you and inspire you for a new video!

[@spaceranger3728]

When I was in grad school taking classical mechanics, quaternions were mentioned briefly but we never got into them. But then when I found myself out working on space shuttle simulators, it suddenly became evident how much more efficient it was to describe rotations with 4 quaternion elements instead of direction cosine matrices. In those days, computing power was limited and passing rotations from one processor to another took a lot of number crunching.

[@DFPercush]

Why 4 quaternions? Isn't one sufficient to express the orientation of the craft? Or are those others for things like gyros?

[@drtidrow]

@@DFPercush He's referring to the four elements of a quaternion, versus the nine in a matrix. There's a lot fewer floating-point operations using quaternions than using direction cosine matrices.

[@DFPercush]

@@drtidrow Oh, of course, derp. :P

[@tomrobla8981]

I worked on missile guidance systems in the 1970s. We used quaternions with Binary Angular Measurement (BAM) to manipulate angles. Real time control calculations using fixed point arithmetic.

[@kevinmorgan2317]

Great video. Thanks. Quaternions find their home in 3D computer graphics. Matrices allow rotations about an axis only sequentially and do not work smoothly. Quaternions allow for rotations around more than one axis simultaneously. And they do this faster and use less memory.

[@elderbob100]

I wish I had Kathy as a teacher when I was in school. Kathy loves Physics and it shows. History is actually interesting once you get past dates, names, places and all the other trivia the our educational system fills our heads with. Kathy makes me curious about Physics. She makes me want dig deeper and learn more. Awesome channel Kathy!

[@jameszmuda6362]

Absolutely right. The whole mystery of the meaning of the dot product and vector product is made clear once you see both of them just “pop out” of the operation of multiplying out two quaternions. I always wondered where those (seemingly arbitrary) definitions of vector products came from.

[@pauljmey]

This actually a good jumping off point for the story of Clifford (building on Hamilton) and modern attempts to revive Clifford Algebras for physics (Geometric Algebra, Spacetime Algebra)

[@jasonzavaglia]

Best teaching on quarternions I’ve ever watched. I never really got it previously.

[@AlabasterClay]

Wow---this is an amazing story. Love, love, love, love it. The history, the biography, the math, they physics. Thank you!

[@Kathy_Loves_Physics]

Glad you enjoyed it!

[@Cmurphize]

Dropping out of school for physics 11 years ago left a void. One that you're helping fill. I'm so grateful for your channel. Would have never guessed my interest in history would get me back into physics.

[@keybawd4023]

Your videos are one of the jewels of the internet. As the man, who you may remember spent an afternoon alone in a railway carriage with Otto Frisch, I have to tell you that although -'a long, long time ago' I did-postgrad research on free radicals, your videos make me realize how little I appreciated the background to so many things that, by the time I got to University, I took for granted. Your videos give depth to our understanding of whatever subject you choose. Thank you.

[@HoraceMash]

Wow! This is a tour de force! Thank you so much for opening my mind to the meaning and evolution of quaternions in relation to vector calculus, and the human dimension of these concepts. What a treat to see such a lucid and thoughtful presentation. You are a brilliant historian and science communicator 🎉

[@Kathy_Loves_Physics]

I’m so glad you liked it David. ❤️

[@SciHeartJourney]

Thank you Kathy. I learned about Quaternions doing video game development. They make the math of "rotation" in matrices a bit easier to calculate.

[@susilgunaratne4267]

Thank you very much for giving us the clear picture of vector calculus & its discovery.

[@robertschlesinger1342]

Excellent video. Very interesting, informative and worthwhile video. BTW, Euler, a German surname, is pronounced as one might pronounce "oiler," rather than the Americanized pronunciation sounding more like "youler." Your many videos are very interesting, worthwhile, and a great benefit to many people wanting to broaden their horizons into the physical sciences.

[@mcello47]

Brilliant! As much as I admire all of Hamilton, Maxwell and Gibbs, I wasn't aware of the common historic thread that united them. Thanks for such an inspiring piece!!

[@briannewman9285]

I'd *love* to have this lady as a math prof.. She is enchanting.

[@alwaysfourfun1671]

So delightfull to hear this story. So well told, by a natural teacher. History, as part of Physics classes, could certainly inspire many students, who otherwise would tend to think of the mathematics as difficult and boring. What a wonderful achievement of mr. Hamilton and what an appreciation from the brightest of his time. Deserves to be commemorated with solid teaching in quaternions.

[@jdp9994]

Beautifully done! It would be wonderful if this continuing tale is tied in with Clifford Algebra/Geometric Algebra, which I think gives further weight to Hamilton's intuition of the important role these can have in understanding "the mathematics of the physical universe".

[@ProfessorBeautiful]

Thank you,, this is a wonderful thing to rehabilitate the reputation of the Hamilton family, and the quaternions! This was a treat. And, to reiterate, the Lightning Tamers is a tremendous book, a great achievement.... and FUN!

[@Kathy_Loves_Physics]

Thank you so much! ❤️

[@7sArts]

@@Kathy_Loves_Physics Thanks Kathy, excellent look at Hamilton and the importance of quaternions. Not only did you show how they were important to physics, but also to Maxwell and electro magnetism. I firmly believe that Tesla used the time that he spent ill and bed ridden to master quaternions and later used them in his research and experiments to achieve the success no one has yet understood or matched.

[@Roommate625]

This content should be on PBS to expose a larger audience to your amazing teachings. Your enthusiasm for the material and delivery remind me of the TV show Connections.

[@Kathy_Loves_Physics]

Thanks. I’m game if PBS is.

[@lawrencedoliveiro9104]

Not sure if you knew this, Kathy, but _Connections_ was a documentary series on the history of science and technology hosted by James Burke. The first lot of episodes was from 1978. There were two more called _Connections II_ and _Connections III_ , as well as _The Day The Universe Changed_ . The key motif was the progress does not proceed in straight lines, but lurches in all kinds of different and unexpected directions. All still well worth watching.

[@therealpbristow]

​@@lawrencedoliveiro9104 I loved that series! James Burke was another fabulous explainer, coming to the world of science and technology via a humanities background. One of several guys deliberately recruited by the BBC to present and "translate" anything that might be too technical for audiences, when they were covering things like the Apollo Space program (which he famously did alongside Patrick Moore).

[@clay.tennis]

I love the historical perspective, the life you breathe into these amazing scientific pursuits are beautifully woven in with brilliance, mathematics, and nothing left to want for.

[@AxisAngles]

Fantastic video. Thank you so much, I never knew how fundamental quaternions were to so much modern mathematics.

[@ErikHare]

Two biographies. One for a quaternions and one for Hamilton. Sounds great!

[@pauleohl]

I was wondering why I never heard of quaternions until you got to the part about dot product, cross product, divergence and curl, all of which I did learn as an undergraduate Mechanical Engineer: Pratt Institute 1965.

[@leehaelters6182]

Yay for Pratt, I loved that steam powerplant that lit the whole campus!

[@pauleohl]

@@leehaelters6182 As i recall, the steam power plant made DC only which was used for lighting and the elevator in the building which powered the steam plant. The spent steam was piped to radiators in that same building for heat in the winter. The whole building burned down in 2013. The floors were soaked with oil from the lubricators on the reciprocating steam engines. When did you graduate? What was your field?

[@matthewhunt6525]

Kathy your awesome I freaking love your mind and your work

[@John-pp2jr]

7:33 Euler is pronounced OILER. Fantastic video❤️

[@Tobascodagama]

My professors were all quite insistent on this as well!

[@Kathy_Loves_Physics]

Wish mine were - would have saved me embarrassment today.

[@amazing7633]

@@Kathy_Loves_Physics This is a standard difficulty for geeks in physics and mathematics who know words only from reading them. Our physics department had two professors who had worked on the Manhattan project and thus had little difficulty attracting guest speakers who were well known in nuclear physics. One of the students pronounced "new-kyew-lar" when speaking to a famous guest and got a withering correction, "Nu-cle-ar, please!"

[@nicholaslederer]

@@Kathy_Loves_Physics I think we all came here to learn and can appreciate the learning process. Keep making awesome videos (and mistakes to learn from)!

[@robbannstrom]

Oh, this is great - thank you Kathy for this video. I remember meeting a member of the physics faculty when studying astrophysics years ago, and he was "well-known" in the department for studying quantum mechanics using quaternions - I never did find out what they were until now! Also, as a radio ham, I look forward to your series on the evolution of wireless - it's sure to be really interesting...

[@petsatcom]

Another fantastic and interesting scientific mathematical video.

[@TexRenner]

I do watch for the history, but you make the math interesting enough I would not want to skip. Nice of you to offer.

[@Cherokee140Pilot]

Kathy, After you passed basic Algebra, you lost me. And yet, I found myself enthralled by your storytelling and enthusiasm for Mr. Hamilton. Great video, and you have a new subscriber! 😊

[@nnfefe9451]

Hello Ms., you too are a mathematician because you able to explain effortlessly what other mathematicians had written. Thanks.

[@NEMOPMORPHY]

So glad I found your channel, this is amazing!

[@shawgeasland2096]

Wow!!!! Another high quality channel to nerd out over!!!! I am so excited!!!❤🎉

[@mariotabali2603]

Great channel. It is a gift that you started it.

[@nathanwestfall6950]

Another great video! Thank you for your dedication, research and hard work!

[@BruceCurrell]

Your enthusiasm is adorable. [And contagious]. Thanks for all the hard hard work you do.

[@nativesun7661]

You’re doing such a fantastic job!!!! Thank you. Great channel for both history and math!!👏🏾👏🏾

[@craigslist6988]

what an amazing history exploring video with such dense information presented but with such a smooth flow, thank you.

[@zetristan4525]

This is so awesome of you! Kathy, you're a force of inspiration

[@dominiquefortin5345]

It was very interesting to see how the development of quaternions lead to matrixes, vectors and geometric algebra. Today, you can simplify Maxell equations in terms of geometric algebra.

[@djredrover]

7:32 I believe the correct pronunciation is "Oil-er". (I used to say "you-ler" before too until my Russian friend corrected me, although the pronunciation is more French-sounding).

[@ralphclark]

Excellent work Kathy. I’ve always found this stuff impenetrable until now.

[@xyz.ijk.]

One of your best videos! This was outstanding. Thank you so much for this work, and for clearing up the misapprehensions about his life.

[@TheAlison1456]

you wrote a book, that's awesome! great video, what a great biography.

[@joehelm5072]

BRAVO! A great job realigning facts and history, connecting the dots on the arrow of math and its applications!

[@ronaldbarrett1506]

Kathy's teaching videos are fantastic. Thanks

[@mortenwesteraa7910]

Extremely good video. Thanks so much Kathy, for sharing this and correcting history. You make the world a better place.

[@hp127]

I simply love your way of enlightening about the history and mathematics. Thanks

[@Cor97]

Thank you for this compelling presentation of a remarkable person.

[@bobdinitto]

Fascinating story, thank you Kathy!

[@h2energynow]

Always a deeper insight into some amazing people who influenced our lives. Thanks for this incredible video.

[@joystickmusic]

Thank you for setting the record straight again! I've watched all your videos and I learn a lot. When I was a kid, my father always used to explain physics and math through te personal quests these researchers and mathematicians were on, so these videos hit close to home, in a good way. (I think you can just leave the bloopers out. They do not add to your stories.)

[@multiplysixbynine]

I love it! As you described how quaternions informed vector analysis, my mind was abuzz with new connections tying back to some math and geometry I have been working on lately while writing software. Like most folks, I learned vectors well before quaternions and some of their connections may have been obscured to me by familiarity.

[@musicalengineer]

Wonderful videos as always. Thank you, Kathy.

[@jbsmathers]

Don't ever stop making these videos or writing your books! History will find such work more valuable than the incomprehensible theoretical gibberish dominating physics today.

[@lylecosmopolite]

A physicist who shares your derogatory opinion of fundamental physics in recent decades, is Sabine Hossenfelder, who KZbin channel has been a very active one for 10+ years.

[@pikiwiki]

Don't ever stop! Ever!

[@lourias]

I really did not grasp all of calculus until I encountered Calculus in the 3 dimensions. Then, boom, I have been "hooked" since. It is beautiful that crochet is calculus wrapped in knots 🪢, stitched knots🪢.

[@tomcarroll6744]

Really over-the-top, great content. Thanks!

[@niconeuman]

Excellent video! As always! Now I understand quaternions much better, thanks!

[@johankotze42]

Another masterpiece! Thank you, Kathy.

[@a4_empty574]

I love and i appreciate the approach used in all your works. Surely people used to Say you that but: you make a good job. Thanks.

[@rgoodwinau]

Wow! Inspired youtube recommendation of an inspiring youtuber! Thanks Kathy.

[@TheChzoronzon]

You are such a jewel, that enthusiasm... I'd have paid to have you as a teacher in my teens I hope you have the most wonderful day :)

[@AT-27182]

Hamilton was incredible and your telling of his story is exciting and helpful. Thank you so much.

[@Kathy_Loves_Physics]

So glad you enjoyed it.

[@jamesbond_007]

Outstanding! Incredibly enlightening -- loved the associated historical narrative of Hamilton's life! Well done Kathy!!!

[@Kathy_Loves_Physics]

Thank you so much!

[@adhisadhaiv1652]

Thankyou for this amazing video , it answered a lot of my questions :)

[@yamspaine]

you are a treasure, well produced videos, thanks!

[@DougMayhew-ds3ug]

Fabulous tour of Hamilton, I love the warmth you bring to this sometimes cold world of mathematics and physics; it’s contagious. Just a beautiful and outstanding approach! You truly bring these people and ideas back to life, and make the subject approachable for the curious but intimidated crowd.

[@Rigel66]

You are awesome Kathy!...fantastic content!

[@Arfonfree]

Years ago I studied fluid dynamics and fell in love with vector mathematics. However, I never followed the history any further back than Edwin Bidwell Wilson. Thank you so much for this video. It has opened a new window for me into this fascinating world.

[@jacejunk]

Spectacular! I always enjoyed the story of Hamilton's discovery of quaternions (and appreciate their modern application in computer graphics), but did not realize how integral they were in inspiring vector calculus.

[@tubbyoneness]

I love your channel. This is my absolute favorite episode so far. Bless you for the work that you are doing. You have combined this fascinating information with your incredible and contagious enthusiasm to make something that is truly wonderful.

[@Kathy_Loves_Physics]

Wow, thank you!

[@yusong1141]

I can't believe my luck when I found thus video. So so so goog! I enjoy every single minute of it.

[@zackstaboy]

Wonderful video. As a supplement, it is fair to say that Clifford/Geometric Algebras are a definitive extrapolation of the true promise of Quaternions (along with the Cayley-Dixon algebra sequence). Either one of two particular 7:04 Clifford Algebras provide a beautiful environment for framing relativistic physics - as emphasized and developed by Hestenes.

[@tedsheridan8725]

What an utterly fascinating video! As someone who's pretty familiar with quaternions (I took a picture with the Broome Bridge plaque during my trip to Ireland last month), I was only vaguely familiar with the history of vector calc., let alone Hamilton himself. I was taught that quaternions were replaced by vectors, and I only actually learned them much later. You do a fantastic job of showing how quaternions were the OG's of it all. Can't wait to see more on your channel!

[@srmxe417]

Amazing refresher :D Thanks so much

[@gregkocher5352]

I love these videos. Thank you from a kindred lover of the wonder of the story of science and and math.

[@daffyduck1486]

This is a wonderful video. Thank you Kathy.

[@wagnerbomfim6266]

Amazing. Thank you for the lecture. Great work

[@fggolding]

As always this a great video. The evolution of mathematic and physics is convoluted one.

[@vitalyrybin5645]

Kathy, you've done a great job! I enjoyed your priceless work very much. Your video is full of amazing historical facts and your way of telling a story is so good that I had no issues with understanding watching it even despite the fact I have an upper-intermediate level of English (my mother tongue is Russian).

[@treborg777]

This was wonderful, thank you for doing this video. I had been confused about quaternions for some time, and this helped me immensely.

[@Kathy_Loves_Physics]

I’m so glad

[@drtidrow]

Indeed, I had thought that vector calculus had been developed independently, instead of evolving directly from quaternion math.

[@coolcat23]

Another masterpiece by Kathy!

[@lucasgroves137]

Great channel, Kathy!

[@jeffm3865]

This was pretty fantastic. I appreciate your research and delivery to educate others. I was searching to see if you had any videos on Fourier, but I didn’t see any. I haven’t found much on Fourier’s origins and early work that eventually led to Fourier series and Fourier analysis. I believe he was once in Napoleons army in Egypt and based some of his early work on maximum rate of firing cannons so they will not melt. Unfortunately, I haven’t been able to find many sources so I’m not sure if the cannot firing source was true.

[@JustaReadingguy]

I love your videos and book. Can't wait for the next books!

[@Kathy_Loves_Physics]

Thanks.

[@khoavo5758]

New subscriber: This is both entertaining and helpful. Thank you!

[@tddybr78]

Awesome, as usual! People should learn more about quaternions as undergrads. Not to mention Hamilton. Best, Ted

[@hectormanuel8360]

Another excellent video to watch. Hooray!

[@roberthayter157]

A brilliant video about a brilliant man. Thank you!!

[@londonmackerel7462]

I love this video and, even though I have a maths background, learned a whole lot of new stuff. It is great to find someone who also loves the history of mathematics and physics. Your enthusiasm is infectious and inspiring. You do a fab job.

[@Kathy_Loves_Physics]

Glad you enjoyed it!

[@ik04]

This was a fun one! Brain crushing details!

[@andrewreeve5537]

Love your histories and perspectives on science. You look great in yellow!