Why can't you multiply vectors?
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- Опубліковано 27 кві 2024
- ..or can you? A deceptively simple question with a complex answer - come join a mathematical journey into madness and wonder, in search of answers that might just give you a new perspective on the mathematical constructs we use in our games
Recorded at Dutch Game Day 2023, October 4th
Timestamps:
00:00 Intro
04:34 Talk Start
05:17 Anatomy of a Vector
06:09 Products
09:14 Numbers of different kinds
13:32 Complex Numbers
18:10 Code for complex numbers
19:10 Complex multiplication visualized
20:22 Multiplying Vectors (an attempt)
24:08 Frustration
24:36 A Leap of Faith
25:36 A divine axiomatic truth
26:40 Squaring Basis Vectors
27:46 The Diagonal of Bases
29:09 Vector multiplication unraveling
29:36 Dot/Cross Byproducts
30:32 Complex Byproduct
31:17 Quaternionic Byproduct
31:52 The Answer
32:52 Beyond the Algebra
33:29 Basis Bivectors
35:06 Conceptualizing Vectors & Bivectors
37:08 Wedge Product
38:46 Curvature Bivector
40:59 Mental Gymnastics
41:56 Geometric Algebra & Multivectors
43:19 What does the multiplication represent?
44:10 Outro
45:53 Q: When is this useful?
48:11 Q: Does your math lib support geometric algebra?
49:22 Q: Do you plan on making tools for other engines?
50:53 AV team kicking us out
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![Math for Game Devs [2022, part 1] • Numbers, Vectors & Dot Product](http://i.ytimg.com/vi/fjOdtSu4Lm4/mqdefault.jpg)
some notes/clarifications!
• "I don't like the title" I thought it was fun, and it's a good conversation starter and defines the whole trajectory of this talk!
• at 9:55, when I multiply natural numbers - snailcats are not "units" they are creatures equal to 1 so im right and you're all wrong get out of here with your obsession with units!!
"but freya it should be 6 snailcats squared" you're square >:(
• at 11:42, there's some nuance in how we define closure for division, since you usually can't divide by 0, so technically there's exceptions here, but the general idea stands if we ignore 0 :)
• at 25:14, the divine truth bestowed upon us by salad, of v² = |v|², only applies to vectors - it does not apply to bivectors or rotors, or anything else! In my approach, this is a fundamental axiom, which is one way you can define a clifford algebra, in case you want to read up on this some more! It is our starting point, the initial assumption, similar to how i² = -1 is the initial axiom/definition for the imaginary unit. There are other ways of defining clifford algebras, as usual in math, definitions can work from many directions, and the math works out the same way regardless, I just happened to pick this approach for this talk, because I think there's particular beauty in how it so cleanly produces many of the products and constructs we're familiar with!
• at 28:27, the reason I'm expanding (x+y)² to xx+xy+yx+yy instead of xx+2xy+yy, is because at this point we don't know whether multiplying vectors is a commutative operation, so we can't say for sure if we can simply swap x and y here. Real numbers are commutative, but in this case we have to be careful, because these aren't real numbers, they're symbols representing our basis vectors! And as it turns out in the end, vector multiplication is in fact non-commutative, as is the whole VGA multivector multiplication
• at 28:57, the rule of swapping xy = -yx only applies to orthonormal vectors, like our basis vectors, which are orthogonal, and of unit length, hence the name orthonormal. This rule does not apply to arbitrary vectors in the general case
• at 43:40 when I apply the quaternion as a rotation to the cube, it assumes the two vectors a and b are normalized, which results in a unit quaternion (a quaternion with a magnitude of 1), which is what we often use for rotations in games. However, for two general vectors, the quaternion/rotor result of multiplying them together is *not* a unit quaternion, and is thus not a valid rotation representation
This should be pinned I think
@@G3rmanGsnLPwait how did it get unpinned the heck. anyways it's pinned now again! I hope!
Yeah editing a comment automatically unpins said comment
@@acegikmo when you edit a pinned comment it gets unpinned
ah, fixed!@@segooglenutzer8907
Freya is a god-level instructor.
If you can make geometric algebra make sense on stage you deserve an award.
University has left the chat
and not just make sense, she made it interesting and truly captivating
She should teach math in school... it's way more interesting this way then the way my old ass math teacher tried to teach us
I think she understands it well because she's working with it. Most people who make a living explaining geometric algebra to other people dont really use it for anything.
Agreed. Her splines video is a masterpiece of visual communication. And actually useful.
You are the 3blue1brown of computer graphics, I wish you made more videos.
I'm trying! burnout has been hitting me very hard, so I've had to retreat quite a lot in my video making after the spline video. I'm only just now starting to be able to work again
@@acegikmo sorry to hear... no hurry, this should be fun not stressful, take your time and thank you 🙂
She is herself and that's great enough. No need to invoke the overrated.
@@acegikmothanks for sharing what you did!
i wish 3blue1brown was the 3blue1brown of computer graphics
Was listening to the talk, and at 10:00 I was like "Two snailcats multiplied by three snailcats is obvioously six snailcats squared" ;D
psh, physicist!
Yep, units matter indeed. The hint is actually already there: squared means planar.. it's a dead give-away really. The bivectors were already there from the start, but disregarded alas.
I am an engineering graduate student. Didn't expect much clicking into this video but after watching a few minutes, I found this is gold.
This video clears up my concepts in relationship between complex numbers and quaternions (which I see often but don't understand) , where they could come from, and introduced an interesting concept of bivectors.
All from one fundamental axiom! The math is beautiful and you elegantly presented it. Thanks a lot!
glad those concepts connected for you!
Several months ago, I watched your video on Splines. It was fantastic and has been a great help to both me and my colleagues. Just yesterday, I was searching for some materials on Geometric Algebra. I noticed this video was mentioned in your Discord, but I skipped over it initially since I was specifically looking for content related to GA. However, when I stumbled upon it again in the GA Discord, I figured I should give it a watch! From the get-go, I was really impressed with your presentation style. You have a knack for presenting information! Instead of bombarding us with equations and jargon, you guide us step-by-step, posing questions and then answering them. I genuinely appreciate it! I hope you weren't constrained by time when putting this together. It's just brilliant how the presenter I admire so much is covering topics I'm deeply interested in. And the timing couldn't be better for what I currently need. It's hard to put my excitement into words!
I'm glad you liked it! I was pretty constrained by time though yeah. Initially I had a lot of stuff related to rotation representations, but in the end I couldn't justify the time for it, so almost all of that was cut. so instead of being a general talk about rotations, pseudovectors and GA, I reframed it as an obsessive attempt at multiplying vectors, to make it a little more cohesive, focused, and narratively structured!
nice
that video is like the the hollywood production for math explanation videos....@@acegikmo you are fantastic and your content is out of this world and I mean it as I say it. Thanks a lot
I would like to validate your feelings by saying that I really enjoyed this talk and that it helped my understanding of quaternions!
I'm glad!
I literally got goosebumps watching this. You just explained something to me that I thought would remain a mystery to my grave. I can't express my gratitude.
hah, I'm glad
For most of the latter half of the talk, I was trying to square the circle with how v^2=||v||^2 doesn't work out for complex numbers, and how that felt wrong with how alike complex numbers and 2d vectors are otherwise.
Then at the end, the reveal of complex numbers being rotors instead came and everything just clicked into place.
Amazing talk, 10/10
I really, really wanted to understand what quaternions meant but never could actually. Then your explanation came and it all felt like it made sense from the start, especially since quaternions aren't the focus of this talk. Color me impressed (as I always am with your content).
see that's how I trick people, lure yall in with a simple question c:
@@05degreesthat video is certainly interesting, but the way quaternions are presented as being 4 dimensional and all the stuff about stereographic projection doesn't serve well as an introduction
you don't even need the 4th dimension to grasp what they actually are and how they work, if you use geometric algebra it becomes clear that quaternions are simply what you get when you compose two reflections using the geometric product
@@acegikmoyou could have spare me a fraction of my life time, by noting your call was gonna be about GA :(
@@2fifty533 Yeah but they also have this 4D nature too. Which comes into play when e. g. rotating in 4D using split-quaternions. (Which can also be understood through geometric algebra too, but their connection with single quaternions would be a bit simpler to navigate.
@@05degrees split quaternions have little to do with 4d rotation, they don't map to SO(4)
4d rotations in geometric algebra work exactly the same as you'd expect for 3d, you can just take the product of 2 4d vectors and that gives you a 4d rotor, and it's applied in the same way using the sandwich product
Man, you manage to capture centuries of math progress under an hour. You did color code, box up, and animate the abstracts to concrete representations. It was a very efficient presentation, your work keeps getting better!
Now this was just incredible! :D To so cleanly present clifford algebra in a way that is accessible from so many different backgrounds, and make it not only coherent but downright tight, neatly tying together all the different concepts you present along the way, is quite a feat ^^ I feel like my understanding not only of bivectors, but of geometric algebra in general, is so much more tangible now than before the talk. Hats off to you, Freya!
thank you!
Thanks for posting this! Love your math videos so it’s nice to get one of your talks aswell
thank you! also thanks to the organizers who let me share it here!
I knew about Geometric Algebra before watching this but even so it was really interesting.
The pacing was really nice and the explanations were quite easy to understand at the same time.
This was an awesome talk! Throughout all of my math education, I’ve had a weird feeling about there being all these different products for vectors. The way you presented it here, also tying in imaginary numbers and quaternions, was incredible. Certainly makes me want to research geometric algebra on my own time.
I'm sorry/you're welcome c:
35:34 This moment is a key insight. I think I finally get what a quaternion is. I had to pause the video for a while to wrap my head around it and pick my jaw up off the floor. A masterful explanation.
I just watched this whole video before sleeping. It's surprisingly relaxing, loved the talk
I'm glad
Always amazed by your content. Keep up the hard work for all of us that don't like math!
As a physicist, I'm used to dealing w/ quantities (number * unit) so two snail-cats times three snail-cats would be six snail-cats-squared... 🙂
Yep, exactly
Really makes me wonder if there is a concrete expression to this
I would say this is quite critical indeed for the purpose of the talk : multiplying two vectors is not the same as "scaling" a vector by a coefficient. Just saying "i have three sets of 2 snails is..." would be better in my opinion.
A square of side length 4 will have perimeter 4x4 =16 meters but an area of 4x4 = square metes. Calculation goes the same but nature is very different.
Despite already knowing the concepts you’re talking about, it was really informative to see them arise from one another in such a natural manner! Really well done!
Omg I loved the cats in the presentation hahahaha gave me a good chuckle hahahah
good!! my children (toast and salad) helped out a lot
That visualization with the circle casting shadows on the xy/xz/yz planes alone finally made quaternions make sense to me. Thanks for another great lesson!
You are such a good technical presenter! You empathize with a person entering the lesson so well.😊
You just single-handedly explained quaternions and why 3D rotation happens in 4D better than any instructor I've ever had. Hat's off to you.
My understanding of complex numbers was always very tied to how you can do 2D geometric algebra with it, so when learning about quaternions I was looking for something similar but never managed to wrap my head around it. After watching this I feel like I'm finally starting to understand it.
omg hi pie!!!!!
So awesome… my gut response was “yea, you can using the geometric product” and then you deliver. Stuff like this will help usher geometric algebra into the mainstream. Well delivered!
I always feel blessed when I see a new upload from you, thank you so much for your work ❤
Literally this afternoon I added your channel to my MUST WATCH Notion page for math in game development. Thank you for all the hard work and sharing, you rock!
I am a student in calc 3 and this has fully blown my mind. I’ve really loved your previous videos about Bézier curves and splines but this felt like a whole other level. Now I gotta go learn a lot more about some of the other concept like what the hell a quaternion is. Thanks
Great talk!
A few months back I tried inventing my own 3D vector rotation trying to avoid quaternions and just ended up back at re-inventing quaternions. Never heard of bivectors before but it all makes a lot more sense now. It's funny how no matter how much you try to go around a solution you just end up back in the same solution in math.
¿Did you try Euler angles and rotation matrices?
@@andresmartinezramos7513 Euler angles have gimbal lock, and you need to convert them from angles to something you can multiply (eg: a quaternion or a matrix) so they're just an unnecessary extra step.
Matrices on the other hand have too many params.
I was looking for a way where I could have a 3D complex number that i could just multiply by a vector.
The problem is:
i*i = -1
j*j=-1
k*k=-1
but i*j=? j*i=? k*i=? etc
I tested tonnes of possible combinations to those solutions and there's no way of defining those multiplications in a way where I wouldn't need a 4th component.
@@spliter88 reasonable
Thank you for everything you're doing! Really appreciate those who love to learn and especially those who love to teach. I'm constantly recalling things from your videos to help me in my daily work.
Your explanations are so good. I was bad at math in school, mostly because of bad teachers and being mistreated, but you inspire me to learn all of it. I also always was obsessed with finding out "why the formulas/axioms/rules are the way they are", i was never content with black box treatment, and im determined to find these things out thanks to you. Thank you
Brilliant! Partway through I guessed what you were building up to and I got really excited as you unfolded the derivation from the axioms. I wish I’d learned this algebra early on instead of quaternions.
At first I thought this would be a basic presentation on the ideas of a first-year linear algebra course. I was thoroughly impressed by the end. Great presentation!
Honestly, the contents of this lecture should be in every linear algebra textbook.
This is really well articulated.
What a coincidence that there was a very similar talk at Strange Loop Conference by Jack Rusher just two weeks ago! Very different audience though. Happy to see Geometric Algebra being introduced to more and more people!
I don't usually comment but I love your videos so much, I'm so excited to watch this as soon as I can
thank you! and, take your time, no rush c:
23:28 when the formula showed up, I was like "hmmm... I think they invented the 'quaternions' or something to handle this thing." Then you explained the bigger picture beyond quaternions for me. Wedge products, differential n-forms and General Relativity are starting to make sense now. Thank you! You are a great teacher. (BTW don't over-stress yourself. Health is important!)
You inspired me to create a playlist of "beautiful math" I've been collecting since the Beauty of Bezier Curves video. I'm delighted to have another OG video for that collection.
This was a mind-blowing talk! Thank you for taking the time to put this together and explaining it as well as you did! Fantastic!
Glad to see you got permission to post the video, I loved this talk (especially the cats) and it was nice to meet you :)
glad you liked it!
😂 Even tho I know this stuff as the back of my hand, I thoroughly enjoyed the presentation. The way you presented the material is so beyond me and Neat. The animations were very well thought out. 🐻
I was at your bézier curve talk in Boston 2015. It was one of the best talks at the show. Glad to see you are still at it. Keep up the good work.
ah gosh that was such a long time ago! happy you liked it though
If you don’t mind me asking, I was wondering where you find out about talks like this happening in your area. I’d love to go to a few presentations like this sometime during my journey in academia, I just don’t know where to look! Thanks!
Awesome talk Freya! mathematical insight and a true understanding of vector space you are gifting to a us, the apparent obfuscation is baffling. Thankyou for treating vectors this way and cant wait to hear about your next project. Love from Australia!😍🥰🤩
I haven't even started to watch this yet, but I have been hoping there would be a video of your talk. Looove your twitter even though I don't understand all the math. I'll just keep banging my head into the wall, perhaps half of it sticks ;) Skål från Köping, Sweden!
Insanely good presentation! I have been struggling for a while now on and off attempting to understand quaternions on a deeper level, and this talk finally solidified that understanding for me. Keep up the great work! 👍👍
I'm glad!
This is absolutely awesome, like always. Your work just keeps on inspiring me to become a technical artist every time!
I want to thank you for all the great videos you've worked on and shared freely for people like me to learn from.
Freya, as someone who also didn't learn about geometric algebra until after my formal education (at which point I was quite excited to discover it), and also having been delighted by your earlier work (especially on splines), I was very pleased indeed to see this lecture by you on this subject. Thank you. 🙂
Extremely useful quality educational content from the get-go !
First INTUITIVE and fundamental explanation of quaternions I have gotten, and I wasn't even expecting that. Love it!
Your work is amazing as always. You are an inspiration. Thank you
i remember the first time i learned the basic quaternion operations and was so shocked that dot and cross products just fall out of quaternion "vector" multiplication. this talk gave that same feeling but this system feels much more complete so thanks!
I knew this talk was gonna be good. But it exceeded my expectations. Thank you Freya. Amazing work.
glad you liked it!
Your the best! these math topics in the context for gamedev are just great additions to the community! you explain them so well and aesthetically pleasing.
Your videos always go deeper and in different directions than I'm expecting. Always worth watching. Thanks for making this!
That's an example of the real beauty of mathematics, you just connect some basic things like i²=-1 and v²=||v||², and voila! Everything turns simple and elegant as long as you understand the core logic behind seemingly complex phenomena. Also I found it cute how your cats take part in the presentation 😸
I have been working my way through "Linear and Geometric Algebra" by Alan Mcdonald. Absolutely fantastic book for anyone trying to learn GA in more depth.
This is such a great talk, fantastic presentation & visualisation! Your work is amazing
I KNEW this was going to be about Clifford algebras as soon as I saw you just... put the two vectors next to each other without any symbol between them, thanks to that having been a bit of a hyperfixation for me for a month or so LOL
The talk was absolutely wonderful, I think that will be the new introduction to the topic I'd send people if they're curious - I really like the exploratory motivation being shown for everything + it seems very easy to follow ^^
As a theoretical physicist, I already did some research on geometric algebra or clifford algebra respectively because it is quite important when it comes to spinors but I really like this presentation. It has a good pacing. Thanks.
Very nice and efficient! For maybe 20 years I have been telling students to dig into Clifford Algrebras and Geometric Calculus. I suspect very few did, which is a shame. It is a most wonderful and striking development from first principles. The man who woke everyone up to Clifford's algebra (And it wasn't easy to get their attention!), Alan Macdonaldl, has a couple books on the subject. "Linear and Geometric Algebra" I don't think requires more than some linear algebra and sines and cosines. It is short and well illustrated and very clear. He also wrote "Vector and Geometric Calculus". Equally good. These were the primary sources for a long time. Today there are loads of books that cover the same material and maybe more recent applications. I know that now there is a formulation of particle physics that is simplified greatly by the geometric calculus.
I love this Intuitive approach to quaternions! Thank you, you're the best
28:13 the confused cat was a nice touch.😺
Hello Freya!! You made my evening!
An inspiring talk!
Let's not forget that in some contexts (e.g., shaders) the meaning of a vector can shift from a "geometric vector" to a generic "array of scalars". By that definition, a vector multiplication is a handy way to multiply more scalars at once. Or, you can mix both and multiply a geometric vector with a scalar vector. One use case of that is an anisotropic scaling of a 3D vector.
Thank you so much for teaching us!
wow this is so incredible. i just took a lot of these things as given, and treated them as a black box. but you explained and visualised all of these concepts and showed how they were all linked together. i never thought I'd get quaternions but now I think i might be able to
Hmm! Very cool! My only question was about whether or not there was some handwavy stuff around 31:23 that xy algebraically works like two "placeholders" ... and then I realized I accepted that this "just works"'idea with i in i^2=-1. More axiomatic magic! Great talk! Thank you so much for sharing!
Thanks for the information, Freya. You have an incredible mindset as far as we are all aware.
Great talk! Always a pleasure to watch your content. Somehow, you always manage to make all this purely technical topics so interesting and fun to watch.
That was exactly the math level I have and the math details I wanted to know.
Thank you a lot.
I think that you have just proven why Geometric Algebra can and ought to be taught in High School ! Also this algebra with a minor variation is a superb tool for calculation in Minkowski Spacetime !!! A most excellent presentation indeed :-)
It's a bit complex of tought rigorously and there is already a lot of math tought in high school (at least in my country) and filling up the curriculum even more probably doesn't make mich sense. I guess this are the reasons why it isn't tought in high schools.
I remember Wikipedia binging, and I found Geometric Algebra/Clifford Algebra. But Wikipedia not necessarily the best place for a hands-on approach to learning the math, I never really wrapped my head around it. But this puts a lot of it into context, thank you, splinecaster Freya!
I found Alan MacDonald's texts, Linear and Geometric Algebra, and Vector and Geometric Calculus, to be good introductions. They were writen as undergraduate texts, and a self-motivated student could get through them with a little help from Discord.
I‘ve been using quats for some years now, still wasn‘t quite sure how it all worked together. Although I love maths (cryptography is so much fun), i‘ve never put in the effort in graphics. It just worked.
Your talk was, again, so much fun and I‘ve had quite a few audible chuckles. I enjoy what you‘re doing sooooo much ♥
You've returned ❤ as always very cool to see a video by you.
Freya, this video is absolutely magical! This is the most intuitive and RATHER SLY introduction to geometric algebra I've ever seen! As someone with a cursory understanding of it I only began to realize (in total shock) where you were heading right before you expanded out the terms for ab in the middle of the video (I did them myself and promptly huh'd at the similarity between this expansion and good descriptions of the geometric product). Sudgymacloe's geometric product video's gave me a good geometric understanding of geometric algebra itself, but this video definitely compensates for how his series has confused many watchers (despite being what I consider to be the best intro to geoalg to date) who could not understand where exactly geoalg's systems _arise_ from. This video combined with sudgymacloe's amazing series on geoalg may literally constitute the best possible intro to geoalg for the public/laymen who are interested in math but neither have the time or the money to pursue a higher education, especially far enough into the niche folds (heh) of math that you begin to consider "esoteric" theories of vector multiplication like the geometric algebra. You are, truly, a master of pacing and education (but any regulars of your channel will already know that ;P). suno pona, suno pona!
thank you! I went through a long period of frustration with how most sources introduce GA with unexplained grounding of exactly how the dot product and the wedge product works, it wasn't until I found this approach from clifford algebras that it all clicked to me, because we only need one very simple axiom, and the rest follows. now, the dot product and the wedge product are downstream from the axiom rather than axiomatic themselves, which I find much more satisfying
Love your content. The talk was slowly edging towards geometric algebra, so I was pleasently suprised you covered it.
What is your thought on geo. algebra being adopted in 3D software, at least under the hood? It scales to many dimensions a lot easier and it is good for switching between positions, planes and volumes.
I don't know yet as I've mostly been looking into VGA, it seems to me that a lot of the big questions are around whether PGA can/should replace effectively all of our matrix operations and the usual intersection tests/joins, etc. I'm not sure yet! it really depends on how it performs, not just how the math works out. And in most cases, having support for 4D+ games is generally not necessary, and so you'd have to make sure it doesn't bog down the library with unnecessary abstractions
@@acegikmo it feels like the extra data that it carries may be too high a cost vs the performance gained. unless you use it to simplify collision or shader calculation with it
This is amazing. 20+ years in the games industry and this has finally filled in a few gaps :)
Ah I love seeing the back of my head in the audience lol. This was such a fun talk to be at and I'm glad it is posted online now so I can share it with other people :D
The exterior product (also known as the wedge product) is not the same as the cross product, they are really only different when your talking about more than 3 dimensions though so the difference isn’t super relevant to game design though. (Edit: this first point is addressed around 37:20 in the chapter labeled wedge product). Also there exists an outer product (also known as the tensor product) which could be relevant but probably isn’t in most cases. Also at around 34:00 there is a diagram of axb, the a and b vectors should be swapped as what is shown is bxa. The correct orientation is shown around 20 seconds later when talking about the xy bivector but is incorrect again when showing the xyz axis and their bivectors, this could be fixed by swapping the x and z vectors and the yz and xy bivectors. This is also why one of the areas is of opposite sign at 35:34 as we are seeing the vector -x and the bivector yx.
Finally someone paying attention to detail. Not to mention the square-snail-cats lol
Awesome Video!
Freya, do you have any recommendation of some easy/ready to use, software for doing a 3D animation of the movement of a point in 3 dimension (just showing the trajectory of a particle in space by a forcefield).
hmm, not sure, I would recommend p5js for 2D things, but for 3D I don't really know what's good out there. I just use game engines since, well, that's my background!
Thank you so much!!!
Very beautiful videos btw!!!
@@luisaim27 thank you!
This is my first video by you. Ive been deepdiving into maths the same way you have. Its glorious
Very few manage to make a talk about math comprehensible, informative and funny. Really good stuff!
I absolutely love seeing beautifully constructed presentations on _complex_ topics such as this. Especially because it's actually quite simple once you connect all the dots (and tie it all together). Though I already understand these topics, I often find it difficult explaining it so well to others. Now I can just send them this. I swear, math really gets a bad rep because of poor presentations - and I hope this will all change soon.
P.S. One thing I was hoping to see was a mention on how you got to the divine truth "v^2=||v||^2" - I am referring to using the geometric interpretation of the dot product "ab=||a|| ||b|| cos(alpha)" and deriving that axiom from it. I mean no offence to the great oracle Salad for his wisdom, just some extra dots I would've wanted to see to connect the entire topic better (at least for me)
Thanks for posting this talk. I had several expectations on where it was going to go, but then it went in a different direction. I was pleasantly surprised!
As someone with a background in mathematics, with a minor in CS, I can appreciate your words about "abstract nonsense". I've gotten that remark far more often than I'd like to admit.😓
When you pulled out the proper name for the pointwise product as the Hadamard product, I smiled. I'm far more familiar with it as the Schur product of matrices.
Oddly enough, I've run across the cross product of vectors as a Lie bracket, rather than an exterior product. I'm interested now in the difference between the two. 🤔
You did lose me briefly when you said that i^2=-1 axiomatically. This isn't wrong, per se, though when I was a teacher, I found that students weren't kind to "true because I said so" responses. Also, philosophically, it's not really a satisfying response either. Instead, I looked for a way either to construct or to motivate why an axiom should be taken. Indeed, your use of the Euclidean norm pointed in that direction. Though, for a 50+ minute talk, you might not have time to take that journey, given everything else. 😅
When you pulled out the Euclidean norm, I thought you'd go to the conjugate operation for the complex numbers and quaterions, but I did not expect bivectors. I haven't studied bivectors, though Clifford algebras did appear in my recent review of quadratic forms and inner products for a research project. 🤔
I am curious. What are your thoughts on the octonions, given your interest in the quaterions? I have a research colleague who has been pitching the octonions to me over the last year.
Also, what are your thoughts on the convolution product? While this is usually used on an infinite dimensional space, it can be modified to work over a finite dimensional space by using a group or modular arithmetic.
I'll look into Clifford algebras and exterior algebras! 👍
Thanks again for your presentation and insight! 🙌
I haven't looked into convolutions or octonions much! mostly because they haven't showed up in the things I've researched. I did stumble across dual quaternions though, but I've yet to dig deeper!
but yeah, students hate "true because I said so" statements, but I think it's better than "I don't want to tell you because it's too complicated", especially because it's true. The imaginary unit doesn't have any other definition, it doesn't "come from" anything really, it's an axiom we made up and started exploring, just like many other concepts in math!
@@acegikmo I respectfully disagree that an axiom doesn't come from anything. Please allow me to elaborate.
Philosophically, I've been in a continual process of recontextualizing my understanding. In graduate school specifically, I really dove into abstract definitions and axioms, formulating much of my mathematics around those. However, in the last few years, particularly as I've begun to study the logical foundations, I have come to accept that each axiom has a motivation from some observed phenomenon. For example, the axioms of set theory are used to model how collections of objects work in reality, and the axioms of a total order were chosen to model how rankings work in practice.
You are correct that we formulated these axioms abstractly, but there is a concrete reason each was chosen. And, that reasoning itself could be interesting to explore.
I rather liked your constructive building of the numerical systems, and each one has a motivation and a generalization, including the transition from the real numbers to the complex numbers. Below I offer my perspective on each transition, a motivation for each, and a generalization for how it manifests elsewhere. I apologize for the length. ^^;
1) The natural numbers can be constructed from set theory, modeling how sets can be enumerated. Viewed categorically, addition is the binary coproduct, multiplication is the binary product, and exponentiation is the exponential bracket. By their construction, the natural numbers have a total order, which can be translated into set containment. Equipped with these operations, the natural numbers is an ordered semiring.
The category of sets is the motivation for more abstract notions in topos theory. Personally, I would argue that topos theory is a natural setting for combinatorial questions for this reason.
2) As you correctly pointed out, the integers arise from the natural numbers by appending the additive inverses of each non-identity element. Indeed, constructing subtraction is the goal here, resulting in an ordered ring.
This process is a specific case of the Grothendieck group construction, which is heavily used in K-theory and other realms of algebra. In general, the Grothendieck group takes a commutative ordered monoid or semigroup and returns an ordered abelian group.
3) As you correctly pointed out, the rational numbers arise from the integers by appending the multiplicative inverses of all non-zero elements and reducing the resulting fractions by "equivalence". "Equivalence" here is the relation that r/s=n/d if and only if rd=sn. Indeed, constructing division is the goal.
This construction generalizes to the localization of a commutative ring, or the "field of fractions" of an integral domain, which appears in algebraic geometry.
4) As you correctly pointed out, the real numbers arise from the rational numbers, but this transition is a very subtle process. The rational numbers are both an ordered field and a normed field, which induces both a norm/metric topology and an order topology. Thankfully, the two coincide, but yield both a (Cauchy)-incomplete metric space and an (order)-incomplete lattice. That is, there are sequences of rational numbers that seem convergent as limits, but do not converge to a rational number. Specifically, key suprema and infima fail to exist. Indeed, the construction of limits is the goal.
Limits are desirable because certain operations, such as the square root, can be formulated as inverses to (topologically) continuous operations, such as the squaring function. Tools like the Intermediate Value Theorem and Inverse Function Theorem allow for such continuous functions to be inverted. Thus, for example, one can construct the real-valued square root by inverting the continuous squaring function through these tools. Limits are also heavily used in approximation of such functions (e.g. splines), and in formulating more advanced operations like the derivative and integral, which are foundational tools for the other sciences, particularly physics.
The construction of the reals can be done many different ways; my favorite uses Dedekind cuts. The result is a new field that is both a complete metric space and a (conditionally) complete lattice.
Indeed, the real numbers are provably the only complete totally ordered field, up to isomorphism. They hold a central role in mathematical analysis and topology for this reason.
There are two canonical generalizations of the transition from rational numbers to real numbers. One is the metric completion of a metric space, which appears in topology and analysis, done specifically to ensure that limits exist. This gives rise to Banach spaces, Hilbert spaces, and many more structures. The other is the Dedekind-MacNeille completion of a partially ordered set, which appears in order theory, done to ensure that suprema and infima exist.
5) While the real numbers are quite powerful, they lack algebraic completeness. That is, there are polynomial equations, such as x^2+1=0, with no real number solutions. Gauss saw many of these appearing in his differential equations for electricity and magnetism, among other places, and he imagined a root that he would notate as "i". Indeed, constructing roots of polynomials is the goal.
This process of manifesting and appending roots of polynomials is known as a field extension, which appears in many other places. In particular, algebraic coding theory uses field extensions heavily on finite fields, and is the basis for error-correcting algebraic codes in computer science. This is also the mathematics behind linear-feedback shift registers, which are used for encryption. The field extension is always an algebra over the parent field, which is why you have vector-like behavior equipped with multiplication, though the multiplication becomes increasingly elaborate as the polynomials become more complicated.
As an aside, I recently reviewed finite field extensions since I am transitioning careers from pure academics to a more applied workplace. :)
The relation "i^2=-1" is an artifact of the field extension of the real numbers using the irreducible polynomial "x^2+1". The explicit construction is to build a polynomial ring over the real numbers, identify the ideal generated by the polynomial x^2+1, and take the quotient ring of the polynomials by that ideal. Since x^2+1 is irreducible, and the polynomial ring is commutative, the result is a field again. The element "i" is precisely defined as the image of the indeterminate "x" from the polynomial ring through the quotient map. The fact that the complex numbers are algebraically complete, i.e. the Fundamental Theorem of Algebra, is nontrivial to prove, but can be approached from either mathematical analysis or abstract algebra.
Sadly, there is a price to be paid.
As the real numbers are the unique complete totally ordered field, the complex numbers are provably not able to be ordered in a manner coherent with their algebraic operations. This is why one often hops back and forth between the real numbers and the complex numbers. The reals have all bounded suprema and infima, but the complexes have all roots of nonconstant polynomials. Both are necessary in many applications, such as the differential equations needed for electricity and magnetism. Indeed, electrical engineering was where complex numbers first garnered serious attention, though electrical engineers use "j" for the imaginary unit rather than "i". It's one way to tell an engineer from a mathematician. ;)
I think you might be interested in finite field extensions. They are quite interesting and appear very often in computer science, particularly when manipulating memory or representing numbers. :)
5a) The quaterions and octonions are notably not commutative, so they are definitionally not fields. Hence, they cannot be field extensions of the the real numbers or the complex numbers. Moreover, while the quaterions are associative, the octonions are not even associative! Given your interest in them, I think I might spend some time to understand them better. :)
Thanks to you, I have an interest in the notions of bivectors, exterior algebras, and Clifford algebras. I'll look into them! \o/
Thanks again for your video and engaging conversation!
@@evilpii my point was just that axioms don't have a "proof" or a "reason for being true", because axioms are true by definition. That being said, there's of course numerous *reasons* as for why we choose to use any given axiom! usually around its usefulness and its consequences
This was really interesting and educational. I’ve always been intrigued about how things actually work and getting to the bottom, and you just made it a lot more clearer. As you said, teachers only teach cross products or dot products and that’s only a part of it but now you explained it as a hole and that makes a lot more sense. I’ll really appreciate if you make more videos about this
Freya, thank you for the mother of all lectures! This opens my eyes to this enormous subject. You've been spot on in making my curiosity and creative juices flowing !!
Z is for the german word "Zahl" which just means number.
you are such an inspiration for Women in STEM thank you for all your work 💜
Nothing personal, really, and I was about to leave this video quietly, but I can’t leave this comment unnoticed, because there are enough actual women to be such an inspiration. There’s only so much propaganda that can be shoved down our throats before we reply. There’s currently a war on women, with some men making a mockery of them, their looks and behaviors, because social power is now on the side of the underprivileged, so I’m siding with women. Things have gone too far. Besides I wouldn’t take scientific or logical lessons from anyone who would argue that a woman can have XY chromosomes. If any theoretical construct can be true, then math is doomed.
@@Edouard16 you should have left the video quietly
@@dominiquenascimento4647You should not have made this video political.
@@Edouard16This video isn't political, you're just being a weirdo. Please go away if you have nothing respectful to say!
@@Edouard16calling a woman a woman is political now 🙄
Another golden video from Freya. Numbers slides are no less than astonishing! wow again!
This is the single best explanation I've seen of how all of these vector products are derived. Bravo!
Wait didn't you say you were gonna wear fox ears? :(
I CHICKENED OUT i even brought my cat ears to the event but I didn't wear them for the talk ;-; im sorry. anyway I have just ordered new ears and I'm gonna be more brave with those I think
What a mathematical morning to start with: both Vi Hart and Freya Holmer have posted a video, nice! :)
I heard about bivectors before, but never learned what they represented. The visualisation with projection of a directed area is great and allowed to understand this. Thank you!
It's been a while since I haven't gotten this little "click" in my head when being explained something. This is such a brilliant talk.
I could listen Freya talking about math for like 15hours /day and I'll be happy! :D
Ive watched so many videos on quaternions and never felt close to understanding them.
You are brilliant, both in terms of your expertise but also your ability to teach so much better than most professors.
Please make more content!
Really amazing talk, this was a lot easier to understand than I was expecting and I really feel like I have a much better intuitive understanding of vectors and quaternions now. You are so talented and amazing for being able to communicate such abstract concepts in a way that only requires small logical steps in understanding from one phase to another! Thank you so much for making this!
Please don't ever stop i cant express how much i value these videos A