A tale of two problem solvers (Average cube shadows)
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- čas přidán 4. 05. 2024
- What's the average area of a cube's shadow?
Numberphile video on Bertrand's paradox: • Bertrand's Paradox (wi...
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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?
• But why is a sphere's ...
If you liked this topic you'll also enjoy Mathologer's videos on very interesting cube shadow facts:
Part 1: • The cube shadow theore...
Part 2: • The cube shadow theore...
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
ia600208.us.archive.org/27/it...
The artwork in this video was done by Kurt Bruns
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
Hindi: rajeshwar-pandey
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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
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At last, Alice and Bob are doing something other than sending cryptic messages to each other.
Is Eve the cube or the light source
@@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.
ahha from the Bitcoin video I see
@@hoppingturtles Alice and Bob are just standard cryptography names, not specific to bitcoin
Ok
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".
@@JohnDoe33408whyd it take 5 minutes? had to think about all possable last digits of pi?
@@TheAdhdGamingseems like lol
Try not😮t😮 r
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?
@@aguyontheinternet8436 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)?
@@arlenboi7360 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
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.
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
@@tomepsilon As a quantum chemist, Bob is how you do things, Alice is how you report it.
@@tomepsilon 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.
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.
@@toxic-LI would rather not believe that there exists such a community that doesn't intersect with that particular video xD
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 CZcams - 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
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
@@ivarangquist9184 fair enough.
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.
@@tezzeret2000 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
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.
You have to be on your p’s and q’s to use Alice’s method.
Your Einstein’s of the world can pull this off due to their vast understanding but your more introductory student certainly has to use a bit more caution for sure.
The issue is that many times the explanations you get from people are plainly bad and inaccurate and the math itself is simple. Many people who did not get to a deep level of understanding will talk with metaphors to explain it to you, while you understand it much better than the "internet teacher" or "colleague teacher" - all you need is to read the math.
It's especially true for ML, what I do.
I don't always understand what's being said, but I do enjoy when a particularly astute blue pi gets angy.
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.
@@x0cx102 I have the same opinion
Ah yes, can the computer determine there is a solution without determining the solution.
@@x0cx102 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 :)
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.
Now if only Alice and Bob had a way to share their proofs, maybe by sending messages that no one else is able to read?
[Eve Lobachevsky has entered the chat]
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
34:30 One of the reasons is that most people, even for those mathematically inclined and consuming mathematical content during their spare time, do NOT want to exercise their brain to a degree that those tedious calculations would demand, and let's be honest, I don't, unless I am REALLY interested in the problem at hand. As a result, those videos that actually dive deep into the calculations would get buried; and the "slick" methods can get people's attention or even shares. It's almost like natural selection that promotes this bias rather than any creator's fault.
I am so unpretty 😭 When I go to the bank, they turn the cameras off. At least I am a big star on CZcams. So don't feel too bad for me, dear mat
As you said thats the way humans have evolved over time.
The quicker more broad solution has prooven to be less energy consuming than having to calculate and check everything
That's true, most of my own videos aim to be more like Alice than Bob not because one is better, but because Alice-like videos somehow seem more natural of a fit for a CZcams audience.
I agree. Its hard to talk about "most people"... I particularly like to exercise my brain with those tedious calculations lol. But taking in cosideration the way math is taught at schools and more generally how we are evolving as a society, the rather creative ways to solve problems comes as candy for the minds that have had enough of systematic approaches.
I wonder if the same effect applies to math cirricula. It would be interesting to chart the number of calculation drills done by primary school students over time.
I would have just taken the area of the smallest shadow, the square, the largest shadow, the hexagon, taken the average and called it a day
that's exactly what I said
@@simple3555 Hi. I have a question. Got a sec?
@@nenmaster5218 uuh sure?
@@simple3555 Sorry for being random but i love to always keep learning and never stop, so do you have some Recommendations for me?
I do have some for you and i LOVE recommending sci-channel and edu-youtubers, but right now, i mainly wanted to ask for some myself. Got some?
Yeah, but there are 8 possible hexigon shadows (one for each corner)- and only 6 possible squares (one for each side)…so your average would skew in favor of being smaller than the average (of all possible orientations were equal).
I think your idea would have better weight with 8h+6s/14 than h+s/2
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
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.
@@DrAlexisOlson 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.
@@DrAlexisOlson 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.
One thing I learned in my IMO days was that while there is often an Alice-style elegant solution, there is also a time and a place for giving up on finding it.
Geometry was my weakpoint and I would always be willing to say "Right, I can't find the elegant solution, time to try a coordinate/trig bash". Or outside geometry, case bash solutions.
The 1999 IMO had a beautiful Alice-style solution to Q3. I sure as hell couldn't find it in the exam, but being able to say 'right, that's fifteen minutes spent looking for a clean solution - let's start bashing cases until we solve this little mongrel' was the difference between solving it and not solving it.
I need help finding a song that goes
Put your poop on my shoulders (oh oh oh oh) and let your worries shut away
I don’t remember what the second last words was, it was either shut or shit. Thanks!
Damn, I'm in the last year of my IMO days. I really want to pass the first stage of the exam on 9th January :(
@@adarshmohapatra5058 Good luck with it! My IMOs were a long time ago, 98 and 99. I later discovered academia wasn't for me, but no regrets.
@@sirgog Thanks!
Maybe grab yourself a beer.
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👍.
@@renanokten6058 Why do you think that could be right?
@@renanokten6058 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)
@@japanada11 you are absolutely right👍. Thank you!
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.
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
a shape made of a perfectly clear material, so it has no shadow ;)
I just want to express how beautiful your animations have become. If this is all still done with your own Python script, then I have nothing but awe for what you have built! Your videos have always been top tier, but lately they have become some of the best produced and most interesting videos on CZcams (in my opinion). Keep up he good work!
honestly so true
I agree. Great explanations & impressive animations
My thinking as well; I hope mathematicians have a greater respect for video editing in general as an art and skill.
I mean I think it's really impressive from his side but since he made Manim public.. anyone can do it pretty easily and it's not that hard anymore.
@@justafish5559 As a person quite familiar with how Manim works, I’m sure this was made with thousands of lines code. Not really easy.
33:32 There is even more subtlety here: you can't just pick an arbitrary sequence of geometric approximations and get the same surface area every time. See the "Schwarz lantern" for an example of the difficulties of approximating surface area this way. So you have to make some statements at least about the type of polygonal solids you are approximating with and justify why that gives the correct limiting result, and why that result is the same as for the true sphere.
Another classic example of this sort is "approximating" the length of the diagonal of a unit square with a sequence of staircases having progressively finer steps. The length of the staircase curve is always 2 and never changes no matter how many steps you add; it fails to converge to the true length of the diagonal sqrt(2).
Thank you, this was bothering me
I recently grappled with your example while driving, might as well take 2 long legs instead of many turns because the distance is the same BUT in driving there's a "cost" to turning lol!
yup. just pick faces that are planes tangent to the space and you’re okey dokey
Yeah, you have to be careful because there can be infinitely meny corners that add up to a non zero surface area, area can fit into zero volume
Also the pi=4 proof is a great example of this
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! 🤷
@@Dogzz13 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.
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.
legit shouted for joy when he brought up using a sphere to find the universal constant, that has to be the most beautiful piece of math I have seen all year
When he pointed out that Alice’s method could be generalized across convex shapes, my eyes actually went wide and I started grinning and mouthing the word “Sphere”.
@@bioboygamer same lol
i genuinely had to pause the video and i mouthed "oh my god is it gonna be the surface area of a sphere?" and i was almost gonna tear up
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.
@@MartinVillagra 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
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!
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'm not great at math by any means, but as is often the case with this channel I find little bits of indication that I'm not entirely without mathematical intuition either. In this video it was that I always defined convexity like at 15:55 for myself, from elementary school onward. This channel has a great way of simultaneously teaching and reassuring the viewer about their abilities.
in the spirit of looking at things from different perspective, this is not the only possible definition of convexity. it takes a bit of working out to formally proof, but the following is also equivalent:
"Any tangent line to the boundary of the set touches the set only at boundary points" and
"The intersection of [any known convex shape (say a circle) tangent at any boundary point of the set], with [the set itself] is exactly 1 point".
some even define a set convex if it is equal to its convex hull, defining the convex hull using the lines method, which I find really backwards but its still exactly as valid.
I've always focused on the edges and angles, personally- A convex shape is one in which all of the inside angles are less than (or equal to?) 180 degrees. There's some geometric truth somewhere that could explain why focusing on the angles and focusing on the points would give the same answer to the same question (something something... make triangles with disparate vertices of the shape...)
@@ferociousfeind8538 in some cases it is not clear what the relevant angles are. E.g. what if it is a smooth shape in many dimensions?
The “are all line segments between points in the shape, contained entirely in the shape” can still be applied even to talk about an infinite dimensional thing, and I think it would be difficult to do that by talking above angles.
So, I think the way OP/the video define it, is probably overall best?
I think applying the bob method first and then the Alice method to gain deeper insights can be really useful, because the Alice approach seems great after the fact, when someone has already discovered the insight, but my experience has been that there are too many possible places to look for clever insights. Having analytical results can guide the insights. As soon as bob got the answer, you can see there must be a simpler answer, but you have some basis to go off of now-a target. For example, the other day I encountered the lunes of Alhazen, and I was sure there was some clever insight, but had no idea where to start, so I just did the most direct computation approach and found (spoiler warning) the area of the lunes was exactly the area of the triangle. This lead me to discover the fact about the Pythagorean theorem that was necessary to finding the clever insight.
This right here is why studying Real Analysis in my undergrad math curriculum was hell. It felt like so many of the proofs relied on having the right kinds of insight to generalize problems that you can't necessarily acquire outside of an organic process of exploring different kinds of problems of a type. And that kind of mathematical learning was really difficult to engage in while also trying to balance all the other commitments and responsibilities of approaching post-undergrad life
Great video, one of your best I'd say. The conclusion you come to is applicable to almost ANY skill, not just math. Doing the rote basics over and over is what allows you to take a higher level, abstracted approach more easily - you can see this in math, cooking, video games, sports, programming, etc! It's a great feeling to reach that point of being able to take the "top down" approach in whatever you're doing, but it has to be built on a strong foundation.
One of your main qualities in my opinion is your ability to make things clear as water. This is precious for us to reflect and develop new understandings. I grew learning how to find new patterns, new ways of interpreting what I know, getting contact with a huge variety of subjects. This makes me very proficient at Alice's approach, and I noticed how I relate a lot to it, but I rely on it to the point I avoid laborious approaches to solving problems in general, sometimes in unhealthy ways, as I've noticed some moments when my justifications for my insights weren't as solid as they should. It's ever important to manage that.
I'll hear the podcast you mentioned, it seems very insightful in order for me to value the importance laborious work can have on being able to find and make great generalizations, cross patterns.
You may be indeed guilty of promoting the beauty of Alice's approach, but the reflections you bring today are also truly invaluable. Thank you, Grant.
I considered myself a Bob when I started grad school, but my advisor is definitely an Alice. There was a lot of conflict for me at first in the sense that I wanted to get the details right & model it all; I didn't really see the point of giving qualitative answers if the thing we're studying is so sensitive to parameters. But I think I've grown to see the merit in both approaches now - it's easy to lose yourself in the weeds if you're a pure Bob, and a pure Alice can require a lot of insight/luck. I've often found myself needing to be a Bob first before I can be an Alice and vice versa. I still have lots to learn, but I think being able to switch hats is an important development. Great video!
My name Adam
@@constante7510 ok
@@constante7510 My name Jeff
@@eltyo340 ok
As an artist looking for a way to empirically paint/draw shadows. This video has been very helpful.
Same, my blender cube is now an ultimate shadow-making cube
Ah, turns out there's a third perspective on the problem. Alice, Bob, and the artist.
@@zbieramnakartonowyprzycisk8026 What if computers suddenly stopped existing. How will you draw your cube shadows then?
@@vigilantcosmicpenguin8721 When the light is closer? I guess using ray vectors would be easier if the light source was non-uniform.
“What if computer suddenly stopped working” Don’t worry that isn’t happening anywhere soon.
I found quite an appealing physics-oriented proof using Gauss law for EM fields. Hope you enjoy :)
1 - Imagine the cube standing in the Euclidean 3D space, and imagine a physical sphere of radius R (where R >> 1) whose center coincides with the center of the cube. Further assume that the sphere carries an overall electrical charge of 1C, uniformly distributed over the surface of the sphere.
2 - Now, look at some differential area da on the sphere. It generates an electrical flux dF through the surface of the cube. Obviously, since the cube lies inside the sphere, it contains no charge and dF=0. But we can further decompose:
dF = dF_in + dF_out = 0
When dF_in is the positive contribution to the flux (entering the cube), and dF_out stands for the negative contribution to the flux (exiting from the cube).
3 - There is another symmetry to account for: for each differential area da around some point x on the sphere, we can associate the differential area da' around the antipodal point -x also lying on the sphere. Denote by dF and dF' the corresponding contributions to the flux. From the problem's symmetry we get:
dF_in = dF'_in and dF_out = dF'_out
4 - If we move a differential charge dQ from the sphere to the inside of the cube, its contribution to the flux jumps from dF_in + dF_out = 0 to -dF_in + dF_out = -2 * dF_in
5 - Now, using (3) and (4 ) and Integrating the flux over the whole sphere and moving the overall charge inside the cube, we get the following relationship between the total flux after the shift, and the total incoming flux before the shift:
|F_shifted| = 4 * |F_in|
Now we can let the magic happen:
6 - From Gauss Law, we get:
|F_shifted| = 1
7 - ...But taking R to infinity, and using the definition of the flux, it's quite obvious that |F_in| (up to some irrelevant physical constant) is exactly the averaged normalized shadow area of the cube!!!
Indeed you can think of each differential dF_in generated by a charge at some point x on the sphere as the relative contribution to the averaged shadow associated with the angular position of x relative to the cube's center.
8 - Therefore,
E(S(shadow)) / S(cube) = |F_in| = 1/4 * |F_shifted| = 1/4
as expected.
Besides requiring no integration and giving a good intuition for the final 1/4 factor, this proof technique also generalizes to *any* orientable 2D surface (say a n-holes torus) since Gauss Law doesn't care about the the fundamental group of the enclosed surface. Therefore, 1-connectedness is easily shown to be un unneeded assumption.
Lol. I immediately check out your comment after you mentioned. You could have just leave me your comment link :D.
Anyway, I was able to find your comment after using browser search in this page.
So glad that I find someone who think alike. I'm no physicist, and I don't have friends to talk about things like this.
My approach is a little bit different but use the same electric flux principle (basically Gauss' Law). I'll just share this in the same thread.
1. The initial question only assumes parallel shadow projections, so I'll admit initially I didn't start as generic as yours. I immediately assumes there is a uniform charges very far away, causing the electric fields to be parallel. This is equivalent with your charges in sphere's surface with R infinity.
2. Electric fields will penetrate any kind of object (it doesn't have to be cube, but we can start with cube if we want to), but it will have net zero flux on the object (no charge inside in the object/cube).
3. The shadow is equivalent to flux penetrating this flat shadow casted by this cube.
4. Since flux on the object is net zero, that means the top half of the object and bottom half of the object have equal absolute flux value, which is the same as flux in the shadow. F_shadow = 1/2 * (sum of absolute value of the flux). Note that, single instance of shadow or average of shadows, this formula doesn't change.
5. For each small section of area dA in the cube surface, if you want to calculate the average flux over all the possible orientation, you iterate all the possible position and orientation of dA. It is just conveniently happens that when you rotate dA to all possible space, it will form a sphere.
6. Conveniently Electric field vector has the same value and orientation in this case. The total absolute flux is just twice the top half of the sphere, which is equivalent to twice the flux of the shadow. Shadow of a sphere is just a circle, with radius r which is the distance of dA to the center of the cube. total flux = 2 * pi * r^2. Since we are calculating the average over all orientation, we divide by our possible spaces, which is the sphere surface of dA. we have 4*pi*r^2. So the average total absolute flux is just 1/2.
7. Sum the average to all section of the surface dA. We got: average(F_shadow) = 1/2 * 1/2 * average(total absolute flux in all orientation)
8. What we want to have is the relationship between the shadow area and the the surface of the cube. Because the electric field value is the same, we can factor out E from both sides. We got average(shadow) = 1/4 * surface area of objects
As you have said, the 1/4 factor is just stems out from the fact in step 6, and irrelevant on the object shape itself. It's just a ratio between the shadow and all possible spaces that makes the shadow (which is conveniently a sphere because our rotation is a sphere). The convex criteria is just so that the possible spaces is always a sphere (if it's not convex, some orientation doesn't have a flux).
You can also generalize if the light source is a point near by the objects. Just treat it as charge source, then the electric field value and orientation will follow Gauss Law (spherical instead of parallel). That means the average shadow is only multiplied by the coordinate transform scale size of the sphere and the projection.
@@RizkyMaulanaNugraha very nice!
Btw, I had no idea you can actually share a yt comment link, so thanks for the tip :)
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.
For the last question: Bob is obvious and Alice was hard. Bob used it in the derivation of integral involving sin factor, and Alice used it when she changes the order of double sums. i.e. the rotation is uniform so that the weights of each face's contributions are all same.
Rings true especially considering the sphere case, where we take the limit of a sequence. Gives off the same vibe as swapping a limit and an integral
as soon as you combine three faces (areas) with indepentent normal vectors, and assume that the sum of their averages is the overall average, you already have to use this orientation distribution. this assumtion is the key step!
the distribution which is used is the only one that is the same for every face, no matter how it is oriented relative to the other faces.
(just what Pa Pa wrote, but stressing that it is not the infinities where the distribution gets fixed, but the finite steps)
I thought so too at first. But wouldn't both orders lead to the same result, no matter how the samples were picked? If I used a biased way to pick a thousand rotations, I could still average the shadow of each face and the sum of the averages of the faces will equal the average of the shadows of the entire shape.
@@CaesarsSalad not quite. at least not if you have the same constant by which the average scales the area. this constant would be different for different orientations of the faces.
@@CaesarsSalad But you can pick biased orientations which don' have the average for each face being equal. For example, if you pick a biased set so the top face must be at the top (i.e. a point on it must be the highest point on the cube) and can never be at the side, then it will contribute a larger amount.
So a key part is going from adding up the faces to multiplying the average for a single face by the number of faces.
I remember a quote:
"There are two kinds of scientific progress: the methodical experimentation and categorization which gradually extend the boundaries of knowledge, and the revolutionary leap of genius which redefines and transcends those boundaries. Acknowledging our debt to the former, we yearn nonetheless for the latter"
A cookie(not really) for who got it.
Not exactly the same, but quite analogous: Bob being the methodical experimentation and categorization, and Alice being the revolutionary leap.
Chances are Bobs' boring but practical progress is what dominates. But Alices' leaps are fun and have a higher impact individually.
Prokhor Zakharov! That quote has remained with me for more than half my life since then.
A beautiful quote
God that game was so good
Nakamoto is the Alice of the 21st century
@@Corwin256 Hey, that is Alpha Centauri. I didn't remember the quote, but I faintly remembered the name Zakharov. Good game. I haven't played it in years.
Thank you so much for continuing to make very accessible and very interesting math-related videos.
I studied math at university, and to this day I love the way it enables insights into the world around us (or into other worlds). Unfortunately my day-to-day work doesn't involve math and its difficult to sit down with a text when I have so many other responsibilities/things to do, but your videos still let me engage with this subject that I love so much, and the simple, elegant animations make it so much easier to grasp a problem, think about it for a while, and then come back to view the explanation when I have the time.
Truly, thank you.
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.
This video is long, mathematically detailed, costly to make and on a pretty obscure problem.. all characteristics that run counter to the typical growth strategy on CZcams. It's so nice to see Grant investing in an excellent video like this. It sets a much needed trend. Love it.
But on the other hand, the problem being solved, although specific, reveals a very general perspective of what it means to think mathematically and how that can vary from person to person. Admittedly, the audience here is relatively niche (i.e. those who know linear algebra and calculus), but that’s his target audience anyway. There would be an extremely high chance of anyone with this demographic to click on the video. But even then, the issue at heart here, i.e. problem solving, is enticing to anyone who is mathematically inclined.
@@vlogsbyrow Good point - it's not like this video isn't doing well. Just nice to see a heavily invested product for a relatively niche audience.
Interesting. Though I've always thought that Alice and Bob were more into cryptography.
Oh, but they have to use cryptography to make sure the other one doesn't see their problem solving.
More like, to share their solutions with each other, and nobody else.
Flashback to Khan Academy information theory XD
@@ekisacik And to detect when their solutions have been tampered with in transit.
It's impressive how the visual and verbal explanation helps me understand a language I'm bad at (math). Since I became a machinist, I've only done the main four kinds of math in my head - addition, substraction, multiplication & division. I still remember the Pytagoras rule but anything else is forgotten as wiser minds devised a programing language which only needs dimensions & angles. Yet I still comprehend a lot of what you're explaining, even if I don't know the more complex language of math. Terrific work.
I love how both of them had to resort to the same Spherical shape irrespective of their way of thinking the problem.
The plethora and quality of the animations used, make this problem so much more comprehensible.
Everything looks so fluid and intuitive. I can't imagine the work it took to create this video.
I think the point when the distribution of rotations gets picked by Alice is when she goes from taking the sum of each face's average over all rotations to taking 6 * the average of one face overall rotations, since if you were using a distribution heavily biased towards the cube being nearly vertical (as in a horizontal top and bottom side, and 4 vertical sides) the top and bottom sides would have much larger averages than the other 4, and you couldn't just take one face and multiply it by 6. Also, the 3d animations in this video look much better than before! (aside from some z-fighting in the shadow when the torus rotates at 16:49)
Or to borrow from physics/the PhD approach: She imposes arbitrary rotational symmetry which locks her solution to SO(3).
Another way to explain (or possibly justify) her distribution here is to argue that for any given orientation of the cube, there are symmetrical orientations where each face can take the place of any other side with equal probability, therefore every term in one sum must appear in each of the other sums as well, or in other words each sum would be a permutation of the others and therefore be equal.
I don't think this is the complete answer. I'll use an intentionally terrible distribution to illustrate: if you sample random rotations by tossing a six-sided die and observing how it comes to rest on the table, the six faces of a cube are still symmetric under that distribution (with area of 0 or 1, 2/3 and 1/3 of the time respectively) and can be considered independently. It's only with the third constraint (asserting that it can be applied to other convex polyhedra) that the symmetry gets broken; tossing a six-sided die does not generate a face-symmetric distribution for an icosohedron, for example.
@@TC-cq7oc no, but rolling an icosohedron WOULD generate face-symmetric distribution for an icosohedron
@@phiefer3 - It would, but that'd be a different distribution of orientations than rolling a cube was using - it'd have 20 possible outcomes instead of 6. If you apply the icosahedron's distribution back to a cube, you'd get a constant factor on surface area much closer to 1/4 (matching the "correct" distribution) than my distribution which gives 1/6.
This is a Masterclass on Problem Solving approach. Your hardworking and dedicated attitude is so clearly visible in the video. And then, you added a crisp explanation on squares summation at 35.20 . Brilliant
I really appreciate the little coda putting the Alice vs. Bob question into perspective. As you say, it is very easy to look at problems like these and get the idea that problem solving is all inspiration, no perspiration.
This video is a great chance to say thank you to Grant for all of his work. I have always loved mathematics, but this man's work gave me the inspiration to become a teacher and start using explanations commonly ignored by other teachers throughout the educational system. So many teachers (and most teachers in my country, Russia, aren't an exception) just throw the formulas and equations at the students, which in my opinion is something that makes math extremely boring and remote from what the science is about.
I think of myself as more of an "Alice", because I am dumb at arithmetic and I can easily make stupid mistakes while computing something easy, so instead I am trying to use creativity to simplify the problem and make its solution as elegant as possible. I used this approach in the mathematical olympiads a lot, and as a high school student at my final year at school and an active participant in lots and lots of math contests, I still do. Although indeed, most math out there is not something that requires creativity, so doing stuff the "Bob" way isn't a thing to neglect.
I am a high school student and one of the translators of 3blue1brown videos into Russian. I first saw Grant's work as a 9th grader when I stumbled upon the calculus playlist on this channel. With a little bit of pausing an pondering, and a fair bit of practice, I quickly understood derivatives and could apply them in many different aspects, such as physics and computer science (later as a 10th grader I tried machine learning, and then gave up on it for not having enough time). As I couldn't find anything like that in the Russian section of CZcams, I decided to translate the entire playlist into my native language.
And I did. It's a long story, but as of now, the entire series is available in Russian. The translations don't often receive too much attention, but I believe we should appreciate the work done to make this content available to anyone in the world, regardless of their level of English. The translators are in the shadow of the original creators, but they invest as much time and effort to help spread the knowledge.
I am now learning Manim to start my own CZcams channel (I mean, it is already started, it is the account I am commenting from, but it's abandoned), but I lack the time, because when you're a student at your last year at school, usually you don't have that much free time for making large educational videos. I will probably use my channel as a dumping ground for whatever I find interesting, and then later find some format that's right for me. I have a few scripts already written, but I have no idea when they will be online.
This is not a request for help, it is just a bunch of thoughts that I decided to express here.
i dont know if the content youll be making will be in russian, but im really genuinely interested in it. ig being curious about everyhting does have its perks haha (besides that, props to you for the channel! i hope you soon get time to make smth you like!! good luck)
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.
Thank you for this video! I’ve been working on an application of this problem for a while now. This video has been helpful for figuring it out, and gave me something to think about with the differences, pros, and cons with the route Bob (me) vs Alice route.
This reminds me of speeds and feeds in machining, drilling, lathing, and tapping. Most of our calculations were most likely derived AFTER thousands if not millions of tests on materials, speeds, depths of cut, coolants, cutting tools, clamps and vibrations, durations, constant cuts or interrupted cuts, etc etc.
Many many factors and each one tested and changing the current formula. I really am thankful for a video like this.
It has been more than a decade since I saw this problem for the first time. This problem was one of the previous problems for the university admission exam. Back then, I was someone closer to Alice, and solving these kinds of problem was quite fun.
Years have passed and now I am trying to get a PhD (but not in mathematics). Problems I encounter day by day are not that different, but the academia requires me to be someone like Bob; I need to calculate quickly (sometimes, it's even not exact, but using numerical methods) and construct a valid-ish intuition as fast as possible. I do not need to generalize the problem, because the nature itself has a lot of restrictions that prohibit such generalizations.
It is fun in its own ways, but I often miss the old days of being Alice. For that, I am very thankful that we have you. Your videos stimulates and encourages me to see in different, sometimes more wider perspective. I am sure there are a lot of people who feel in the same way. Please, keep up the good work!
I definitely find myself using bob's approach often when doing math, I've always found the theory and generalization aspects of math confusing, but I will admit that on the rare occasion when I remember a theory or method to apply to a problem I'm doing, and it works, it is a very nice feeling, knowing that I didn't have to do all of those calculations, even if it would still be satisfying to see a series of calculations get me to the answer.
I really appreciate this in depth video explaining the nuances and differences in approaches to the same problem, and how we can learn from using aspects of both methods to help us.
Beautiful video. I love the mix of the two ways of solving the problem with the differences of teaching and learning about it. I myself like to explain certain interesting subjects to people, but I'm also interested in how to actually explain those things in ways that take in to account the listeners point of view. It's a whole different challenge on its own since it's very social compared to the facts I'm trying to explain, but it also makes it more personal and engaging.
This is an absolutely amazing video! I love how you take some many disparate mathematical concepts and apply them together for a single problem, it feels so "non-wrote" in a way that I never got in school. I love the beauty in math, and it feels so good to finally see content that teaches math in a way respectful of that beauty.
I would like to say though, I did do a double take at 30:01 when you said "What is it that Alice does to carry out the final solution". In my head, I heard n@zi words, took me a second to reel myself back into a reasonable context LOL.
Still though, absolutely wonderful video! Keep up the amazing work!
It's funny, after what feels like a drought of 3b1b videos I start to internally complain about the lack of new content and then the next one is released and I think "oh that's right, he's creates masterpieces every time". Thank you for the continued quality in both production and substance (especially the substance) over the years!
The probability distribution hypotesis is made by Alice when she say that the average area of the shade of a given face is indipendent by the face. Infact, if we chose a different probability distribution, for example one that is peaked in correspondence to the state "two faces of the cube parallel to the ground, the other perpendicular" the average shade area of the two parallel face would be 1, the average area of the others would be 0. Great video!
Indeed. I was doing the exercise in parallel to the video (as an undergrad student this was delightful practice), and I did it in a very similar way to Alice. I defined the normal vectors of the different faces of the cube as "a series of linear transformations on the normal vector of 'face 1'." It quickly became clear that only with uniform probability distribution I could ignore those extra transformations and apply the formula for average shadow.
This was my first thought as well, but I think that's not it. I thought about a probability distribution similar to the one you suggested -- force two faces to be parallel to the ground, like rolling a die on the table. But the average shaded area of any given face is the same: 1/3, corresponding to the probability that the face is pointed up or down.
I think the assumption is made when she generalizes her formula from the cube to other shapes -- around 21:16 in the video.
Unlike a cube, an arbitrary shape can have faces pointing in any direction. So to apply her "1/2 * c * A" formula to each face, she must assume that turning the face around in space does not change the probability distribution. That is, the probability distribution must be invariant under rotation.
@@biggnate yes, you are perfectly right, thanks
@@Brainth1780 very clean way of doing it!
@@biggnate You make a very interesting point, which prompted me to reevaluate a lot of what Alice did. Long wall of text incoming, the rabbit hole went deeper than I imagined. *TL;DR:* It's up to how you interpret the steps, both answers are right in their own way.
What she claims for the "shadow of a cube" step is _not_ just that the shadow corresponds to "a constant times the surface area of the cube", she reaches the conclusion that that constant is 1/2 of *the same constant **_c_* that she got from calculating the average shadow of a square. As we know, for a Uniform probability distribution, c=1/2.
Now, things get interesting. Using a different probability distribution would mean that a "side face" has a different average shadow than the top face, but you could get around that and say that "a face has a 1/3 chance to be one of the two that are parallel to the ground." This would still allow you to factorize in order to find a number for which the formula "1/2 * c * A" applies (meaning _c_ is independent of A), but _c_ would be particular to a cube and, arguably, not the same constant as the one you'd get for the shadow of a square.
I say arguably because if you define the probability distribution p(θ) in a convenient way, the constant *can* be the same one. That is, using the same argument that a square has a 1/6 chance of being any given face of the cube, and modelling the distribution accordingly. Say that p_1(θ) is the base probability distribution of one of the faces, it's possible to define all p_i(θ) distributions, then average them for a "general" distribution. For the previous example it would look like this:
> Let p_i(θ) be the probability distribution for "face i" of the cube.
> p_1(θ) = { *1* if θ=0; *0* everywhere else}
> p_2(θ) = { *1* if θ=π/2; *0* everywhere else}
> ...
> p(θ) = Sum(p_i(θ))/6 ; i∈{1,...,6}
=> p(θ) = { *1/6* if θ=0;
*4/6* if θ=π/2;
*1/6* if θ=π;
*0* everywhere else}
I personally don't like this too much, as I believe that the distribution should determine the position of the cube's faces, not the other way around. Still, the result is consistent, so there you go. What's cool is that from here it's very easy to point out that "uniform distribution => p_1(θ) = ... = p_6(θ) = p(θ)" which gets rid of the geometric dependency of _c_ and allows you to keep going from there. It's longer than assuming uniform distribution earlier on, but it's a valid alternative so that's cool.
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.
The section on "Which one is better?" was even deeper than the whole video, thanks for explaining it so clearly.
These kind of stuff give me such a beautiful sense or warmth, depth, intellect, passion, and wholesomeness.
And that's why I love 3b1b more than anything else.
Oh wow, I encountered this exact problem trying to figure out the drag coefficient (which is proportional to the area) of a cubesat in low Earth orbit. I rendered every orientation at discrete step sizes and counted the pixels in each render, then averaged across all images. As an engineer, I didn't need a closed form solution, just something representative - keep in mind any orbit life projection will also be driven largely by how strong the sun is, which will expand the Earth's atmosphere and increase drag. So any projection will only be as good as the sun's strength forecast is years from now anyway.
Just btw, depending on how much drag variation between orientations you have in your cubesat (if it's a 3U, quite a bit), this problem is not a great analogue. The amount of time it spends in any given orientation depends on the aerodynamics of your satellite. Even in a 1U will likely have some preferred orientation depending on the center of mass placement. That said... it's a cubesat, if it lasts more than a year, it's probably done whatever you sent it up there to do.
@@JMurph2015 We were specifically targeting proximity operations with our 3U. The simulations I was running showed how long we could tumble before we were too far apart from our cohort to get back in formation (utilizing the 1U vs 3U faces) before the orbit decayed.
this one of the best videos I've ever seen on this platform. I keep coming back to watch it.
I remember when you had 100k subscribers and I recommended you to my university friends telling them how good your channel was. You explained determinants of matrices.
When I was a child teaching myself algebra I'd try some manipulations, then I'd check them with actual numbers. This seemed perfectly natural to me. This was a good way to find errors in my derivations (in basic algebra), and taught me a lot. I was fascinated by the way different methods got you to the same result. This video seems to recommend something similar. Using a number if examples to build intuition
I seem to recall this was a thing that Richard Feynman was infamous for: he would listen to a discussion of an abstract calculation and surreptitiously do it with some real numbers to see whether it made sense. The example which comes to mind was when a suggested formula for the radius of the observable universe would have come out to about half an inch or so…
It's years since I read his book, so the details are fuzzy, maybe someone has their copy to hand to double-check me?
@@PhilBoswell that's fascinating.
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??
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.
Thank you for this. I've always been insecure about my inability to quickly see abstractions. Normally, I'll try to solve the problem the hard way first. And in doing so I gain a deeper understanding into the subtle details.
I think it's useful to have these subtle details before insights can be found. And that most people are capable of noticing insight if given enough information and time.
On the distribution of Bob vs Alice math videos on the internet: I watch math videos on the internet for the Alice insights, I go to work and do Bob work. Those insights serve as inspiration for better methods and deeper understanding for my Bob work. Can I use Alice's insight to cut down the time complexity of this algorithm or verify it's unbiased? Sometimes listening to you just recharges my math batteries haha.
Thanks Grant
21:27 The moment everything clicks together into its place... I will never stop loving these moments. They are the reason I did a math major and post-degree, and the reason I fell in love with it.
My sincere congratulations to 3blue1brown for illustrating and explaining this problem and the two problem-solving mindsets better than anyone could ever do.
So awesome to see this! I tried to solve this at work awhile back for a spinning satellite, to determine the average surface area of the solar panels in the sunlight
The illustration of Bob and Alice at the beginning is brilliant! Very elegant 🙂
Taking me through Linear Algebra, Calculus, and general problem solving is quite literally the bulk of my math minor. Thanks for showing me the multiple ways a problem can be proven!
Ooh, I like that challenge at the end to identify when Alice locks down a specific distribution of orientations. I think it's when she generalizes to all convex shapes. In particular, all convex shapes must include all possible rotations of any specific convex shape. If the solution for all rotations of any specific convex shape must be the same, then the distribution must be independent of orientation - which is to say, uniform.
This is a fun challenge and an excellent video. I agree that you've nailed where the assumption is being used *first* in Alice's proof. Spinning the cube (and other shapes) around and claiming this preserves the constant is fixing the probability distribution.
However I think @3Blue1Brown could've been even more devious. Imagine the cube didn't spin but was kept in a fixed orientation. Then you could still build polygons with more and more sides that still "point" in the same direction and get the approximation to a sphere used later in the video. This would give c=1/2 no matter what the initial orientation of the cube is. Now where is the probability distribution being assumed?!
To me, this is your most beautiful video so far. Amazing job Grant. 👍
Why are you so great, man? Like seriously why? I've been watching your videos the past few months, and it's changed how I see mathematics.
At 34:00 note that theoretically you should have defined the specific polyhedra you use, since for some of them their surface area would not approach the sphere's. For example, take smallar and smaller cubes from the vertices of a cube encompassing the sphere, and the surface area would stay constant no matter how many cubes you take, ehile the shape will approach a sphere intuitively. This is the 3D analogy to the "proof" that π=4.
You can’t do it with convex polyhedra, though.
Isn’t the limit increasing the number of sides of one (regular) polyhedron? The complaint was that the previous sum over faces operator cannot be exchanged with the area operator for spheres because spheres don’t have faces.
So the limit was just necessary to show that yes, the same area and sum operators can be exchanged even for spheres by showing spheres are infinite sided regular polyhedrons, not 0 sided shapes. It shouldn’t matter that dividing up cubes don’t become closer to a circle
I guess I don’t understand how dividing the cube into smaller cubes becomes a sphere in the limit
While you have a valid point, the tiny cubes stuck together to form a jagged surface, and they do not form a convex shape.
@@coolcax99 Imagine the 2D case of a circle in a square. The circle's perimeter is 2πr, by definition, and the squares is 8r. If I take off the corners of the square, up to the circle's circumference, I'm left with a dodecagon that looks like a "+" with a bit of thickness, with a circle at the centre of the plus. The perimeter of the new + shape is still 8r. If we keep cutting off corners, we get the same perimeter 8r each time. In the limit, it **appears** as though this jagged shape approaches a circle, and its perimeter is therefore 8r. Therefore, 8r=2πr, so π=4. Yoav is extending this to the 3D case (which would get the bogus result π=6) and using it to poke holes in the proof (which is good practice), but as Vaibhav points out, it would cease to be convex.
The reason why the 'proof' of π=4 is false would most likely have to do with cantor sets and the fact that the resulting curve would not be smooth (I've seen the staircase paradox, but this was the first time I've heard of the π=4 paradox, an interesting thought)
Words can’t describe how much I enjoyed watching this video! Visiting various approaches, learning how to think, do computations and put all the relevant pieces into a number of insights - brilliant!
This video and the narrative around Alice's and Bob's approaches has just become my new favorite
This was super interesting to watch, even for so done with little experience in the subject. I was still able to understand and follow along
Examinations made me prefer bob's way of solving problems, it is much more consistent, I can foresee the answer before doing the calculation
Schoolwork (and most achedemia) is geared around Bob. Alice's of the world get shoved to the side, much to our detriment.
For the interested: Alice is on her way to discovering Hadwiger's Theorem for convex bodies
Does she exist
absolutely phenomenal content. i really enjoyed the story you told, and the chance to compare it to how i might have approached the problem, and have approached similar ones in the past
Best maths videos on youtube. Hands down. Thank you so much for making them.
Perfect timing for some quality content ❤️
15:00 UTC?
@@zbieramnakartonowyprzycisk8026 yes
@@ardaehi well, I can't disagree
3Alice1Bob is the best combination for an ideal thinker after all!
Your video had layers upon layers of significant elements. But, to address one which might statistically be prone to less attention, I really enjoyed the way you showed the fact that coming up with a seemingly faster and smoother answer, like that of Alice, almost always demands more and deeper cognitive activity...
Such an approach is after all the thing most probable to cause a paradigm-changing reduction towards unification in each and all sciences; and, that's something to be dreamt of for now...
This channel actually reminds me of my now-social teacher, who explains all the topics in an intuitive way. And most of the time, he acts just like a helper while we, the students, try to answer the questions. This video has a lot of "pause and ponder" moments, which can be compared to that teacher of mine. I love these types of teachers
Really insightful video! I really appreciate your reflection on the topic!
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!
The little details man. At 31:00 on the dark side of the globe there are lights at population centers. Lovely
Every time I watch one of your videos, I am amazed by how well they are made, how nice your animations are and how much work must have gone into it. Thank you!
Beautiful, beautiful, beautiful!! From the animation to the parallelism to the story, everything is absolutely incredible. One of the best videos I've ever seen!
I waited for a long time. Finally waited, it’s really the best Christmas gift for me. Thank you to all the people in the team!
It was at introducing the sphere when Alice solved the "what does average of rotations mean" puzzle, isn't it? Exactly when she got a sphere by taking a limit of n-hedron she managed to include all possible rotations of infinitely many sides. It's amazing how this "averaging sphere" popped up in both methods but each one in completely different context.
No, I think it's very early in the video - when she's considering the linearity. Imagine if the probability distribution of rotations was bonkers and valued certain angle much more than all the others. It's not at "limit of this shapes is a sphere", where things break down. They break down way in the beginning, when she says that the average shadow of the single face is linearly proportional to its area.
You can remember that she has a rotation matrix there. Chosing a random orientation in her case boils down to chosing a random matrix. And the probability distribution being "uniform" plays into preserving the linearity there.
@@DukeBG But how exactly is the uniform probability tied to the linearity? I don't see any contradiction in choosing one particular rotation matrix and simultaneously not assuming that this one is as likely to occur as the others. Moreover, Bob also proves the linearity right in the beginning but he needed the sphere of normal vectors for the probability to be uniform.
@@krcprc Ok, I thought about it more and I was wrong. The uniform-ness (uniformity?) of the distribution plays a role later - more when the "rows into columns" trick is happening - we take get a "random choice" of N rotations and take a limit as N goes to infinity. And while we have a sum of 6 things N times, we turn it into 6 sums of N things and argue that the limit is applied to "both sides" and we get the resulting new formulae.
Well, the limit of both sides needs to _exist_. I mean that the "random choice" of N rotations shouldn't change how the sequence of an average over them converges to the limit. That's what Grant is talking about in the numberphile video - how the rotations should be invariant under symmetries.
Let's consider again what exactly breaks if we make the distribution favor some rotations over the others. Like, assigning some rotations a bigger probability than others we con construct a sequence that won't converge to a limit. Sorry if this still feels handwavey.
I think this is a great way of looking at it because the distribution of normal vectors Bob integrates over is exactly that of the normal vectors on the surface of the sphere.
I think this goes to show that you would actually gain more insight as Bob if the question were posed as "what is the average shadow of the sphere", but you forced yourself to go through the integral calculus. I think you would reach a lot of the insights gained by Alice along the way.
@@DukeBG I don’t see why you wouldn’t get a limit if you chose a lopsided distribution?
Taking the expectation of a random variable is linear.
If we let R be the rotation to be applied, which is a random variable with some distribution, and S(R(F_i)) is the projected area of the rotation R applied to face F_i ,
and S(R(F_i)) is also a random variable, (but this time is real-valued, rather than matrix-valued) and E is used for the expectation, then, for any finite number of faces,
E[sum_i S(R(F_i))] = sum_i E[S(R(F_i))]
I think the issue is in concluding that the average for each surface depends only on the area of the surface, and not on the orientation of the surface.
For example, suppose our shape is a very short box, like, a 1 by 1 by (1/100) box, with the (1/100) dimension being initially the up/down dimension.
Then, suppose that in our distribution over the rotations, we essentially only include the orientations which are very close to the original orientation of the box (perhaps someone has written “this side up!” On the top of the box? Haha).
It is still perfectly valid to distribute the operation of taking the expectation across the finite sum over the 6 sides, but we can see now that when the distribution isn’t uniform, the average for each side will depend on the initial orientation for that side.
When all rotations are equally likely, then we can disregard the original orientations of the different parts.
OMG, I saw the title of the video and I tried to do it by myself before watching it. One of the best videos i've watched, it has helped me to order my ideas abut that problem. Also loved the final reflection, I clearly relate to Bob :DDD
I think Alice's method is a great way to translate the actual math to something anyone can understand. I have very little knowledge of math more advanced than college level intro to stats and Alice's thoughts drew me a clear path without drowning me in numbers and calculations I barely understand. So for the medium of CZcams where your channel is meant to demonstrate interesting mathematical concepts to a large number of people, that's perfectly acceptable. Showing how math can be applied to real objects can help demystify and help students become more confident in things like calculus and trigonometry; which are subjects the average student often dreads. You take the abstract and make it concrete.
That being said, it's good to acknowledge that knowing the abstract allows you to deal with less...tidy problems. So the intuition will get you forest, but the math will help you chop down a specific tree.