Alice, Bob and the average shadow of a cube

At 13:00, the subscripts for f(R_{10}) and so on look a little wonky. Not sure what the manim syntax is like, but in LaTeX this looks like what you'd get by typing $f(R_10)$ instead of $f(R_{10})$.

Dan Kinch

Personally, I wouldn't have minded a single-line justification that a sphere can be approximated by a sequence of convex polyhedra. It's just the 3D version of the Method of Exhaustion often associated with Archimedes.

Alexis Olson

Thank you so much for this beautiful explanation of two very different problem solving approaches. As a software engineer, I'm often toggling back and forth between Alice and Bob approaches. When I interview candidates, I try to lean in with Alice insights while if they are a Bob and toiling away - it's hard to have the Bob discoveries in a 30 minute exercise. I also loved the showing of Newton's notes to remind us that toil leads to insight. Sometimes it just takes effort to see the pattern.

Really excited to watch the video, Grant! Looks like a fun one. One small non-math point that’s just a pet peeve of mine that I wanted to share is how tired I am of seeing Alice and Bob as the go-to names for all A/B things in math/physics. Maybe think about mixing it up a bit next time (e.g., Arturo and Barkha, Antoni and Belinda). Just a thought. Can’t wait to watch!

Thanks very much for this, it really made me feel less stupid. A lot of the time watching Maths explainers I can follow the arguments well but the ease with which the answers fit together compared to working on my own problems of interest is intimidating, but seeing more of how much hard work goes into working these things out in the first place makes the slow rate of progress when doing my own experiments less daunting.

James Matheson

I found myself wondering if the "replacing the sphere with polyhedra" stuff is really as simple as described in the video. It's not hard to come up with a sequence of (non-convex) polyhedra get "closer and closer" to a sphere, but whose surface area doesn't converge to 4 pi r^2 . There will, of course, be convergence for any series of convex approximations, but if pressed, I'd be tempted to do something Bob's side of the problem to show that that's true.

Nate

Spectacular! I clearly recognize this from software development. Hard work is sometimes necessary in order to re-approach a problem with new insights which produce an elegant algorithm.

My hunch is that it's field-dependent. In string theory (my old field, so this may have changed) there was definitely a sense conveyed that "real theorists" reasoned in a purely Alice style, looking at a problem and seeing its analogy to more generic problems, and if you were sitting and doing actual integrals it was a sign of your weak theoretical abilities and that you were being a mere :sputter cough: _phenomenologist._ (String theory is a kind of pathological field, socially speaking) In CS, it's improved a good deal in the past 10-15 years, but you can still see a difference between departments that grew out of math departments and ones that grew out of engineering departments. The former tend to think you're only doing "real computer _science_" if you're proving theorems and bounds and being as Alice-like as possible; being Bob-like is seen as a way not to get tenure. When people come from those departments into industry (which is where most of the significant research has been getting done for decades!) it's often quite a shock for them, as the Bob approach tends to work _much_ better in most of the active problems in CS today, to the extent that people will consider trying to do an Alice approach actively wrong.

Yonatan Zunger

I'm not sure the "rows-to-column" switch is the problem (that's just pushing an expectation inside a sum). It's in a combination of a couple of later steps that the distribution becomes fixed. (I don't know if you want the answer posted here, so I've refrained from spelling it out.) I would say that your "Alice" approach is not entirely analogous to the "Bob" approach. Bob ends up with an answer. Alice ends up with insights and a great idea for a more general proof, but still must work through the details of the more general proof. It is skipping the hard steps of being exact about a more general derivation.

Indeed, just a little "speak-o". I tried something different with this video and went through with no script, so I guess I didn't comb through carefully enough while giving it a final listen.

3blue1brown

Such a good book. Don't ask me why, but I have 5 copies of it on my shelf :) Another great piece of writing in that vein that I just saw this morning is this Alon Amit Quora answer: https://www.quora.com/How-do-you-find-all-positive-integers-N-for-which-the-vertices-of-a-regular-tetrahedron-inscribed-into-the-sphere-x-2-y-2-z-2-N-have-integer-coordinates-see-also-comments/answer/Alon-Amit?ch=15&oid=325718663&share=62abf6c6&srid=36bB&target_type=answer

3blue1brown

It's a great point! I hope by raising the question at the end, and by highlighting at two points that there's a hidden assumption in the rows-to-column switch. The point distribution is a great example, thanks for bringing it up. It's fun to think through why that breaks down the whole rows-to-column switch in the first place.

3blue1brown

Good point. The reason I left it like that it's easier to generalize to other convex solids which don't necessarily have congruent faces.

3blue1brown

Thanks! Thought to be fair, I still ended up not _really_ getting to the interesting question of how exactly Alice specified this distribution. So tempting to talk about Haar measure, but the video was already 40 minutes.

3blue1brown

That's interesting to hear. Would you conclude what's implicitly taught for a lot of PhDs is to aim for elegance over the messiness of specifics? That seems strange, given that the best theorists, historically, also seemed happy to immerse themselves in extreme specifics.

3blue1brown

I'm not sure what you mean by defining randomness; you mean specifying a distribution on the space of all orientations? In the case of the "hardest problem" context, it was choosing four points uniformly on a sphere, and we know that that means because we have a well-defined measure of area for subsets on the sphere (well, measurable subsets anyway...). Defining it on SO(3), the space of all orientations, is not as immediately straightforward, because you have to ask how it is that we're defining volume in this space.

3blue1brown

And to add that point, it often feels like in programming the path to a final result is to first do some quick and dirty implementation that works, but maybe not elegantly, then after the fact think of whether there are cleaner and more general ways to do it.

3blue1brown

Another great video! It gives me Kahneman's System 1 and System 2 vibes from Thinking Fast and Slow.

Benjamin Bailey

Thanks Grant! The video is great and the point at @37:00 is really important and under-expressed. "You usually need to grind through examples before you get the insight. Don't expect it to come without work!" Many students watching these videos or reading textbooks of all the accumulated knowledge of centuries just end up feeling dumb. "I couldn't have done that." This leads to imposter-syndrome and turning away from math (or science, engineering, etc). It's not nearly as sexy, but I'm really starting to prefer books like "Measurement", where the reader is encouraged to tinker around with example after example until the insights start to develop internally.

Gabe

You mention it at the end, but the distribution over SO(3) that Alice assumes is so hidden that she is likely to think her result holds for any distribution over SO(3). A quick examination of this shows it must be false (consider a point distribution over SO(3)). And now Alice is left wondering where she made this assumption (assuming she did this check at the end... or actually took the steps of making the proof formal). This isn't a bad thing (provided she does the check at the end), and in general I'm a fan of Alice's approach (well, really the goal is to do Bob's approach looking for the higher-level version at each step). However, it might be useful to point out that with higher-level arguments comes the need to be even more careful about the premises and assumptions to each step. There are still details to be filled in. These details are more on the level of "does this limit exist?" than "what's the integral of |cos(x)|sin(x) dx?" So, Alice still has work left ahead of her at the end of this video. That's been, unfortunately, left out of the description.

One thig I noticed around 19:00: the formula jumps from the sum over the average shadow area of each individual face (with the face index still included) right to the entire surface area of the cube. It might have been a valuable insight to note that this average is the same for each face, so you can actually get rid of the sum and index and just multiply by 6

Lionel Pöffel

Very beautiful. I totally loved how the time was used to really dive into the nitty gritty that I often feel is glossed about (think missing details about the probability distribution in *every* puzzle about random value of xyz)

Lionel Pöffel

Wow, I really like this. The "which is better?" section is the real gold in here -- it's the thing that we never hear in videos like these, for the exact reasons you say. As a [string] theorist-turned-engineer, I feel this really acutely: I often feel like I've "lost the ability to think like a mathematician" since I've moved so much from being an Alice to being a Bob over the years. I even spread this: usually one of my first tasks if I get a fresh PhD in my group (... or occasionally even a tenured professor) is to disabuse them of Alice-type thinking, because they're about to be in situations where this kind of search for theoretical purity will be a very slow way to get nowhere, and simply messing about with real data for a while will quickly illuminate the path -- but it feels difficult to do that, because I don't think we have a good language for explaining _why_ that kind of shift is often so important.

Yonatan Zunger

I love the art and animation in this one! And what a fun story! Another connection that comes to mind is the "hardest problem" video with the tetrahedron. Could the approach there be used to define randomness here?

Totally excellent!!!!!! Thank you....

Richard Hackathorn

From this, I think I'm mostly a Bob with the ability to wear Alice's hat once I've immersed myself in the details of a problem. And also, it's been a while since I've done either of those, so I think it's time to look for something I can get immersed in. You know, for fun. So thanks!

Frank Wales

Thank you, enjoyable and informative. To add to your point about drilling problems: I'm a computer programmer, and the nature of problem solving (or if you prefer, "debugging") is the same mix of nitty-gritty details and flashes of inspiration. And ultimately, inspiration's training comes from the nitty-gritty. I'd go so far as to say that intuition is, at its heart, the summary of your brain's prior training on the details. So when you drill problems like that, you're improving your ability to figure things out - not just to figure out the same things faster, but to actually be able to grasp new classes of problems, because the field of complexity becomes larger. Having the light source closer would introduce many new complexities, it's true, and calculus would be a good start; but if you were to ask brilliant mathematicians to tackle the problem, I wouldn't be surprised if they start with an intuition-based approach of "let's look in this direction". (Me, though? I'd look blankly at it and go "uhh........ maybe there's a 3b1b video about that". :) I am not a brilliant mathematician.)

Rosuav

At 19:35, I think you mean to say "multiply the total by one half", not "divide..." Dividing by a half would double the total.

Frank Wales

"What is it that Alice does to carry out the final solution?" (30:00) o_O wording... On the general idea of the video I find that it's useful to first find an ugly approach that gets you to the solution and then let the laziness to carry it out motivate you to find the elegant way. So start out with Bobs mindset, but try to become Alice.