Why 4d geometry makes me sad

[@3blue1brown]

There were several other puzzles I considered for this theme, and I made a quick bonus video for Patreon supporters showing two more: www.patreon.com/posts/115570453 Please feel free to share more puzzles like this below in comments, I love this stuff!

[@tehb13]

Another awesome video as always! Not sure if you remember me, but I was the was the contestant who mentioned Monge's theorem at 10:48! (although I had no idea how to salvage the proof lol) About the origin proof from 16:30, a similar idea is used in the proof of Casey's theorem in Math Olympiad Dark Arts - Goucher (2012), so it might be worth looking into Casey's theorem.

[@computerzero2681]

Hi, you can also slide the rhombus in the hexagon.

[@iota_i_1]

Sir , I had thought of a really nice solution, finding a ((1 single mathematical equation which tells about the life or universe or what ever "happening" "happens" "happpened" )) That is :- if we look at the universe which is expanding per unit time ... and considering universe as a system (by thermodynamics) .The universe is trying to attain the chemical and physical equilibrium . I am saying this because, before "big bang" the universe or the system was concentrated at a single point have no physical motion which in terms of physics we call it as equilibrium, it was disturbed by the external work or energy ,done or applied one the system , hence the universe is trying to regain it's constant equilibrium. Causing this all what ever we are seeing today happening beyond our consciousness. Will this help us to get that single mathematical equation ??? If said anything wrong please help me to correct it ,sir !.

[@computerzero2681]

@iota_i_1 this guy trying to answer the meaning of life

[@michaelharrison1093]

I am an electrical engineer and in Power Systems engineering three-phase electrical power can be thought of in terms of 2D fields of rotating quantities. I.E., a three-phase voltage can be viewed as a rotating voltage field on a 2D plane. In a lot of the mathematics that is used in Power Systems engineering it is easier to project all these 2D quantities up into a 3D space and perform all the calculations in 3D space and then finally project the solution down into 2D. There are certain calculations that are almost intractable if you insist on trying to calculate in the native three-phase (I.E., 2D) reference frame that become trivial once projected into 3D. A good example is in Symmetrical Sequence Components which represents the three-phase quantities based on three Symmetrical Sequence Components where the additional of the third dimension provides the computation 'space' required for these three Sequence Components to exist.

[@JonBrase]

4:37 I don't even have to squint my eyes here: it's harder for me *not* to see it as a cube stack.

[@mathmachine4266]

Yeah, I saw it right away, as well

[@gernottiefenbrunner172]

Although after each rotation, at first an impossible one.

[@bwayagnes]

Same I immediately saw the cubes even before it was brought up

[@spitzhorneule]

Omg thank you. Was about to write it. I had huge problems NOT seeing it before. Every rotation was like the cube disappeared or more like the volume of the cube got inverted

[@ccost]

yeah i was like "thats just isometric, wait, rotating it removes a cube??"

[@akoszahorsky1071]

Thanks for the shoutout :) The rest of my investigation about the solution´s origin went on like this: I remembered I heard it at a lecture by my highschool math teacher, Balint Hujter, who said he first heard it from his math teacher Sandor Dobos way back when, and Mr. Dobos cannot remember where he first heard or seen it. So at this point we agreed on labeling the solution "folklore". Hopefully someone who found it by themselves will come along in the comments. BTW Great video, as always

[@ancientswordrage]

Let's boost this comment so 3b1b sees this

[@bruh-zn8ju]

awesome!

[@xentarch]

That's super cool!

[@myrec8883]

I just want to add that I heard that solution too, also in the folklore of Slovak Math Olympiad (around 2005).

[@gloopiee]

Mr Dobos has forgotten more about mathematics than I will ever know.

[@insouciantFox]

24:57 Bold of you to assume such a nerdy cocktail party would have people that don't watch 3b1b

[@KSignalEingang]

This is why I only go to multi-disciplinary nerdy cocktail parties. I might learn something new about botany or mineralogy or whatever, and I can be pretty sure the mineralogist hasn't heard all my dumb math jokes.

[@abbasuccess3155]

@@KSignalEingang 😆

[@malagoke]

it's true, i learned so much on this channel!

[@xentarch]

1 thousandth like

[@kirillsukhomlin3036]

Could work for places where most people speak their local language instead of English.

[@jamesmnguyen]

25:36 4D creature: "Just squint your eyes and you'll see it's basically a stack of hypercubes"

[@arvinderbali]

See ya what you just did there 😂

[@elektr1x7881]

and then when they're leaving after you ask them for help they say "bye!" and shrink into nothing

[@phnml8440]

To me it's really interesting how especially dimension 4 is so special since many phenomena become somewhat trivial or uninteresting in too high of dimension and "peak" in dimension 4

[@animowany111]

I just love something about the 4D sphere packing solution. It's such a nice, neat and intuitive 4D crystal, and all the distances work out just perfectly.

[@GeorgiiPotapov]

I often wonder if it is because dim 4 is special, or if it is only because 3-dimensional humans have troubles coming up with questions that are not inherently low-dimensional in some sense

[@wewladstbh]

@@WindyHeavy You also have the super strange exotic R^4 and the fact that the group of equivalence classes of spheres is abelian except for n = 4 - I wonder how the latter is related to what you've said...

[@Elitekross]

​@GeorgiiPotapov that is a very interesting question that will haunt me for life now 😂

[@mr_rede_de_stone916]

In physics, dimension 3 is often the hardest: statistical models like the Ising model often reach some kind of simplicity in 4 or more D (look up "higher critical dimension") but are super hard to solve in 3D.

[@rubensleite5838]

I recently fixed a bug in a dual quaternion library in python. The bug happened when I tried to interpolate between two quaternions that were at 90° to each other. After a month I realized that if you walk 90° on the surface of the 4D sphere something bizarre happens, you are teleported to the other side of the sphere. There's no way of imagining this continuously, but the video of the Hopf's fibration is very useful. When I saw it, I pointed my finger like in the Leonardo Dicaprio meme, because that's exactly what I felt without being able to see it. Somehow, walking continuously on the surface of the sphere, the signal inverted wildly. It's incredible to be able to describe something and gain intuition about an object that we can't conceive of, for me there's something spiritual about it.

[@benayers8622]

Imo this is how things defy physics or vanish effortlessly, its like standing to the left of mario in 2d space then walking around him 90 degrees to his '3dleft' he could never comprehend where you went even tho ur right there still :)

[@abebuckingham8198]

Yeah in four dimensions rotations and reflections become indistinguishable. You can sort of see how for any plane of reflection in 4D that there is a whole circle of rotations that are orthogonal to it so you can move along those instead of reflecting.

[@bryanreed742]

​@@abebuckingham8198I must be misunderstanding. It sounds like you're saying there's a way to compose a bunch of determinant 1 linear operations to get you to the same place as a single determinant -1 operation.

[@abebuckingham8198]

​@@bryanreed742 if you define rotations to be members of the special orthogonal group rather than geometrically the two rotations for a reflection no longer works. However if you do this then the with the central inversion being C_2={-I,I} we have that SO(4)/C_2 is not simple, unlike every other even dimensional case. So left and right isoclinic rotations are not conjugate to each other.

[@logantodd5943]

I am an electrical engineer, not a mathematician. Despite this difference I adore math to no end. With respect to the "There are not many uses for these things" This is not true! I have personally used Monge's Theorem building laser pointing systems! Although when Building it I did not know what Monge's theorem was I more or less "figured it out" (I kinda brute forced a simulation to confirm all coordinates I cared about were fulfilled) to create a viable I/O control scheme and now I am learning it has a name! It is things like this that truly make me happy. Keep up the great work 3B1B!

[@michaelharrison1093]

I am also an electrical engineer and in Power Systems engineering three-phase electrical power can be thought of in terms of 2D fields of rotating quantities. I.E., a three-phase voltage can be viewed as a rotating voltage field on a 2D plane. In a lot of the mathematics that is used in Power Systems engineering it is easier to project all these 2D quantities up into a 3D space and perform all the calculations in 3D space and then finally project the solution down into 2D. There are certain calculations that are almost intractable if you insist on trying to calculate in the native three-phase (I.E., 2D) reference frame that become trivial once projected into 3D. A good example is in Symmetrical Sequence Components which represents the three-phase quantities based on three Symmetrical Sequence Components where the additional of the third dimension provides the computation 'space' required for these three Sequence Components to exist.

[@AlanTheBeast100]

@@michaelharrison1093 I'm devastated that you don't use quaternions.

[@DuckGia]

Haha. Maths is more powerful than that. I'm not a fan of those broken "applied math" kind of thing, tho.

[@thebigfew4261]

​@@DuckGiaas an electrical engineer who wanted to be a mathematician as a school student, I can tell you that it'd be amazing if you could stay in the pure math field and build your career there, It takes a lot of persistence and passion. Still, applied math is also amazing. You'll find applied math beautiful in many ways the further you go there. What 3b1b is doing is also applied math in a way. He takes a problem from pure math and programs a simulation of it to teach everyone about it, and I find it fascinating.

[@AlanTheBeast100]

@@DuckGia Math has applications, funnily enough - and that is how most of it came to be: to solve practical problems. Engineers have to be strong at math, but they don't wander down the myriad avenues of mathematical abstractions except where it is a tool for their work (and yes, some engineers are also into "math" that has nothing to do with engineering).

[@StarkRG]

The problem, and what's really, truly sad, is that four-dimensional beings would never be able to make those leaps to solve 3D puzzles in 4D because four-dimensional beings universally hate math and geometry. It's not that they can't, they just won't.

[@FireyDeath4]

How did you get that information

[@feetfungus19]

@@FireyDeath4 Met a 4D guy once. He HATED Math.

[@FireyDeath4]

@@feetfungus19 Great. There's been quite a few trionians who hate math, too How did you figure you were talking to a tetronian specifically, anyway

[@killerbee.13]

it's because knots don't exist in 4D, without knots, there's just no point in doing geometry. truly tragic

[@FireyDeath4]

@@killerbee.13 4D knots are made of 2D surfaces

[@janosostor3291]

It's of course worth mentioning that there *is* such a thing as analytic intuition, intuition for which possible logical move to make in pursuit of a proof. This can be entirely devoid of the "geometric" type of intuition, and it is no doubt the type of intuition which best translates to problems without clear geometric insights.

[@academyofuselessideas]

I was precisely thinking about this. The advantage of 3D intuition is that it's build in (I am not sure but I assume that parts of animal brains are pre-wired to make it easier to "see" 3D). On the other hand, building mathematical intuition for objects that you cannot see takes some deliberate effort. This is somehow related to Grothendieck's Rising Sea metaphor: You can either fiercely attack a problem with whatever you know, or you can take time to build intuition about it (and hope that eventually the problem will be easily solved). The two approaches might actually be the same, and perhaps a desirable talent for a good mathematician is to figure out how to break down a problem in a way that allows her to build some intuition about it.

[@Aesthetycs]

Analytic intuition is just machine learnings of proof, finding patterns in syntactic proofs. It does not help you build intuitive concepts.

[@academyofuselessideas]

@@Aesthetycs There is also this type of intuition... but in my experience, finding patterns in proofs only takes you so far... Let's say you see a proof that uses a particular technique, then you can use that technique/pattern to proof a bunch of new things. All this is good but it is not the same as finding a new creative idea. When you spend time with mathematical objects, you develop a different type of intuition similar to how you have a feeling for 3D objects... though, now that i write this, it is possible that all those intuitions are just pattern recognition... very philosophical

[@juliogama2451]

These animations never fail to completely wow me… thank u so much for the insanely amazing content

[@BetaTestingUrGf]

1:05 What I'm told: Three rhombus shapes rotating, what i see: Stacked cubes, being added/removed, with a rotating fade effect

[@Player-Leli]

YES!

[@anstropleuton2]

YES! As someone who played Minecraft, the first question looks very obvious Just think of it as looking a 3D stack of cubes from an isometric (orthographic?) view The rotating of those hexagon looks like removing or adding a block

[@anstropleuton2]

Wait, he just revealed my way of thinking

[@portalwalker_]

I thought this was trivial but then he explained it at 4:30 and I thought how someone couldn't immediately notice that? Aren't human brains designed to infer depth onto 2d images automatically?

[@liuby33]

seriously I cannot see this as 2D as my brain doesn't allow that

[@craiggersify]

One thing about the recent 4d [mini]Golf game that came out was how quickly one gained certain intuitions for navigating 4d space; it’s kind of like the WASD keys are the sliders on the 3b1b hypersphere video, except locked into one’s own intrinsic sense of 3d navigation. It was kind of miraculous - you could feel yourself internalizing the geometric relationships at the level of moving objects instead of just intellectual analysis. It was like it bootstrapped me into having more direct intuitions about 4d space; although I must confess I still have trouble with some rotations, so really it’s like my mind’s eye can be 3.5D instead of some 4d hyper-eye.

[@mihailpramatarov]

the same with me and the magic cube 4d game.

[@perhapsyes2493]

I'm guessing it's to do with the decades of videogames, but when I look at that first example around 1:35 I inherently see depth. I see a level a character could jump about on, like Q*Bert.

[@Patterner]

Q*Bert was my first thought 😂

[@ProjectionProjects2.7182]

I know I saw it as 3D immediately.

[@Mightyzep]

@@ProjectionProjects2.7182I kept trying to not see it as 3d because I was worried it would confuse me later on. I didn’t think it would wind up being part of the solution!

[@ProjectionProjects2.7182]

@@Mightyzep Yeah thats the same thing that happened to me. I thought "oh I better stop doing that it will confuse me". Then 3B1B went like "on you can visualize this in 3D". 😂

[@freyc1]

That's just the way human vision works.

[@sender1496]

I'm amazed at the number of people who don't realize that the first puzzle was solely CHOSEN because it's so easy for us to see. People are making it sound like there is something special about immediately seeing cubes. It is the perfect example of higher dimensional intuition because nearly anyone can figure out that it's equivalent to stacking cubes.

[@jademonass2954]

its just that he makes it sound like we WERENT meant to see it, at least during the explanations

[@sender1496]

@jademonass2954 He definitely did, but on the other hand, had he said that it was obvious, that might have made quite a few people who actually didn’t see it feel dumb. I’m by no means saying that it was a calculated move, but if he did in fact aim for the reaction that people would feel smart if they got it, then that would be a genius way to make people engage in the video.

[@SirNobleIZH]

17:23 missed opportunity for a pi-ramid

[@ragunanthan787]

Wow 😂

[@Grayson_Wu]

That's some Parker joke right there

[@andersjjensen]

I take it you'll see yourself out? :P

[@Excalibaard]

That's pretty (pira) mid

[@Xboy1207]

Unrelated, but I think the reason that it’s hard to grasp 4D structures is that we graph them in 2D. Imagine trying to graph a 3D object in 1D, then you’ll see why it couldn’t work.

[@Kishblockpro]

I have had this thought for years and years, it’s nice to see someone say else mention it, we should have some kind of 3-d space to do this

[@Overcrox]

I suppose that’s a consequence of the fact that a lot of math is done on 2D surfaces (paper and screens). We need real-world 3-dimensional math aids for 4D math.

[@blinded6502]

Yeah I'm working on it

[@PaoloLammens]

I've thought about this too, but the problem is that even if we had a 3D model, since we are 3D creatures, we can only ever see a 2D slice of it. When you draw a picture on a plane, you can see everything all at once, but if you have a 3D object, your eyes can only see its surface from different angles, and cannot see its "insides". We would need to be 4D creatures for that :(

[@benayers8622]

@@PaoloLammens shrooms n ket will get ya there 🤣

[@devd_rx]

if we were able to think in 4D naturally, we would find ourselves complaining about not being able to touch into 5D, and that would be true for all the higher dimensions, such is the insurmountable burden that no creature can overcome

[@uhrguhrguhrg]

4d hogs a lot of the regular shapes though. And it kinda gets samey after that

[@FireyDeath4]

Can you break the cycle in Hilbert space as an apeironian

[@fgvcosmic6752]

Hilbert space beings go brrrr

[@good_brake]

Just learn to think in nD, and specialize to a particular value of n when necessary.

[@good_brake]

​@@fgvcosmic6752Banach space beings: "oh you are still just a baby."

[@saniancreations]

Here's how I intuited my way through problem 3 without doing any fancy math tricks: At any point you can rotate the whole scene to make the white line perfectly horizontal. Suddenly, the horizontal line has become an actual horizon and the circles have become spheres in a 3D space. Now imagine the spheres as all being the same size, they only look smaller or bigger because of how close they are to the camera. The only configuration where that is possible is if the spheres were laying on a flat plane (the plane which disappears at the horizon). Because the spheres are exactly the same size, any two lines that connect them must be parallel lines in the 3D space. And now it makes total sense that the lines converge on the horizontal line, because according to the rules of perspective, parallel lines always converge on the horizon!

[@timothemalahieude5076]

I thought the same thing! But it doesn't work in the counterexample 3b1b mentions because one of the tangent lines appears behind the horizon

[@johnjameson6751]

I am also a fan of converting a problem in affine/Euclidean geometry to one in projective geometry, then using a projective transformation to send a crucial part of the configuration to infinity. Then many other lines become parallel, making the problem easier to solve.

[@jpe1]

5:53 the moment you “remove” the last cube, my mental representation of the space into which the little cubes are “placed” switches from looking like a depression into the screen and instead looks like a projection out of the screen. Then, once cubes are “added” back in, they look like they are sitting above the projection until a critical number are added (about 12) at which point the whole thing snaps back to looking like a depression into the screen with little cubes inside.

[@jayeff6712]

This happened to me too. The brain is a curious device.

[@TheElexec]

Yes! That starled me a little lol

[@cameron7374]

I always need to do really weird mental gymnastics to flip one of these the other way round when I'm already seeing it one way. (No matter if colored or not. If anything, colors make it harder.)

[@angeldude101]

At some point you realize that the cube is both filled and empty at the same time, and that adding a cube in one universe is removing a cube in a parallel universe. With just a single cube, you can't tell if it's full or empty, and with two cubes, you can tell that one is full and the other empty, but not which is which.

[@julianbaader4351]

It would have been really nice to always have a little rotational oscillation to reaffirm the 3dness

[@SpiritOfHugs]

Thank you so much. We really needed this to bring us a sense of wonder and excitement once again in these trying times.

[@Gears_AndGrinds]

If anybody was wondering, the "sliding cubes through the origin" move described in the 5th puzzle can be described more formally with a central inversion. Think of it as taking the vectors and negating all of their coordinates (sliding and centrally inverting are technically different but they give the same result due to symmetry)

[@columbus8myhw]

I'm pretty sure you could _also_ do it with a boring 60 degree twist (assuming the parallelopipeds are indistinguishable)

[@evildude109]

I was thinking about this, but it doesn't look like the inversion operation. The cubes come out of the operation purely translated, not an improper rotation you'd expect with an inversion.

[@Gears_AndGrinds]

​@@evildude109 Sure, but under the parallelopiped's symmetry, doing both actions makes it look the same way in the end. I guess I should make my comment clearer regarding that though

[@johnjameson6751]

I find it more natural to describe it as a central inversion, because it explains why a 60 degree twist clockwise or anti-clockwise have the same effect.

[@charles.e.g.]

Your videos are always so mathematically creative and visually stunning that they never fail to leave me with this extraordinary sense of childlike wonder and awe. What a gift. Thank you. ❤

[@ericthiel6553]

I found monges theorem becomes very intuitive, when you think of the three spheres of beeing equally sized spheres under perspective projection. the points of intersections can be thought of as vanishing points and the line they define as the horizon

[@dmdeemer]

This was my thought as well, I was very surprised when he drew the tangent planes to different-sized spheres.

[@DeanCalhoun]

right, that’s what I saw at first. plays on the viewing angle trick of the cube stack as well

[@beautyofmath6821]

Yeah I agree, still under the assumption that we can way three circles to any three circles under a homography

[@timothemalahieude5076]

I thought the same thing! But it doesn't work in the counterexample 3b1b mentions because one of the tangent lines appears behind the horizon

[@ansiwen]

Came here to write exactly this comment, but I was humble enough to check first, if someone had the same idea. 😅

[@AbhinavKumar-nh8dl]

18:05 now this puzzle becomes really easy for us 12th grade students in India as we are taught to find vlomue of parallelopiped and tetrahedron in vectors so basically if you have all four points of a Tetrahedron take one of the four points as A and describe 3 vectors along AB , AC AD where B C and D are the remaining three points now just take cross product of any two of the three vectors and take the dot product to the cross product and find the magnitude you get is 6 times the volume of a tetrahedron which is the volume of a parallelopiped we also described this as [ AB AC AD] which is (AB×AC)•AD !! Which gives us the volume required now The problem is our professors were not really able to visualize us the proof so I will be waiting for you to help us out thank you!!!

[@QiYueWang-l4j]

Very glad to have the chance to hear the first 3 puzzles in the actual conference hall of 3b1b and it genuinely inspired me so much during IMO! (I still remembered the moment when the girl asked what if the three spheres don't have a common surface) Thank you for another amazing video and all the work you have done❤

[@aaryamangupta]

Were you a participant?

[@nicezombie8054]

yeah I immediately was like, ow I remember these puzzles, but from where, and then I recalled that it must've been from the lectures in the IMO, amazing to have been there and keep it up

[@nicezombie8054]

I still vividly remember spending the night then solving the puzzles

[@isaac10231]

​@@nicezombie8054how was it at the imo?

[@mrazo]

Do we all agree that Grant is not only an exceptional communicator but also a true artist? After following this channel for years, I’m continually amazed at how far he’s taken his craft. The way he merges rigorous mathematics with captivating visuals has profoundly shaped how I understand math itself. Many of the abstract concepts I’ve encountered now effortlessly fall back into the visual experiences I’ve had watching 3B1B. Thank you, Grant, for shedding light on the beauty of abstract reasoning and making it feel both intuitive and accessible.

[@rolfvdhulst]

As a PhD student working in optimization, I would like to say that there are actually many applications of this idea in practice. Many optimization problems from practice can be formulated as finding extreme points of very complicated polyhedra that have an exponential number of vertices in some high dimensional space. Sometimes, it may be possible to model this very complicated polyhedra as a much simpler polyhedra with only a polynomial number of vertices, in an even higher dimension space. One can then solve the optimization problem on the complicated polyhedron by finding an extreme point for the simple one in a higher dimension. A group of researchers even won the Gödel prize in 2023 for showing that for the travelling salesman problem, where ones attempts to visit a list of cities in order using the shortest cycle, by showing that there does not exist such projection for this problem.

[@jackm.1628]

Do you have any example where there exists such a projection into a higher dimensional space?

[@AZALI00013]

this might be the coolest math video ive ever seen

[@jwjustjwgd]

pov every 3blue1brown video

[@litterbox019]

azali spotted on 3blu1brown??

[@jwjustjwgd]

@@litterbox019 fr

[@litterbox019]

@@jwjustjwgd someone tell him math videos get even cooler

[@Kishblockpro]

AZALI OMG

[@roy04]

I might be wrong here, but for the second puzzle, I found it simpler to think about non-parallel strips as strips that are overlapping - they add to the total width, but will always add less area to the total area (due to overlaps) than parallel strips.

[@cameron7374]

That makes sense intuitively, but how do you know that there's not some odd, counterintuitive solution that manages to do better with a little overlap.

[@geoffreysimms2520]

@@cameron7374 The planks must AT LEAST cover the circle. If they are not all parallel, they must also have some overlap. QED. I don't see why this one needs more complication.

[@cameron7374]

@@geoffreysimms2520 Because overlap or not says nothing about the width of the planks being optimal to fill the shape. This happens to work for a circle, but if you'd need to fill a U-shape the optimal solution would involve overlap in the 'corners'.

[@geoffreysimms2520]

@@cameron7374 wasn't the problem statement for a circle and not a U-shape?

[@xenontesla122]

@@geoffreysimms2520 They're explaining that your logic works for this particular case, but in math you have to be very specific about what cases your proof works for (I think in this case the shape has to be convex). That's why I'm an engineer and not a mathematician 🤣

[@amaury31415]

At 17:15, when you say that these 3 centers of similarity always must fall on a line, I suddenly remembered my drawing classes from high school : This line is actually the horizon line, when drawing 3D objects in perspective.🙃

26:00 "not a lot of direct utility". Your statement is incomplete. We just haven't found the utility yet.

[@chocoblocko9]

fast forward to 2078 where this is like the backbone of some new encryption method that's used everywhere

[@damjandenic]

Okay nerd

[@stardragon8585]

Spoken like a true software engineer

@@stardragon8585 how do you know how I spend my paid time?

[@simonpfennig1203]

After the first puzzle was intoduced i realized, I had to rewatch the first minutes since the animations were to mesmerizing for me to pay any attention. I absolutly love your videos.

[@vytah]

26:54 Why that error correction code works and is unique, is nicely explained in Another Roof's video "Why Do Sporadic Groups Exist?" I'm somewhat surprised it's related to sphere packing as well, although perfect correction codes are somewhat related to spheres, so maybe it shouldn't be so surprising.

[@karolakkolo123]

Group theory is a beautiful thing, and something that you don't truly appreciate until you work with geometry/tilings to solve a problem. The reason why they are related to sphere packing is because highly symmetrical infinite groups can be related to lattices in N-dimensional spaces. I used to wonder why do mathematicians bother with space tilings other than for curiosity, but actually studying them introduces you to the ubiquity of symmetry in math and the natural world, and often abstract representations of stuff found in the natural world form highly symmetrical tilings of space. It's just so fascinating

[@DoctrDoc]

18:30 I am not being factitious this is just the only way I can think of to do this. Step 1: Translate the triangle so one of the 3 points lies on the origin, let this point be (x1,y1) reflect the triangle as necessary so all 3 points are in the positive quarter of the X,Y plane. Step 2: let sides A and B connect to the origin, let B be defined as the line segment with the greater slope. Let (x2,y2) be the second point of B. let line segment C connect (x2,y2) to (x3,y3) Step 3: calculate the slope of A let it be m, B let it be n, and C, let it be o Step 4: f(A) = mx, f(B) = nx, f(C) = ox Step 5: Use a definite integral from the origin to coordinate (x2,y2) calculate the areas under B as b and A as a, subtract a from b, let this value be T1 Step 6: Use a definite integral from (x2,y2) to (x3,y3) to calculate areas under A as a and C as c, subtract a from c let this value be T2 Step 7: add T1 to T2. Step 8: Place a curse upon Ancient Greece for inventing geometry (okay this step is a little bit factitious) and I "know" the integrals have to be related to your linear algebra determinant prompt just no clue how to connect them.

[@Lord_zeel]

22:20 I was so excited to see the rhombic dodecahedron. It's my favorite non-platonic 3D shape! A really nice aspect of it is that if you use it to make dice, they are just as fair as platonic solids. I never liked that a d4 is hard to roll, and alternatives to a tetrahedron typically are either also hard to roll, or are just another existing die shape relabeled making them hard to pick out from a bunch. But the rhombic dodecahedron is perfection, you can label is 1-4 three times producing a fair d4 that rolls very well and doesn't look like any other die. I actually 3D printed some of these, and they're super nice.

[@karolakkolo123]

Cuboctahedron, which is a dual of that shape, is also cool. They are both two different logical generalizations of hexagons into 3d space!

[@BrooksMoses]

@@karolakkolo123 The juxtaposition of cuboctahedrons with the idea of non-platonic dice that have multiple faces with the same number makes me wonder: Can we make fair dice with non-platonic shapes that have two (or more) different face types, using multiple faces with the same number? A cuboctahedron only works for a d2 because the numbers of faces of each type don't have much for common factors, but perhaps there are versions that work for higher numbers?

[@Skiddla]

18:10 "i'm not going to show you the full answer to this puzzle" i am the pi guy on the left

[@drag0nblight]

This channel still proves I'm a visual learner. Kudos for the animation skills you have.

[@wyboo2019]

for the volume problem, i don't understand the need to express it as a 4x4 matrix determinant. if you consider the tetrahedron as being half of a parallelepiped, you can calculate the volume of the parallelepiped using a simple 3x3 determinant. pick some vertex v1 and translate the whole tetrahedron s.t. v1 lands on the origin. then, the sides v2-v1, v3-v1, and v4-v1 determine the rest of the parallelepiped. since the determinant is the (oriented) area of the unit volume under the linear transformation where the basis vectors land on the columns, if you make a 3x3 matrix where the columns are v2-v1, v3-v1, v4-v1, then you calculate the determinant, you'd get a formula for the (oriented) volume of the parallelepiped. then, cut it in half and take the absolute value and you get the volume of the tetrahedron. no need to add and extra dimension, and it's not hard (i havent tried the challenge in the video, so maybe the 4d version is more elegant or something, but this is an easy solution)

[@galoomba5559]

It's one sixth of a parallelepiped, not a half, but otherwise yes that seems correct

[@Aldrasio]

Another way to think about the rhombic dodecahedron tiling while remaining in 3D space is this: Imagine you draw 6 planes, each of which goes through the center of a cube and connects each edge to its opposite. These planes divide the cube into 6 identical 45 degree pyramids, with the tips of the pyramids all meeting in the middle of the cube. Since the cube tiles 3D space, any subdivision of the cube must also tile 3D space. So these pyramids, when arranged this way, must also tile 3D space. Now we arrange the 3D tiling so that it alternates between cubes and 6-pyramids in the cube arrangement, such that the face of each 6-pyramid is touching a cube and the face of each cube is touching a pyramid's base. Again, we have something that tiles 3D space. Now take one of those cubes with a pyramid on each side. Since the pyramids are 45 degrees, we know that a pyramid adjacent to another on the cube must have one parallel side, and so we can say that the 2 triangular sides become one rhombic side. Each pyramid has 4 faces, and 4 * 6 / 2 = 12. So you have a rhombic dodecahedron now. Since the cube/pyramid combination tiles the space, and since you can use the same arrangement of cubes and pyramids to construct a rhombic dodecahedron, it follows that a rhombic dodecahedron must also tile the space.

[@beansprugget2505]

This makes sense to me but is really hard to visualize. I can see it if I draw a line across each rhombus of the dodecahedron: that forms a square which is a side of the cube.

[@ilmionomenonloso]

It's just incredible to me how high you set the bar during your youtube carreer, and how you always manage to keep the level so high, and possibly even getting better over time! Hands down the best math channel, by far

[@stefanklass6763]

OK, I think that I have a very intuitive solution to the 1st puzzle, so I'm writing this comment before continuing to watch. I think you can just imagine the hexagons as cubes, then see how you add or remove one cube from the whole scene (My brain interpretets what it saw as 3-dimensional right away). The "no hexagons" tiling is just another isometric projection of that, isometric projections don't need to conserve angles so you could think of it as the same thing. The extreme states are: a completely empty cube and a completely filled cube. Every step adds or removes one small cube from the large cube and there are n^3 small cubes so it would take n^3 steps. Let's see if I missed something or my interpretation doesnt hold up😁. Edit: Alright, i'm pretty proud of myself right now. the 1st thing that came to mind was correct.

[@BenSpitz]

The work you put in on these visualizations is incredible! Congrats

[@annmirren6132]

for 4D specifically i have found that i developed some, if rudimentary, intuition for higher dimensional geometry by playing the game 4D Golf by CodeParade. of course this doesn't quite solve the problem of developing general intuition for how and when going from dimension n to n+1 might help, but it is still something.

[@Ze_eT]

For me, I gained some intuition from 4D Toys; it is surprisingly enough still getting updates.

[@rokaslokusevicius3810]

I gained some inituition (also for 5d) from 5d chess with multiverses and time travel. If you swap the y axis (doesn't actually exist in the game) with the time axis, you still get a 3d space, where the pieces move in any direction. You can imagine 1 square as a cube, where the piece is located. 5d works the similarily, you just imagine a line of 4d boards where the pieces can also move across the boards like they move through time and space (it's how you get 2 or more kings and other pieces). With 5d you can move a knight 2 turns back in time and 1 board in the multiverse (or vice versa) without moving it in space. The human brain can't understand 4d without learning it for a long time separately, so the only way to completely understand it is being a 4d entity and seeing in 3d, meaning you have to see everything in the observable universe at once. That would drive any human (or other 3d being) insane.

[@xiphosura413]

What 4D golf really helped me grok was actually what 3D space might look like to a 2D being. In 4D golf you are stuck looking at a 3D "slice" of the 4D course, and you can change this slice to be entirely contained within the course so it appears closed to you on all sides. It struck me that a 2D perspective of a 3D golf course would be similar, from one angle (lying flat on the golf course) it looks entirely sealed off from the world, but tilted from another angle suddenly you can see how short the boundaries of the course are.

[@newmerek]

@@rokaslokusevicius3810 We don't see everything in one 2 spatial dimension plane slice of the observable universe at once so what you are saying doesn't make sense. You could have 3 spatial dimension vision without seeing everything in the observable universe. Our current vison actually has some 3 spatial dimension aspects because while our light receptors are on a two spatial dimension area, that area is on the surface of our 3 spatial dimension eye and our eye is curved, if we had a flat 2 spatial dimension plane at the back of our eye then the signal sent to our brain would be different.

[@rokaslokusevicius3810]

@@newmerek we see 2 2d planes, but from different locations. Our brains receive 2 images and combine them for depth perception. Seeing in 3d would be seeing every possible 2d slice of the universe at once, and probably impossible in this universe. A 4d entity's eyes would be 4d, see in any 3 dimensions and probably have depth perception for the last one.

[@EannaButler]

13:30 - no waaay! What's amazing is the video are this point showed me the reason why they are all on the same line in 2D space, moments before you explaining it in words. But I had already just conceptualised the scenario before you got a chance to say the words, because of your brilliant video content. And then, you said the words! Which put into English the confirmation of my anticipatory understanding. You "collapsed the wave function" of my understanding, why going to a 3D projection is the mechanism to prove a 2D conjecture, in my head in a moment 🙂 This approach is revelatory!! Amazing... Thank you! I had already solved this problem in my head tho, of minimising the widths of pairs of parallel chord strips, by a reasoning in 2D space. For me, the solution had to be the base case of a single pair of chords that are tangents to the circle, having a width of two. Any other 'random' combination of parallel chord-pairs involves duplication of the area inside the circle, due to the necessary overlaying the areas in any non-parallel chord arrangements. In 2D, if you want to minimise a quantity like area, you have to minimise the "amount of area" covered by the strips. You can't afford to have overlapping strips if you want to minimise the strip widths - it wouldn't make sense. So, to minimise the total area of the circle covered by the strips can only hold when all strips are parallel - which in the simplest case is the two tangents to the circle, meaning the answer to the total stip width that covers the circle has to be 2, for the unit circle. But - that feels nearly like a qualitative answer, a logical reasoning over the 2D projection of the broader problem. "Going 3D" gives the Mathematical proof.... Wild!

[@rms_txrx]

The 1st puzzle was a cool reminder of the GameCube animation!

[@patrickskelly8517]

If you hold z you get monkey sounds

[@matricepeinard7879]

I still have no experience in solving or translating geometry puzzle to mathematic demonstration, and struggle to understand the formulas. But the presentation and demonstration is so clear the 29 minutes just flew by. This is fascinating to look and think about. Thank you for translating it

[@dominikmuller4477]

"Pi-ramid shape". I see what you did there.

[@StaphyJr]

This might be your best-looking video so far. Great stuff!

[@jovi_monet]

Your visualization of the perpendicularity of the pairs of random vectors in higher dimensions at the end screams entropy to me. Wonderful video, thank you!

[@Icefox297]

Can you elaborate more? I am curious

[@yoavravid7893]

About the Tarski Plank Problem there is a solution that involves strips that aren't parallel if you allow 1 dimentional strips. You could take a strip that covers the entire circle other than a single point on either side and cover these 2 remaining points with a 1 dimensional line saving 1 point of width in the process.

[@samuelwaller4924]

could that not be repeated indefinitely?

[@yoavravid7893]

@samuelwaller4924 no. The next time you do it you trade 2 points for 2 points.

[@anjansharma8631]

I have watched the 4π² proof video and remembered why area of strip would be πd

[@tuatarian6591]

Really good video. The second porblem is completely insane I really enjoyed that one

[@mashmachine4087]

I think I managed to prove the circle with strips puzzle a bit differently, and I wanted to share it When you put down a strip, it will cover a certain amount of the EDGE of the circle The edge of the circle MUST be covered (otherwise there will be a small amount of the circle not covered) Each strip's width is always greater than or equal to a chord whose points intersect the circle at the edges of the strip Each strip always accounts for two sections of the radius (where it goes in and where it goes out), and those strips must be of equal length -Strips that go off the edge of the circle can be ignored, as it would be inefficient to make the strip go any further than the edge of the circle So, instead of a full circle, you can imagine a semicircle, and instead of strips, you can imagine putting down a bunch of chords which go along the edge. What we want to find is the shortest path made of one or more chords, such that each point on the edge of the semicircle is underneath a chord And so you can see that any detour we make to the edge of the semicircle that isn't a diameter will make the path longer, because the shortest path between two points is a straight line It's maybe not as nice of a proof as the one in the video, but I was proud of myself for getting my own answer! :3

[@Reydriel]

This was along the lines of my thinking as well. As soon as you angle a chord strip against another, that creates a "gap" along the circumference that is always bigger than whatever width you might save by shortening that angled chord strip

[@ritahall8148]

Instead of going up 1 dimension to solve it you went down 1 dimension.

[@jwjustjwgd]

I can't follow this line of reasoning at all, none of the sentences make sense to me, I'll try reading it again later to see if I can understand it then

[@mashmachine4087]

@@jwjustjwgd I'm sorry, if I could post a diagram, I'd have drawn something in MS paint to illustrate

[@BraydenIsThatGuy]

Couldn't you just use a square to find the answer. In the video it says to use a half sphere, which could be turned into a full sphere, which could be turned into a cylinder. But couldn't you turn the cylinder into a rectangle by unfolding it and laying it back into the second detention. Btw the first thing I though about in this problem was to turn the circle into a square.

[@harryeast95]

4:51 "if you squint?" You mean it's possible to *not* see it as a 3D representation?

[@thephysicistcuber175]

22:34 one of my oral exam questions for SNS admission was to prove that rhombic dodecahedra tile space. The professors first asked me to consider the shape obtained by a cube by gluing regular pyramids of square base to each face such that their heights are half the cube side length. They first asked me to count the number or faces. Then they asked me to prove that these shapes tile space, which was quite simple at this point.

[@Spammedbyparker]

Aye bruh can u teach me math

[@Notthatkindofdr]

Another astounding and mind-opening video! Thank you

[@antonios-st.frantz6523]

Great video, I think everything would be easier with some projective geometry ideas placed in another video. Desargues Theorem pretty much explains the processes and the limitations. Great video!

[@fierydino9402]

Yeesss!!!! That numberphile episode is one of my favorites too!! Wow I really like this video. Thank you👍

[@javierdaher825]

Spoilers ahead for Puzzle #4: The answer is 1/6*abs(det(x1 y1 z1 1; x2 y2 z2 1; x3 y3 z3 1; x4 y4 z4 1)). Analized it through (3D) parallelepiped volumes, but I couldn't give you an ingenious solution thinking in 4D. I assume it has to do with the fact that "extending" the tetrahedron into 4D by sweeping through 1 unit in dimension 4, yields a 4D volume with the same numerical value of the 3D volume of the original tetrahedron. Looking forward to the follow up video. Great work as always, Grant! Truly the best content on YT.

[@Thiefy_]

Certainly not as elegant as the ideas proposed in video but for those of you wondering here a quick way to think about puzzle 4: Begin by letting your points be letters A, B, C, D. Now to find the volume of a tetrahedron defined by four points, A, B, C, and D, we can use a combination of dot and cross products with vectors between these points. First, pick one of the points-say A-as a reference. Then, form three vectors from A to the other three vertices: AB⃗, AC⃗, and AD⃗. These vectors give us directions and distances from A to each of the points B, C, and D in space. Now, to calculate the volume, start by taking the cross product of two of these vectors, such as AC⃗ and AD⃗. This cross product, AC⃗×AD⃗, creates a new vector that points perpendicular to the plane containing A, C, and D, and its length represents the area of triangle ACD. This is because the cross product between two vectors captures both the area and orientation of the parallelogram they form, and since the triangle is half of this parallelogram, the vector length matches that triangle area. Next, we take the dot product of this perpendicular vector with AB⃗. The dot product in this step essentially "stretches" the area of triangle ACD in the direction of AB⃗, giving us a volume that corresponds to the 3D shape (parallelepiped) spanned by A, B, C, and D. Since a tetrahedron occupies exactly one-sixth of this parallelepiped, dividing by 6 gives us the volume of the tetrahedron: Or mathematically: V = 1/6 ∣AB⃗⋅ (AC⃗×AD⃗)∣ This method essentially uses vector operations to capture the entire volume, leveraging both direction and magnitude of the vectors formed by points A, B, C, and D.

[@Nevermind445]

*3B1B at* 4:57 : the curious viewers might enjoy taking a moment to pause and ponder and convince themselves it goes the other way around. *Me:* aww he called me curious

[@prismarinestars7471]

During the Monge's theorem segment I was so sure you were gonna bring up a different intuition from 3D: Perspective! Rather than thinking of the setup as three similar shapes in the plane, we can instead consider it as a _perspective drawing_ of three _congruent_ shapes in 3D space that appear to be different sizes because they are at different distances from the viewer. Under this view, the set of converging lines drawn by connecting corresponding points on two of the shapes is really the projection of a set of _parallel_ lines drawn by connecting corresponding points on the shapes in 3D space, and the point of convergence is their vanishing point. Under the perspective projection, all lines parallel to a given line are mapped to rays ending at a common vanishing point, and all lines parallel to a given plane are mapped to rays with vanishing points that fall along a line. In perspective drawing, the most famous of these lines is the _horizon,_ which contains the vanishing points of all horizontal lines in the 3D scene. We know that the centers of each of the three shapes in 3D space lie in a common plane, as any 3 points in 3D space lie in a common plane. Any line drawn between corresponding points on two of the shapes is parallel to the line between their centers, and thus parallel to this plane. Thus, the vanishing points for every set of parallel lines drawn by connecting corresponding points on two on the shapes in 3D space must fall along a line in the 2D projection. From this, we can also see that the theorem would not hold true for four shapes, as four points are not guaranteed to lie in a common plane.

[@poke7661]

25:04 the faces of the rhombic dodecahedron are not 60°-120° rhombi. they have different angles

[@yyeeeyyyey8802]

Oh you're right, if the big angle was 120 then 3 of them in a vertice would make it flat. Do you know what the angles are?

[@crb233]

You can find it on Wikipedia. The acute angle is arccos(1/3). We could also describe such rhombi as having a long diagonal exactly √2 times the short diagonal.

[@geoffreysimms2520]

This caught me too, I started to reach for my box of polydrons and then had a bit of a "wait a minute"

[@karolakkolo123]

@@yyeeeyyyey8802 I don't know what the angles are, but if in doubt, you can always use the dot product

[@maths.visualization]

Hello 3Blue1Brown team! I'm a huge fan of your videos, and they’ve been incredibly helpful in understanding complex math concepts. Your visual explanations are top-notch! 🙌 It would be amazing if you could consider adding a Hindi audio track or subtitles. It would help a lot of viewers from India and other Hindi-speaking regions who want to learn but may find it challenging in English. This could make your content even more accessible and impactful for many people here. Thank you for all the incredible work you do! 😊

[@treborhuang233]

22:37 This might not be true! In fact quasicrystals come from projecting/cutting higher dimensional tilings to lower dimensions. For example a specific angle of a 5D cube to 2D will produce regular pentagonal "quasi-symmetry", which is isomorphic to Penrose tilings.

[@galoomba5559]

Yeah, the jump from "tesseracts tile 4-space" to "rhombic dodecahedra tile 3-space" is not justified. Two adjacent tesseracts meet at cubic *cells*, not faces. Although if you only look at every other tesseract in a checkerboard pattern, you should get the tessellation of rhombic dodecahedra.

[@Bageer1]

In what way is what you are saying contradicting what he said?

[@galoomba5559]

@@Bageer1 You can project a 5-cube into 2D or 3D space with pentagonal symmetry. But no periodic tiling of 2D or 3D space has pentagonal symmetry, so those projections don't actually tile, at least not in any obvious way. The closest you get is quasi-periodic tilings like the Penrose tiling.

[@Bageer1]

@@galoomba5559 Penrose tiling is a tiling so not really sure what your point is. That is why I am confused. Is the complaint that his very brief mention on how the rhombic dodecahedron tilings can come up or be seen didn’t go into enough detail or rigor for your taste?

[@blblblblblbl7505]

What he said is fine. If you take the tiling of 4D space using hypercubes, and then project them onto the 3D subspace perpendicular to (1,1,1,1), then all of the hypercubes become rhombic dodecahdrons. The only bit that you might want more justification for is why these tiles are guaranteed to have no overlap after the projection.

[@arpadkomar3148]

22:20 I'm so happy you mentioned this, I watched it like 3 times

[@taukakao]

And the four-dimensional creature thought to itself: "Isn't it sad that I can't imagine five dimensions?"

[@C134B]

I love how this takes me from understanding basic math, shows me simple math and anagrams to get to tough solutions! I started teaching org chem in college to people because it seemed simple to me and used simple examples to explain complex reactions too, cool to see this in another light

[@ooqui]

I agree to the difference between analysis and intuition being a dilemma. For example, normal trichromatic color vision is 3-dimensional. Its color space can be represented by a 3D cube. However, there are animals and a very few people with tetrachromatic color vision. A tetrachromat's color space is 4-dimensional because color vision scales in dimensionality (n+1) with each additional functional distinct cone class. This means that a tetrachromat- with just 1 extra cone class - can see an entire dimension of new colors, which translates to an unbelievable more amount of new colors in the right lighting conditions. However, understanding tetrachromacy and acutally seeing through tetrachromatic eyes are two very different things. I wanted to SEE for myself how a tetrachromatic color space and 4D colors look like. I achieved what I sought (and even more) after many years of tinkering and I'm now an artificial tetrachromat. So I can directly comment on the "analysis vs. intuition" dilemma. Analysis is good, but intuition is better, although both are married and two sides of the same coin. No amount of analysis has prepared me for the tetrachromatic colors that I can now perceive, but the analysis helps me in understanding what I see excatly and how to understand tetrachromacy and categorize its colors.

[@ToyTrainAdventures-z3e]

How did you become an artificial tetrachromat?

[@ooqui]

@@ToyTrainAdventures-z3e I've made a KZbin video about it on my channel "Ooqui", named "This is how I turned myself into a tetrachromat (VR)". This is a shameless plug, but it explain this condition better than I could ever do in KZbin comments.

[@trueriver1950]

​@@ooqui justifiable plug, especially as it's in reply to a direct q

[@baptisteprevost6835]

Loved the narration for the Monge problem

[@aloosh1375]

For the second puzzle I assumed to start with any random general strip that passes the center (since in the end atleast 1 strip has to pass the center of the circle). Then due to the symmetrical nature of the circle we can rotate it so the strip is standing vertical. After that there are only parallel strips you can place next to the first strip if you don’t want the second strip (or any strips after that) to pass over area that is already covered by the first strip. And as such we get our width

[@3blue1brown]

Why is it necessarily bad to have two strips crossing over the same area? It's possible to have two strips such that one covers more total area, and the other adds more total width, e.g. if the thin one is closer to the center and the thick one is closer to the edge.

[@aloosh1375]

@ Hmm, I’ve given it some more thought so here is a (possibly) better constructed argument. We want to maximize “new” area for a certain width, let’s call that certain width dx. This is a natural conclusion since we want to fill the area pi with the least width. Now we return to our circle with the vertical strip running through it and we move dx out from the edge of the strip that is closest from the center. If we now want to cover that part of the circle as well as covering the most “new” area it would be another vertical strip right next to it. In essence for every strip we are adding we want to maximize new area for the width dx and as a consequence each new strip will be vertical next to the previous strip (since that is the closest to the center without passing over previous strips hence maximum new area) Repeating the process will eventually fill the entire circle with vertical strips, no? Since placing a strip in the outer edge or similar would create less new area for the same width since it is further out from the center. We also cannot place non vertical strips next to our vertical strips close to the center since then they would pass over “old” area hence not maximizing area.

[@Noname-67]

​@@aloosh1375 it's not necessary that we have to maximize the area after every step. It could be the case that the area achieved by 2 consecutive maximal steps is not maximal. In the same vein that a greedy algorithm is not necessarily the best algorithm.

[@Thiefy_]

Certainly not as elegant as the ideas proposed in video but for those wondering here a quick way to think about puzzle 4: Begin by letting your points be letters A, B, C, D. Now to find the volume of a tetrahedron defined by four points, A, B, C, and D, we can use a combination of dot and cross products with vectors between these points. First, pick one of the points-say A-as a reference. Then, form three vectors from A to the other three vertices: AB⃗, AC⃗, and AD⃗. These vectors give us directions and distances from A to each of the points B, C, and D in space. Now, to calculate the volume, start by taking the cross product of two of these vectors, such as AC⃗ and AD⃗. This cross product, AC⃗×AD⃗, creates a new vector that points perpendicular to the plane containing A, C, and D, and its length represents the area of triangle ACD. This is because the cross product between two vectors captures both the area and orientation of the parallelogram they form, and since the triangle is half of this parallelogram, the vector length matches that triangle area. Next, we take the dot product of this perpendicular vector with AB⃗. The dot product in this step essentially "stretches" the area of triangle ACD in the direction of AB⃗, giving us a volume that corresponds to the 3D shape (parallelepiped) spanned by A, B, C, and D. Since a tetrahedron occupies exactly one-sixth of this parallelepiped, dividing by 6 gives us the volume of the tetrahedron: V= 1/6 ∣AB⃗⋅ (AC⃗×AD⃗)∣ This method essentially uses vector operations to capture the entire volume, leveraging both direction and magnitude of the vectors formed by points A, B, C, and D.

[@thabomsiza2502]

5:27 😮 This realization blew my mind. I love this channel very very much.

[@pizzarickk333]

You never fail to make me appreciate your existence.

[@varunvaswani4562]

10:35 I am writing this midway, while viewing the video. Might edit it later, might add on in the replies. I am absolutely mind blown by this video! Very honestly I got the first one(to my surprise!), but didn't get the second one. Yet, it's just so beautiful. Yet to see the later ones. But huge thanks to Grant!!!! Love from India. Your videos have been one of the key motivators for me to continue taking hard problems.

[@varunvaswani4562]

Wow. Replying after viewing it completely...... I loved it! It's one of the videos that make you fall in love once again

[@TheSmegPod]

Unless I'm missing the bigger picture surely the solution to the "strips inside a circle" puzzle is obviously 2, since otherwise you're essentially asking "is there any way to fill a circle of width 2 in a way that adds up to less than 2" which obviously can't be possible

[@geoffreysimms2520]

exactly. That one has a far easier and more intuitive solution. I'm shouting at the screen like at a horror movie: DON'T GO UPSTAIRS! THE MONSTER IS THERE!

[@amagicmuffin1191]

that's the fun and frustrating part of proof based math, it might be obvious as all hell but you don't know until you prove it :p they're plenty of really obvious things that are actually false, I'm sure

[@danielyuan9862]

Yeah the "obviously" is the problem. You can't just call it obvious.

[@silvesteronono4157]

The deck of cards is a very useful model. After much struggle, finally that is the closest I have come to imagining a dimension perpendicular to the three axes in three dimensions and indeed provides a visualization we would see on screen.

[@4g4m3n0n]

Many things once thought as useless brain teasers or curious facts were eventually developed by the right people at the right time into very useful things. I imagine cryptography started as somebody doing fun math exercises. In storage engineering the concept of a parity (used in hard drive RAID 5 for redundancy all over the world) is nothing but an implementation of the curious mathematical nature of the XOR logical operator. I can't help but to imagine that someday, some curious mind somewhere will be able to make sense of 4D geometry so intuitively that they'll be able to find practical solutions to problems that we may not even have conceived yet. Thank you for the amazing work you do in spreading your love for mathematics to the next generations from which such a mind will surely one day emerge.

[@cameron7374]

I imagine a large part of cryptography started as being at war with the Nazis, but boolean algebra (with XORs and the like) did actually start as fun math exercises by one guy, way before the computer or anything like it. He was called George Boole.

[@karolakkolo123]

Highly symmetric N-dimensional lattices can be used to visualize the relationships between musical chords, although the limit to human imagination is probably chords with a collection of 5 pitches, as they can be projected down to 4d space, which could further be intuitited by cracking down on the symmetries! Although I'm not aware of anyone who has fully mastered this, if I ever have some money and free time, I will take a shot at this. It's interesting stuff, there's all kinds of symmetries embedded in the natural world

[@yichenwang7440]

there is a simpler solution for Puzzle 2: 1. put a strip with an infinite small d on the middle of the circle, this cut the circle into two halve (this is necessary because the center must be covered anyway ) 2. for each of the two halve, add an infinite small d strip to cover the most area as possible, and it would be on the side of the strip in step 1. 3. repeating step 2 until all area are covered. now every area is covered with the only solution that uses the smallest possible d.

[@Player-Leli]

I don't seem to understand why the first problem is impossible in an infinite plane... For me it seems obvious that this tiling is like a set of cubes, where every rotation is equivalent to taking away or adding a cube. As long as it's infinite, shouldn't it always be possible to get to any other cube stack? Edit: ok I suddenly understand why... its impossible to get from a flat plane of cubes to anything else... it still seems interesting that technically you can get from any position to a flat plane but not the other way round 🤔 Edit 2: apparently I should've waited for you to tell me about the cube method before commenting

[@ZeroPlayerGame]

you "can" get from any position to a flat plane but it takes an infinite number of moves (and you cannot do an infinitie number of moves in reverse order).

[@-tera-3345]

Yeah, you're not just adding or removing cubes anywhere you want. If you pay attention to the difference between the two tiles, you can only add a new cube in a place that borders three other cubes. That's why the animation going from the empty room to the full one goes in that order. So you need three "walls" somewhere to even be able to add the first cube.

[@freetolisten]

i love that content like this exists on youtube and it's watched by so many people, makes me feel hope

[@michaelgreer7306]

I can't believe I got to see one of my favorite theorems of all time in a 3B1B video! I love the Cayley-Menger Determinant specifically because it not only generalizes to a simplex of any degree and dimension, but also because it even applies to hyperbolic and spherical spaces. In a similar vein, I'm actually currently working on a way to generalize the "minimum norm" problem to any degree and dimension simplex.

[@BlameThande]

That first example is a very good one because it's the kind of thing that will make sense to plenty of people who've had that idle observation of the rhomboidal tiles, even if they don't know much about maths. I remember your older video on topology really blew my mind at the time because I didn't know higher dimensions could be used as a problem-solving exercise as well as to actually describe spacetime models and so on.

[@Living_Murphys_Law]

Last time I was this early Euler had only one thing named for him.

[@null-0x]

after*

[@error_6o6]

lol

[@rolan638]

what i'm most impressed by is the fact that you managed to make your rendering software work with all this

[@JP_Hatecrew]

1:20 I'm just seeing cubes

[@OrionRatzlaff]

Me too

[@pafnutiytheartist]

Keep watching, the cubes are the answer

[@DesistoDeSerFurry]

Me too

[@channelgigas7042]

Me too

[@ProjectionProjects2.7182]

Awesome video! This reminds me of a problem that I figured out about 3D that used 4D to explain how it works geometrically.

[@signordoge8721]

petition to ban 4d geometry because it makes Grant sad. this cannot be allowed.

[@-YELDAH]

Grant and 4D geometry are contradictory theories, and Grant is clearly superior in all aspects

[@thechocopi2146]

When thinking about Monge's Theorem, I imagined that each circle was the cross section of some cylinder. The smaller of the two circles in each pairing was the side of the cylinder that was farther from the viewer, and the tangent lines approached some vanishing point, as in a perspective drawing. As an artist, it was easy to understand that all of those vanishing points would fall on one single horizon line. That's how perspective drawings are created, but sort of in reverse. You start with the vanishing points and work your way back

[@Inspirator_AG112]

Random thought in the comments, but Minecraft has so much in common with 3-dimensional tilings that it is not implausible to discover a math proof in Minecraft.

[@issholland]

Speedrunners already use trigonometry for an added edge so yeah

[@eagle32349]

@@issholland Math is math, you are literally praising someone for doing some quite simple.

[@issholland]

@@eagle32349 #1 you're trivializing thousand sof years of human progress just because you happened to be born on the latter end of that progress. #2 the scope of OP was to find mathematical proofs in Minecraft, wherein I prove if one thing can be accomplished in Minecraft that translates to real life, then there can be others.

[@eagle32349]

@@issholland 1. Trigonometry is trivial by nature, it is literally just the ratios of the sides of a triangle, later made possible to figure out by angle as well. Hell even the complex uses are simple enough, because trig is inherently…not that complex. 2. If you have a triangle, you can do trig no problem. Praising someone for racking their brain in the specific train of thought to think of connecting the dots, which are, stronghold location (with a ton of constraints which narrow it tf down), player location and angle. Anyone with knowledge of trig and its uses is eligible to be that guy. Praising, in this case, is just glazing for no other reason than your own personal lack of experience in the field dictating how you see people are, indeed, partially informed.

[@issholland]

@eagle32349 thanks for proving my point number one. It's only trivial because you have thousands of years of proofs to look to. Something people of the past didn't have, just like someone from the future will say "relativity is so easy the people in the year 2000 were quite dumb" but we didn't have access to future knowledge. Riddle me this, if it's so easy and intuitive why did the people that outlined the theory to trigonometry become heroes in the math world today? Was it because everybody already knew the information? Thirdly, science and theory are based on reproducible experiments that have a founded math basis. Reproducing experiments in different environments is just as useful as finding new proofs because it still forwards human progression - making sure we didn't waste our time learning something flawed. Tldr: if trig is possible in Minecraft then Minecraft is a great vector for people to think about higher proofs just as religion was a thousand years ago but maybe even better.

[@x0cx102]

Yeah after watching through to the end I think this is a really beautiful video. Thanks, 3b1b!

[@PauLtus_B]

5:52 my brain somehow immediately started seeing this as an upside down cube instead of a hollow shape, which made really weird when more cubes were added.

[@-hc__]

same, as soon as he removed the cube the structure suddenly flipped

[@CollinWilliams-by5cs]

Yes, when he started putting them back 😮

[@PauLtus_B]

@@-hc__ I don't know why brain did that...

[@PauLtus_B]

@@CollinWilliams-by5cs It was suddenly weird spatial dissonance, if that's something that exists.

[@abuzarov]

Such a beautiful video! I really enjoyed it. Thank you!

[@emre_ez]

0:51 hexagons are the bestagons

[@adamdzavoronok5396]

My favorite example of such stepping out (in this case multiple dimension) is the Behrends construction of 3AP free progression.

[@duby.industries]

This twirling tiles pattern seems like a really ingenious way to design video game levels. If levels were laid out like this you could rotate the hexagons (made of rhombuses) to give each level a new feel without having to design a new level over from scratch

[@risingsun9064]

Thinking about how to show the that every valid tiling admits a cube stacking in 3D space, the other direction is easy. The center of similarity proof was very elegant, thanks for sharing this!

[@TheMichaelmorad]

This stratagy of looking at a 2d question and making it about three dimensions Has a really nice Hebrew name, מִרְחוּב (mirkhuv). the russian name for this is Выход в пространство. I guess the English equivalent of this would be "spaceification" BTW The hebrew word is made by taking the Hebrew word for space (merhav), taking the consonants and shoving them into the causative verb structure. this means you make it space.

[@Fun_maths]

Did not expect such a comment, ממש מגניב.

[@angeldude101]

Neat! It's fun to see how languages mutate words to represent new concepts, and what concepts are considered important enough to the language for their mutations to become standard accepted words. If the Hebrew word is a verb, then the English equivalent would be "spaceify". "Spaceification" is then turning that verb into a noun representing the action, so it's really a double-mutation.

[@TheMichaelmorad]

@@angeldude101 It doesn't really matter, mirkhuv specifically is technically not a verb but a gerund (like talking). it is just a gerund of a verb which is in a causative verb structure

[@angeldude101]

@@TheMichaelmorad Ah. Thank you.

[@horrorspirit]

linguistics? in my math video comment section? what is this a crossover episode?

[@gamingcat6668]

4:55 the transition between the isotopic tiles and normal perspective tiles was ssooooo satisfiying

[@MrZadeak]

Yes