3Blue1Brown - 2021-12-20
A tale of two problem solvers. Numberphile video on Bertrand's paradox: https://youtu.be/mZBwsm6B280 Help fund future projects: https://www.patreon.com/3blue1brown Special thanks to these supporters: https://3b1b.co/lessons/newtons-fractal#thanks An equally valuable form of support is to simply share the videos. There's a small error at 19:30, I say "divide the total by 1/2", but of course meant to say "multiply..." Curious why a sphere's surface area is exactly four times its shadow? https://youtu.be/GNcFjFmqEc8 If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts: Part 1: https://youtu.be/rAHcZGjKVvg Part 2: https://youtu.be/cEhLNS5AHss I first heard this puzzle in a problem-solving seminar at Stanford, but the general result about all convex solids was originally proved by Cauchy. Mémoire sur la rectification des courbes et la quadrature des surfaces courbes par M. Augustin Cauchy https://ia600208.us.archive.org/27/items/bub_gb_EomNI7m8__UC/bub_gb_EomNI7m8__UC.pdf The artwork in this video was done by Kurt Bruns ------------------- Timestamps 0:00 - The players 5:22 - How to start 9:12 - Alice's initial thoughts 13:37 - Piecing together the cube 22:11 - Bob's conclusion 29:58 - Alice's conclusion 34:09 - Which is better? 38:59 - Homework ------------------ These animations are largely made using a custom python library, manim. See the FAQ comments here: https://www.3blue1brown.com/faq#manim https://github.com/3b1b/manim https://github.com/ManimCommunity/manim/ You can find code for specific videos and projects here: https://github.com/3b1b/videos/ Music by Vincent Rubinetti. https://www.vincentrubinetti.com/ Download the music on Bandcamp: https://vincerubinetti.bandcamp.com/album/the-music-of-3blue1brown Stream the music on Spotify: https://open.spotify.com/album/1dVyjwS8FBqXhRunaG5W5u ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted on new videos, subscribe: http://3b1b.co/subscribe Various social media stuffs: Website: https://www.3blue1brown.com Twitter: https://twitter.com/3blue1brown Reddit: https://www.reddit.com/r/3blue1brown Instagram: https://www.instagram.com/3blue1brown_animations/ Patreon: https://patreon.com/3blue1brown Facebook: https://www.facebook.com/3blue1brown
28:30 "And we can simplify that 2π/4π to simply be 1/2"
Me: Finally something that I could've done myself.
I FELT SAME ASÖALKSMAKAND
Me: Pauses the video and scribbles on a piece of paper for five minutes. " Yes that checks out".
@John Doewhyd it take 5 minutes? had to think about all possable last digits of pi?
@Oli Tickayseems like lol
Anyone else incredibly impressed just by the process of drawing Bob and Alice?
not just that, but in every video every animation is so well made that understanding the math behind it gets way easier
Another note: Alice's result is more generalizable than Bob's, while Bob's method is more generalizable than Alice's. (You can see this by thinking about the harder problem of a nearby light, where Bob's method keeps working while Alice's doesn't!)
This is one reason why combining the two approaches is so valuable. You can start with something you know will work but may not unlock a great mystery, and then look for patterns that clue you in to a wider story.
this really is the essence of what my experience felt in olympiad mathematics, understanding the two methods is crucial
I’ve always been a Bob kinda guy and my inability to find Alice like patterns is why I didn’t pursue a PHD. I wish I could learn though
That's a really great way of putting it
Oh wow. This is a well put way of describing this. I'm gonna steal that thought pattern. The generalizability of the method vs the result.
dont try to be a genius why so serious
😡 you
To me this seems like the difference between what we in software call "Top down" versus "Bottom up" problem solving. Bob takes the "bottom up" approach of looking at the specific problem he's attacking, going through the motions of solving it, and through that, he might stumble into some generality that he can later come back to. Alice on the other hand starts from the top. She notices that if she manipulates and connects the abstract pieces of information to finally arrive that the simplest form of the problem, which she then solves.
One of my teachers had a nice saying about it: "Always solve the problem top down, except the first time", echoing the conclusion hit here. Top down problem solving is fast and awesome, but it's really difficult (if not impossible) to solve real problems like that. It often while working through the bottom up tedium that we realize what top down abstractions we can manipulate.
It's great to see comments that are valuable and great additions to the video :)
does that mean that Alice is actually a genius?
@A guy on the internet no exactly. as he said , it's incredibly hard to solve real life problems using top down method, the problem here was unrealistic where in real life example you would need do things down up because of sheer amount of variables.
edit: spelling
Kind of like using the pythagoras(top down) vs creating it (bottom up)?
@Arlenboi yeah so precisely because its hard but she still does it she's a genius? whether she can do it everywhere is irrelevant because even in the cases that can be done by this method others aren't able to
Your writing is so, so insanely good. It is a RARITY to find an educator so capable and devoted to the task of creating genuine understanding. You demonstrate an ability not just to expound upon every detail, but to minimize, order, and portion complexity in a way that can actually be digested. You make your motivations very clear, and you execute with a self-awareness that shows just how much you understand your audience. Not to mention the relevance and quality of your ANIMATIONS.
There are many famous video demonstrations that have gone down in history - in physics classrooms, on YouTube - as being particularly eye-opening, particularly effective at conveying a topic in isolation. Somehow, you manage to achieve this quality in every video. I've only chosen to write this here because this is your most recent!
well said! couldnt agree more
Another thing to note about the two philosophies is that Alice's way is beautiful but it requires you to be clever or lucky to connect disparate ideas and exploit the general connection. Bob explores the space with calculation and uses the connections that he identifies. I think there is not a separate Alice and Bob but instead a bob thinker picks away at a problem until he is able to build up to a generalization that equals Alice's. Bob's next question should be what about other shapes? Followed by what about all shapes? Eventually he would come to the same conclusion and probably prove the problem in the same way as Alice.
One of my frustrations with learning (highschool/undergrad level) math was that we only see Alice's brilliant proofs and sometimes it appears as a magnificent logical leap that I would have no hope making if I was in their position.
I agree!
Most of my math teachers have made me feel that if i dont solve problems like alice does im bad at math
Stepping back we can see that that was the (actual) generalization this video meant to derive 😉
Yes I probably would be Bob for solving this for cube or tetrahedron and suspect the fact the answer is a quarter of its surface area then become Alice.
I would suppose that the brilliant, refined mathematical tools we have exist only because years of experimentation and work - sometimes even happy little accidents - were put into creating them. Do not feel bad that an obvious, to the instructor who has spent years reiterating the same lessons, proof doesn't come to you naturally. You only learned about it five minutes ago.
This. I have almost always taken this exact approach. Typically I will go from calculation to insight to verification of the insight with further calculations, at which point I'll either follow the insight through or perform more calculations to connect more deeply with the insight. Never do I ever utilize only one, because even if I it's simply too easy to miss something if following only one method.
Being a very Bob-minded person myself: to me the most dangerous thing about Alice's approach is how easy it is to miss hidden assumptions.
I am sure Alice in this story was fully aware of all the assumptions she made along the way, but someone with less expertise trying to follow her method might not realise it.
Conversely Bob was forced to explicitly address the problem of defining uniform distribution of rotations and from his calculations it is evident that for specific shapes the answer absolutely would depend on the probability distribution.
Just use Haar measure.
Indeed, something even more subtle is that measuring on limits has its own dangers. E.g. the limit of stair cases (when stairs got smaller and smaller) is a triangle. But the length of the stair cases will always be total height + total length while the length of the triangle line will be sqrt((total height)^2 + (total width)^2). Of course, the limit of stair cases is topological something very different (e.g. nowhere differentiable, also never convex) vs a triangle (everywhere differentiable, of course convex etc). - I guess, Alice approach here works because she is assuming convexity (in the approximation of a sphere), but tbh, I couldn't really argument it here beside of a plausibilization and the world of math is full of paradox (like you can split a sphere into two spheres that have both the same volumina) if you miss a subtle point here when applying infinite limits. So, to be sure that Alice solution is right, you'll probably need already be a master in math (and even those have "failed" there). And you still need to do Bob's approach in most cases anywhere to detect "false" results (e.g. physical impossible solutions) or to make a plausibilitization of the results (these magic tricks can always be doubted by non maths, just as there are so many paradoxies, especially in combination with probabilities), while if you can calculate down in a numerical approximation, it's much more trustworthy for non mathematicians and also "easy" to double check via experiments (classic example: Buffons needle can be verified by by every 10year old).
Exactly. It would be easy for someone following Alice's reasoning to try to apply it to, say, a torus, without reasoning that it violates an assumption made along the way.
When he started turning the sphere into a band I was preparing myself emotionally for him to turn the sphere inside out without pinching any points
A man of culture, I see.
Well you see... the outside and the inside of a sphere both have a turning number of one...
lmfaooo seems like it's haunting us all then
I can't believe I see the "outside in" community here.
What I found from studying physics for over 4 years now, is that often times (as with this problem) the Bob approach is what happens first. At least for me I often do the hardcore calculation first for something, because I have difficulties of finding these "nice" solutions, without having spent time on this problem already. It happened a few times myself, that after doing the hardcore calculation, I found ways so simplify it further and further until it became a very pretty "Alice-like" solution. However I couldn't have found the Alice solution without being Bob first.
So pretty much exactly what you say at the end. I hadn't finished the video yet :D
Ditto. Physics guy here, too. Didn't get into much fancy math until a bit later and, when I did, was reminded of some of the tedious integrals from homework problems that ended up with nearly all of the terms canceling one another out, leaving something along the lines of a constant multiplied by an integral from zero to one/pi/2pi/etc with a constant integrand. Wish I had had the benefit of content like this back in those days... seems obvious (in hindsight and w/ Grant's awesome material) that calculations which eat themselves away into almost nothing are good signs of a more abstract method of reasoning about a problem.
Grant and his team and supporters are a national treasure!
I think it only makes sense. Humans are pattern-matchers, not SAT solvers. It is much easier for us to come to conclusions by generalizing over discrete data, and only afterwards finding the justification.
@NXTangl pattern is really true on note of vsauce face recognition
i’m not a physics major in the slightest, but what i did. was take the area of the shadow where it’s the smallest (1.00) and where it’s the largest (1.73) and took the mean of those two. i haven’t finished the video yet someone tell me if i’m either a genius or really really stupid
All of Grant’s videos are good, but this one really stands out to me. Absolutely incredible work.
There were about five times throughout the video where I had a question/objection, or simply had to pause and justify things in my own head, and make sure I was really on board. Without fail, as soon as I unpaused the video, Grant addressed exactly what I had been wondering, with exactly the best and most intuitive justification I had been able to come up with. The video followed my path of thought almost to an unnerving level of precision.
I have never before seen a video that could hold a candle to the layout quality here. The order in which topics were addressed was perfect, as was the level of detail, not to mention the beautifully constructed graphics. And I think that is a rather difficult task for this topic, because at any one point, there is more than one interesting question to be answered. The questions to be explored do not lead into one another single file, but branch out like a tree.
The comparison of the methods at the end was very good as well. A hearty congratulations to everyone who contributed to this video.
An important point: while Alice's steps make sense in this concrete world off 3D shapes, exchanging infinite sums and so on will absolutely get you in trouble when dealing with less regular scenarios. It should be noted that doing Alice's method properly requires quite a bit of delicate technicality and the use of various theorems about infinite sums and whatnot. It's easy to trick yourself with clever re-arrangement tricks when you are working with non-visualizable objects. The best place to see this is that it is genuinely subtle, as you say, where Alice assumes a probability distribution. That should be considered a serious weakness of her method! It is not good to do things in a way that you can be making implicit assumptions without it being obvious to you or your interlocutor, and can easily lead to false results.
Indeed. I do find it interesting that Alice never does assume a probability distribution. What she assumes is a property of the distribution (isotropy, roughly speaking), and to me the fact that her property leads to the same answer is a noteworthy result itself.
I did a PhD in pure maths. The main result in my thesis had a very pretty Alice-like proof. The way it eventually dawned on me was spending a couple of years doing Bob style calculations of specific examples.
Bob is the superior for school, Alice is the superior for the real world
@Tom Epsilon As a quantum chemist, Bob is how you do things, Alice is how you report it.
@Tom Epsilon I think it might be the other way around
Bob is a practitioner, Alice is a theoretician
Sounds a lot like the P = NP conjecture (which is almost certainly false). You use Bob (NP) to find the solution. You use Alice (P) to show that a proposed solution is a solution. Finding the solution in the first place is the hard part. Showing that it is a solution is much easier and fun.
Nice to see Alice and Bob doing something else besides fiddling with weird primes for once.
I don't always understand what's being said, but I do enjoy when a particularly astute blue pi gets angy.
I'm surprised nobody has mentioned this as far as I've seen, but one of the most impressive things about this is the fact that Alice found a solution which used almost entirely linear algebra, and Bob found a solution which used almost entirely calculus, and despite this both found the exact same solution to the problem.
Alice probability distribution definition comes in at 20:07 when she assumes the proportionality constant is the same for all faces, even though the are shifted by initial rotations relative to each other.
By doing that she is imposing a kind of symmetry over those rotations in the probability distribution.
Then, when she does the same for progressively larger amounts of faces, in order to approach the case of a sphere, she imposes increasingly stricter symmetry demands on the probability distribution.
On the limit, infinitely many restrictions leave only one distribution standing, that being the uniformly spherical one.
Not sure that I understand what you are saying, but it’s lovely. Dive deeper 🙂
I literally don't understand this. But dude youre smart. Can U dumb it down for me 🤣
It's earilier than that. Around 17:45, he talks about sampling rotations and this is where probability distribution really matters. The hidden assumption is that it's a uniform distribution over the space SO(3) mentioned earlier.
@Alexis Olson The OP is correct. Everything up to 20:07 has made no assumptions about the set of rotations being considered and would be true for any specified set of rotations. At 20:07 when c rather than c_j is introduced into the sum it requires that all the faces have the same average shadow coefficient under the chosen distribution over the set of rotations. That doesn't work for any arbitrary distribution over the rotations anymore, even for regular polyhedra since the faces start with different initial orientations. It only works for sets of rotations that respect the symmetries of the faces, for example the set of all rotations that exchange two equivalent faces, or (crucially) for SO(3). Oddly, any set of rotations would work again once you generalize to the sphere, but only a uniform distribution over SO(3) works for the infinite series of polyhedra that gets you to the sphere.
@Alexis Olson It's even earlier than that. At 12:20, the notation f(R1)*A already assumes that the coefficient f only depends on the rotation R1 and not the original orientation of the shape. This is not true; luckily later Alice is only using the proposition that the average of the f(Ri)'s is not dependent on the original orientation of the shape. This is only true if all the rotations Ri are distributed in SO(3) uniformly.
That last point is really important. When I was in college I learned about "dynamic programming" in three separate classes, and in all three I was only ever presented with clever solutions that had resulted from the use of DP and their common characteristics. It was never mentioned that there was an actual stable process to use in order to generate DP solutions, so I just had to memorize the methods that had been presented and adapt them to the problems presented on the tests. I didn't learn until a couple of years later that you can actually identify DP problems and take concrete steps to come up with what had previously seemed like arbitrary solutions.
The way it should be taught is that you come up with a recursive solution and then optimize it. A problem has a recursive solution when each nontrivial problem can be split into smaller problems.
@Martín Villagra i sometimes prefer doing dp forwards though
Then you take a control theory course and the dynamic programming in that has nothing to do with your prior understanding and reconciling the two seems pretty hard since the CS version is so inherently discrete.
Very cool, thank
Ok
I really appreciate this video's focus on contrasting different problem solving styles. I think it is important that we all be a bit reflective of our own biases and what we enjoy and what we find natural, particularly because some problems lean themselves more one way than the other. I know for myself I always thought of myself more as an "Alice", but over time I've actually come to really enjoy more computation-centric approaches.
love your vids! They really helped me on my calc 3 final.
Perhaps one uses the generality one is capable of.
As one advances and the problems get harder one inevitably takes a more Bob approach.
I also think of it from a real world problem solving situation. This is a situation I encountered. Say you went to the doctor and your insurance company denied your claim. You call your insurance company and they say it looks like there was a mistake made. You will need to send a letter appealing the claim and explaining the mistake and requesting the claim be reviewed again. I was taking the call for the insurance company and I saw that there was a simple solution to the issue. Writing a letter explaining the mistake would almost guarantee that the claim would be paid. However the person who went to the doctor did not see it as a simple solution, because if it was guaranteed that the claim would be paid if he wrote the letter then why did he have to write the letter. In the end there were two issues. The first was that the claim was denied, that issue was easy to fix. The second was that he had to write a letter, but that was not as easily solved. It involved policies from the insurance company, laws passed by congress, and issues of ethics avoiding potential for fraud. The deeper you went the more you realized that it went even deeper. In the end the insurance company had applied a policy that worked most of the time, but was sometimes inconvenient. In the end I told the guy that we could spend weeks debating about why things are the way they are, and I was quite enjoying the conversation, but in the end the thing we are really wanting to solve is his denied claim and we already have a simple solution for that.
Not necessarily, Michael Micek. Olympiad questions (such as those from IMO) are good examples of brutally hard questions that primarily require Alice's mindset
@Ivar Ängquist fair enough.
This video had everything, from being a engaging movie on who's gonna find the solution and to putting a smile on my face throughout the video. It felt like Alice had made no improvement after so long but every insight came together in an instant to get to the solution.
wow this is incredibly interesting. seriously sometimes a simple question can present the sheer complexity in even the simplest of actions, i love content like this because it allows me to imagine what it was like being asked "simple questions" thousands of years ago with minimal tools to solve them. what things initially go through your mind, and the way everyone breaks down a problem and views it differently. super awesome man , also animations clean af as always
I love how both of them had to resort to the same Spherical shape irrespective of their way of thinking the problem.
38:12 wow now i wonder what the "most" concave shape could be?
edit: actually, you could have nearly limitless by having a sphere-type thing with tunnels going into the inside to make a limitless amount of surface area inside that won't show up on shadows
Probably a volumetric fractal of some sort
My own problem solving style is definitely more along Bob's, and I've always wanted to be more like Alice. But the final part really resonated with me, wonderful video as always!
Can you shere some more channels Like this??
A video from 3B1B, what a wonderful Christmas present! And it's 40 minutes long!
Anyone can see the amazing effort put into each second of these videos, and there are 2405 of those here.
Thanks for yet another high quality production, Grant.
You inspire me to continue pursuing my studies in mathematics and physics, and I can't thank you enough!
Beautiful, isn't it?
Wow, thanks for such extremely kind words. Merry Christmas, and best of luck with your studies!
@3Blue1Brown the comments are well deserved. Merry Christmas!
@3Blue1Brown I'm but a lowly technician who loves physics, and I agree that youtube concentrates on the Alice approach far too much. I would much rather spend 20 hours with Leonard Susskind walking me trough the mathematics of general relativity than watch yet another 20 minute video of an apple on a curved trajectory or a time dilation analogy of a river flowing more slowly near its bank. I wish more university teachers would take a page from Stanford and start putting all of their courses online, for free. That is the kind of thing that will bring on the 2nd Renaissance.
@Michael Bishop i mean you only need one amazing course to be available for free not 10.. and btw idts Stanford has courses in other subjects available as well it was only for a special program Leonard Susskind was running to teach old people who are interested in learning physics
Your channel was one of, if not the deciding factor why i enrolled in in a math degree besides my computer science degree. On some days, when I absolutely loathe my degree program (usually before exams), I watch one of your videos and get immediately reminded why I chose to take more math classes even though I have no intention whatsoever to do anything different than compsci in my career. I really just do it because I find the subject endlessly interesting and just beautiful. In a few months I will start my math masters degree and again, not because I want to have any math-degree-career I really just do it for the fun of it as ridiculous as it might sound even to me (and especially to literally everyone I consider friends and family). Thank you so, so much for what you do and to wake my curiosity!
i felt pretty bad after watching this video cuz i was completely lost, when i know i wouldve understood it a couple of years ago before i had to drop out of university to work. but then a few days later at work, i went through what i could remember about it in my head and managed to get the entire thing, and finished the alice style proof on the back of a reciept - it really helped remind me why i need to try as hard as i can to get back to uni, maths is beautiful nd i need to retrain the alice part of my brain
When you're an engineer you use Bob's approach the first time you see a problem of a given type. If you see something similar, you remember the solution to the first instance, and try to see if it is adaptable to the present problem. Eventually you find a generalized solution. But you don't worry about generalized solutions until you have a valid reason for seek them (at least, not while you're on the clock). And having found the generalized solution, in every application you also take the time to see if the specifics create a special case that lends itself to an optimization.
Man, this is so beautiful. Being a PhD candidate in mathematics, I can fully testify how drilling truly helps you understand something. You can tell you know something, but you don't develop the internal gut feeling, the feeling of knowing the problem truly in your bones until you have fully drilled through the problem.
13:00 it occurs to me here that Alice has saved herself the work that some might accuse her of not having done if she later finds that this infinite sum/average is unfindable or doesn't converge.
Its a shadow! 🤷
@Dungeon Dogs But the sum itself isn't. You can, of course, relate it to integrals because of the 1/n and the limit you're taking, and you can very easily ask Bob to provide an easy upper bound you can use to show convergence, but that's not completely trivial.
I feel like Bob’s approach acts as a launch pad for Alice’s method. If you solve a few special cases, you can then look for patterns that then allow you to hunt down the elegant solution later. It feels rare for someone to see the elegant solution on first sight. It’s something found in hindsight after some special calculations are made to provide a sketch of what is probably true, though math doesn’t have to care if things are pretty.
That's a really nice way to put it. Begin by diving in with vigor to a few representative cases, and put on the Alice hat when you sit back to reflect.
Agreed. I think Bob would actually gain a lot of Alice's insights if the question were posed as "What is the average shadow of a sphere?" and forced themselves to do the integral calculus.
@Daniel King Isn't the average shadow of a sphere just a circle with the same radius? Since there's only one shadow for a sphere...
Exactly what I was thinking ... often it is not so easy to jump to the Alice mode, to get to that mode, it would need a Bob mode to have consumed and understood those special cases.
Lm
It’s interesting we didn’t run into any paradoxes from randomly choosing from a non countable set, like we did when trying to find the average distance between two points on a circle.
Best maths videos on youtube. Hands down. Thank you so much for making them.
I'm someone who has long been much more an Alice than a Bob, and as such I've lately been losing my mind dealing with the never-ending computations of an Analysis course (I just want my Point-Set Topology back lol).
This video came as a pretty important reminder of why I took that course to begin with: regardless of which method you prefer, it's important to have both approaches in your problem-solving toolkit (I often find it's helpful to use one or the other to check the work of the one you used first and convince yourself of the answer).
Thanks!! Your insightful videos are really fun to watch and though provoking. I look forward to them about as much as I looked forwards to new episodes of my favorite cartoons as a kid.
This message resonates with me. I definitely have a bias toward the jump in and calculate approach. I often find I’m more creative after grappling with a problem a bit in that initial computation phase. One of the most memorable pieces of advice from undergrad came from my DiffEq TA who said that when you come across a problem and don’t know where to start, attack it with “brute force ignorance” and that always stuck with me. I also attribute most of my math success in college to my high school years, where my teachers philosophy was to assign every problem on every page. I feel that those years of drilling calculations gave me a confidence in my abilities and experience that freed me up to approach new problems more openly and with a creative mindset.
I think a nice upside to "The Bob approach" that I'd like to emphaize, is that you can make forward progress on a problem without having any particular insight into the problem. Sometimes it's a lot easier to have insight into an answer once you already have a solution.
It is also useful to have Bob's approach on hand when looking for any logical holes in Alice's solution. At least that's how I usually discover & fix mistakes in my "slick" reasoning attempts.
That's true. Often you try something straightforward and after you finish all the work you get a nice answer. That's often indicative that there's a different way to think about the problem. Though sometimes the Bob approach actually doesn't work. Trying the most obvious or "just do it" evaluation of a problem or brute force sometimes gets you stuck in a world of computations that you can't actually compute well. Then you'd try to make more insights and think about how to do the problem a different way that requires more observations and understanding.
@x0cx10 I have the same opinion
Ah yes, can the computer determine there is a solution without determining the solution.
@x0cx10 It's fascinating to me that some general concepts can be argued for/against throughout the history as a lot of comments in this section (like yours) are actually talking about Occam's razor.
I actually don't have anything to add to the discussion just found it interesting :)
The animations are terrific.
The goal of math is to bring in new ideas :)
The illustration of Bob and Alice at the beginning is brilliant! Very elegant 🙂
The part at the end about assigning a number to convexity made me so happy! Any time something discreet transforms into something continuous, it fills me with wonder!
I am absolutely infatuated with abstractions!
This is what I watch when I want to zone out and feel less connected to the real world. I love watching your videos keep it up! :)
I tutor for math below Calc II at my local JuCo and one thing I've noticed is that the students who have a really strong intuition for simple arithmetic are the students who grasp early college math concepts a lot quicker.
I was homeschooled before college and my mom did some stuff for me that I think a lot of people don't get so rigorously in the public school system. One of those things was drilling addition, subtraction, multiplication, and division. Through high school and college I've really enjoyed math, and I think one of the reasons for that is because I don't get hung up on the super-fundamentals.
I see students who take a significant amount of time to mentally compute a simple sum or difference. I presume it takes them a moment longer because they just didn't have the repeated exposure to the drilling of that simple stuff to the point where it was second nature. I can definitely see how having to repeatedly think about fundamental concepts can get in the way of tackling larger concepts and that could definitely lead to a dislike of math, which makes me sad.
I first watched 3blue1brown about 3 or 4 years ago, and even though I didn’t understand it I thoroughly enjoyed it. Now years later when I’ve gone through the majority of high school, I realise these videos are some of the best on youtube
They really are, watching still in my last year of college
I'm in the exact same position
And therefore likely some of the best in the world :)
they are definitely very good (enjoying as a phd in cs)
I'm gonna go on a limb here and throw some "high praise" and say: some of the best in the world. If you've seen the one on Quaternions -- and specifically the webtool they developed to help teach about them -- I wish really really wish down to my heart that schools would adopt that style of teaching. Within hours I got something that I've been struggling to even faintly grasp for years, and not for lack of trying. There's a lot of value in 3B1B's particular style of teaching, and I'm super happy to see he and whatever team may be lurking behind him in his productions are getting the eyeballs they deserve.
I saw the puzzle and immediately had an intuition of the answer, but watching you show me how to apply that was incredible. I love math.
This was actually a really great drawing tutorial that also boosted my 2 brain cells intelligence as a bonus
The fact that this leads you to a ratio of how convex a shape is, is really cool to me.
As time goes on, I think 3B1B is turning from a math channel to a philosophy of education channel disguised as math
I have to rewatch Alice’s solution, Bob’s solution seems like exactly what I would’ve done, and it is what I’m used to doing. Maybe one day I’ll learn to be like Alice
God, I just say through this entire video in one setting. You had me hooked the entire way.
I think this is one of the best videos you've made---if not THE best---and certainly it's the most relevant.
I loved the wonderful aha moments throughout the Alice portions (you had me screaming out loud at certain points because I was so excited about an insight), but the meta-commentary you provided at the end is just as, if not more, important.
Thanks so much!
@3Blue1Brown I am sorry trying to get your attention like this, however would it be right if we would add the smallest surface of the shadow (1^2 =1) and the biggest surface shadow (~1,73 which is the surface when the shadow is a perfect hexagon) and devid them by two (end result ~1,366) to get the average surface shadow? By the way I like the amount of time spend into the animations👍.
@Renan Ökten Why do you think that could be right?
@Renan Ökten unfortunately not, the correct average* is 1.5 (see 31:40). What your approach misses is that a typical shadow is more likely to be closer to the higher end 1.73 than the lower end 1
(*using the rotation-invariant measure)
@J L you are absolutely right👍. Thank you!
This video and the narrative around Alice's and Bob's approaches has just become my new favorite
Spice Master II - 2021-12-20
At last, Alice and Bob are doing something other than sending cryptic messages to each other.
columbus8myhw - 2021-12-20
Is Eve the cube or the light source
Robert de Forest - 2021-12-21
@columbus8myhw Eve is holding the camera filming their math contest. For some reason Eve's camera makes everything look like a wire-frame rendering with hidden line removal.
What I want to know is what Trent and Malory were up to while all this was going on.
Harshil - 2021-12-21
ahha from the Bitcoin video I see
Kvarts314 - 2021-12-21
@Harshil Alice and Bob are just standard cryptography names, not specific to bitcoin
Ishwor Shrestha - 2021-12-21
Ok