Why 4d geometry makes me sad

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  • Опубликовано: 19 ноя 2024

Комментарии • 1,9 тыс.

  • @3blue1brown
    @3blue1brown  12 дней назад +426

    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
      @tehb13 12 дней назад +40

      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
      @computerzero2681 12 дней назад

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

    • @iota_i_1
      @iota_i_1 12 дней назад

      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
      @computerzero2681 12 дней назад +5

      @iota_i_1 this guy trying to answer the meaning of life

    • @michaelharrison1093
      @michaelharrison1093 12 дней назад +4

      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.

  • @insouciantFox
    @insouciantFox 12 дней назад +2229

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

    • @KSignalEingang
      @KSignalEingang 12 дней назад +178

      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
      @abbasuccess3155 12 дней назад +10

      @@KSignalEingang 😆

    • @malagoke
      @malagoke 12 дней назад +4

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

    • @xentarch
      @xentarch 11 дней назад +2

      1 thousandth like

    • @kirillsukhomlin3036
      @kirillsukhomlin3036 11 дней назад +2

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

  • @akoszahorsky1071
    @akoszahorsky1071 12 дней назад +964

    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
      @ancientswordrage 12 дней назад +26

      Let's boost this comment so 3b1b sees this

    • @bruh-zn8ju
      @bruh-zn8ju 11 дней назад +1

      awesome!

    • @xentarch
      @xentarch 11 дней назад +1

      That's super cool!

    • @myrec8883
      @myrec8883 10 дней назад +23

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

    • @gloopiee
      @gloopiee 10 дней назад +30

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

  • @JonBrase
    @JonBrase 12 дней назад +2275

    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
      @mathmachine4266 12 дней назад +113

      Yeah, I saw it right away, as well

    • @gernottiefenbrunner172
      @gernottiefenbrunner172 12 дней назад +38

      Although after each rotation, at first an impossible one.

    • @bwayagnes
      @bwayagnes 12 дней назад +56

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

    • @spitzhorneule
      @spitzhorneule 12 дней назад +81

      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
      @ccost 12 дней назад +48

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

  • @jamesmnguyen
    @jamesmnguyen 12 дней назад +227

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

    • @arvinderbali
      @arvinderbali День назад

      See ya what you just did there 😂

  • @phnml8440
    @phnml8440 12 дней назад +2601

    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
      @animowany111 12 дней назад +159

      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
      @GeorgiiPotapov 12 дней назад +340

      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
      @wewladstbh 12 дней назад +62

      @@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
      @Elitekross 12 дней назад +40

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

    • @mr_rede_de_stone916
      @mr_rede_de_stone916 12 дней назад +100

      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.

  • @StarkRG
    @StarkRG 12 дней назад +317

    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
      @FireyDeath4 12 дней назад +16

      How did you get that information

    • @feetfungus19
      @feetfungus19 12 дней назад +122

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

    • @FireyDeath4
      @FireyDeath4 12 дней назад +9

      @@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
      @killerbee.13 12 дней назад +43

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

    • @FireyDeath4
      @FireyDeath4 12 дней назад +21

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

  • @logantodd5943
    @logantodd5943 12 дней назад +1700

    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
      @michaelharrison1093 12 дней назад +86

      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
      @AlanTheBeast100 12 дней назад +9

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

    • @DuckGia
      @DuckGia 12 дней назад +1

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

    • @thebigfew4261
      @thebigfew4261 12 дней назад +31

      ​@@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
      @AlanTheBeast100 12 дней назад +20

      @@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).

  • @rubensleite5838
    @rubensleite5838 12 дней назад +46

    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
      @benayers8622 9 дней назад +5

      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
      @abebuckingham8198 6 дней назад +5

      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
      @bryanreed742 5 дней назад

      ​@@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
      @abebuckingham8198 5 дней назад

      ​@@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.

  • @janosostor3291
    @janosostor3291 12 дней назад +441

    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
      @academyofuselessideas 11 дней назад +13

      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
      @Aesthetycs 10 дней назад +2

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

    • @academyofuselessideas
      @academyofuselessideas 10 дней назад +3

      @@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
    @juliogama2451 12 дней назад +18

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

  • @BetaTestingUrGf
    @BetaTestingUrGf 12 дней назад +817

    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
      @Player-Leli 12 дней назад +34

      YES!

    • @anstropleuton2
      @anstropleuton2 12 дней назад +90

      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
      @anstropleuton2 12 дней назад +21

      Wait, he just revealed my way of thinking

    • @portalwalker_
      @portalwalker_ 12 дней назад +50

      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
      @liuby33 12 дней назад +37

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

  • @craiggersify
    @craiggersify 12 дней назад +22

    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
      @mihailpramatarov 12 дней назад +2

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

  • @perhapsyes2493
    @perhapsyes2493 12 дней назад +451

    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
      @Patterner 12 дней назад +29

      Q*Bert was my first thought 😂

    • @ProjectionProjects2.7182
      @ProjectionProjects2.7182 12 дней назад +9

      I know I saw it as 3D immediately.

    • @Mightyzep
      @Mightyzep 12 дней назад +20

      @@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
      @ProjectionProjects2.7182 12 дней назад +8

      @@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
      @freyc1 12 дней назад +3

      That's just the way human vision works.

  • @saniancreations
    @saniancreations 12 дней назад +20

    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
      @timothemalahieude5076 11 дней назад

      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
      @johnjameson6751 11 дней назад +2

      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.

  • @SirNobleIZH
    @SirNobleIZH 12 дней назад +440

    17:23 missed opportunity for a pi-ramid

    • @ragunanthan787
      @ragunanthan787 12 дней назад +12

      Wow 😂

    • @Grayson_Wu
      @Grayson_Wu 12 дней назад +32

      That's some Parker joke right there

    • @andersjjensen
      @andersjjensen 12 дней назад +3

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

    • @Excalibaard
      @Excalibaard 12 дней назад +1

      That's pretty (pira) mid

  • @Xboy1207
    @Xboy1207 11 дней назад +144

    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
      @Kishblockpro 10 дней назад +23

      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
      @Overcrox 10 дней назад +26

      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
      @blinded6502 10 дней назад +6

      Yeah I'm working on it

    • @PaoloLammens
      @PaoloLammens 9 дней назад +27

      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
      @benayers8622 9 дней назад +4

      @@PaoloLammens shrooms n ket will get ya there 🤣

  • @Gears_AndGrinds
    @Gears_AndGrinds 12 дней назад +189

    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
      @columbus8myhw 12 дней назад +1

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

    • @evildude109
      @evildude109 12 дней назад

      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
      @Gears_AndGrinds 12 дней назад +10

      ​@@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
      @johnjameson6751 11 дней назад

      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.
    @charles.e.g. 12 дней назад +3

    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. ❤

  • @jpe1
    @jpe1 12 дней назад +143

    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
      @jayeff6712 12 дней назад +12

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

    • @TheElexec
      @TheElexec 12 дней назад +6

      Yes! That starled me a little lol

    • @cameron7374
      @cameron7374 12 дней назад +7

      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
      @angeldude101 12 дней назад +8

      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
      @julianbaader4351 12 дней назад +1

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

  • @SpiritOfHugs
    @SpiritOfHugs 9 дней назад +1

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

  • @devd_rx
    @devd_rx 12 дней назад +555

    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
      @uhrguhrguhrg 12 дней назад +78

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

    • @FireyDeath4
      @FireyDeath4 12 дней назад +7

      Can you break the cycle in Hilbert space as an apeironian

    • @fgvcosmic6752
      @fgvcosmic6752 12 дней назад +14

      Hilbert space beings go brrrr

    • @good_brake
      @good_brake 12 дней назад +31

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

    • @good_brake
      @good_brake 12 дней назад +8

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

  • @mrazo
    @mrazo 12 дней назад +1

    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.

  • @ericthiel6553
    @ericthiel6553 12 дней назад +95

    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
      @dmdeemer 12 дней назад +8

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

    • @DeanCalhoun
      @DeanCalhoun 12 дней назад +1

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

    • @beautyofmath6821
      @beautyofmath6821 12 дней назад

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

    • @timothemalahieude5076
      @timothemalahieude5076 11 дней назад +1

      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
      @ansiwen 9 дней назад +1

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

  • @sender1496
    @sender1496 10 дней назад +28

    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
      @jademonass2954 5 дней назад +2

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

    • @sender1496
      @sender1496 5 дней назад +1

      @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.

  • @QiYueWang-l4j
    @QiYueWang-l4j 12 дней назад +66

    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
      @aaryamangupta 12 дней назад +2

      Were you a participant?

    • @nicezombie8054
      @nicezombie8054 12 дней назад +3

      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
      @nicezombie8054 12 дней назад +3

      I still vividly remember spending the night then solving the puzzles

    • @isaac10231
      @isaac10231 12 дней назад

      ​@@nicezombie8054how was it at the imo?

  • @Aquillyne
    @Aquillyne 12 дней назад +1

    The first hexagon puzzle is probably the first time in any of your videos where you set up a tricky problem and I intuited the answer well before you laid it out. I’m a very visual thinker so that probably helped. A load of times in geometry I’ve caught myself thinking that the arrangement of lines also works as a projection from 3D, but never seen this used as a line of reasoning for a proof before. Loved it.

  • @rolfvdhulst
    @rolfvdhulst 12 дней назад +39

    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
      @jackm.1628 5 дней назад

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

  • @BenSpitz
    @BenSpitz 11 дней назад

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

  • @vytah
    @vytah 12 дней назад +35

    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
      @karolakkolo123 12 дней назад +5

      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

  • @AbhinavKumar-nh8dl
    @AbhinavKumar-nh8dl 11 дней назад +5

    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!!!

  • @roy04
    @roy04 12 дней назад +88

    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
      @cameron7374 12 дней назад +25

      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
      @geoffreysimms2520 12 дней назад +16

      @@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
      @cameron7374 12 дней назад +14

      @@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
      @geoffreysimms2520 12 дней назад +3

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

    • @xenontesla122
      @xenontesla122 12 дней назад +16

      @@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 🤣

  • @AZALI00013
    @AZALI00013 12 дней назад +16

    this might be the coolest math video ive ever seen

    • @jwjustjwgd
      @jwjustjwgd 11 дней назад +2

      pov every 3blue1brown video

    • @litterbox019
      @litterbox019 10 дней назад +1

      azali spotted on 3blu1brown??

    • @jwjustjwgd
      @jwjustjwgd 10 дней назад

      @@litterbox019 fr

    • @litterbox019
      @litterbox019 10 дней назад

      @@jwjustjwgd someone tell him math videos get even cooler

    • @Kishblockpro
      @Kishblockpro 10 дней назад

      AZALI OMG

  • @Lord_zeel
    @Lord_zeel 12 дней назад +24

    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
      @karolakkolo123 12 дней назад +4

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

    • @BrooksMoses
      @BrooksMoses 12 дней назад +1

      @@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?

  • @rolan638
    @rolan638 9 дней назад

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

  •  12 дней назад +63

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

    • @chocoblocko9
      @chocoblocko9 12 дней назад +15

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

    • @damjandenic
      @damjandenic 12 дней назад

      Okay nerd

    • @stardragon8585
      @stardragon8585 День назад

      Spoken like a true software engineer

    •  День назад

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

  • @amaury31415
    @amaury31415 6 дней назад +2

    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.🙃

  • @Aldrasio
    @Aldrasio 12 дней назад +5

    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
      @beansprugget2505 3 дня назад

      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.

  • @simonpfennig1203
    @simonpfennig1203 12 дней назад

    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.

  • @drag0nblight
    @drag0nblight 12 дней назад +21

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

  • @StaphyJr
    @StaphyJr День назад

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

  • @jovi_monet
    @jovi_monet 12 дней назад +9

    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
      @Icefox297 12 дней назад +1

      Can you elaborate more? I am curious

  • @harryeast95
    @harryeast95 12 дней назад +23

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

  • @annmirren6132
    @annmirren6132 12 дней назад +32

    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
      @Ze_eT 12 дней назад +3

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

    • @rokaslokusevicius3810
      @rokaslokusevicius3810 12 дней назад +1

      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
      @xiphosura413 12 дней назад

      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
      @newmerek 11 дней назад

      @@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
      @rokaslokusevicius3810 11 дней назад

      @@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.

  • @wyboo2019
    @wyboo2019 11 дней назад +4

    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
      @galoomba5559 8 дней назад

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

  • @stefanklass6763
    @stefanklass6763 12 дней назад +4

    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.

  • @JTCF
    @JTCF 12 дней назад +1

    I was so excited when stepping up the dimension in the tiling puzzle. At some point something "clicked" and I was excited to see how my mind was correctly predicting what the video will do next.

  • @rms_txrx
    @rms_txrx 12 дней назад +42

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

  • @Notthatkindofdr
    @Notthatkindofdr 9 дней назад

    Another astounding and mind-opening video! Thank you

  • @mashmachine4087
    @mashmachine4087 12 дней назад +20

    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
      @Reydriel 12 дней назад +4

      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
      @ritahall8148 12 дней назад +3

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

    • @jwjustjwgd
      @jwjustjwgd 11 дней назад +1

      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
      @mashmachine4087 10 дней назад +1

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

    • @BraydenIsThatGuy
      @BraydenIsThatGuy 10 дней назад

      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.

  • @prismarinestars7471
    @prismarinestars7471 11 дней назад +1

    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.

  • @thephysicistcuber175
    @thephysicistcuber175 12 дней назад +16

    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.

    • @QuizeyMc
      @QuizeyMc 12 дней назад

      Aye bruh can u teach me math

  • @BlameThande
    @BlameThande 12 дней назад

    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.

  • @poke7661
    @poke7661 12 дней назад +15

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

    • @yyeeeyyyey8802
      @yyeeeyyyey8802 12 дней назад +3

      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
      @crb233 12 дней назад +1

      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
      @geoffreysimms2520 12 дней назад

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

    • @karolakkolo123
      @karolakkolo123 12 дней назад

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

  • @jasonhildebrand1574
    @jasonhildebrand1574 7 дней назад +1

    3:51 "real analysis buffs will note that this circle is not completely covered by chords" :)
    Another excellent video, Grant !!

  • @antonios-st.frantz6523
    @antonios-st.frantz6523 12 дней назад +4

    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!

  • @EannaButler
    @EannaButler 9 дней назад

    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!

  • @ooqui
    @ooqui 11 дней назад +4

    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
      @ToyTrainAdventures-z3e 10 дней назад +1

      How did you become an artificial tetrachromat?

    • @ooqui
      @ooqui 10 дней назад

      @@ToyTrainAdventures-z3e I've made a RUclips 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 RUclips comments.

    • @trueriver1950
      @trueriver1950 6 дней назад +1

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

  • @tuatarian6591
    @tuatarian6591 9 дней назад

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

  • @dominikmuller4477
    @dominikmuller4477 12 дней назад +49

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

  • @freetolisten
    @freetolisten 6 дней назад

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

  • @anjansharma8631
    @anjansharma8631 12 дней назад +10

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

  • @varunvaswani4562
    @varunvaswani4562 8 дней назад

    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
      @varunvaswani4562 8 дней назад

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

  • @treborhuang233
    @treborhuang233 12 дней назад +10

    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
      @galoomba5559 12 дней назад +1

      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
      @Bageer1 12 дней назад

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

    • @galoomba5559
      @galoomba5559 12 дней назад

      @@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
      @Bageer1 12 дней назад

      @@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
      @blblblblblbl7505 12 дней назад

      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.

  • @matricepeinard7879
    @matricepeinard7879 12 дней назад

    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

  • @javierdaher825
    @javierdaher825 11 дней назад +3

    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.

  • @baptisteprevost6835
    @baptisteprevost6835 8 дней назад

    Loved the narration for the Monge problem

  • @Skiddla
    @Skiddla 9 дней назад +3

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

  • @fierydino9402
    @fierydino9402 7 дней назад

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

  • @pizzarickk333
    @pizzarickk333 12 дней назад +4

    You never fail to make me appreciate your existence.

  • @thabomsiza2502
    @thabomsiza2502 10 дней назад

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

  • @michaelgreer7306
    @michaelgreer7306 12 дней назад +5

    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.

  • @BryndanMeyerholtTheRealDeal
    @BryndanMeyerholtTheRealDeal 11 дней назад +1

    A video with 4d geometry in the title, but not talk about said 4d geometry, but this video is still pretty great…

  • @aloosh1375
    @aloosh1375 12 дней назад +4

    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
      @3blue1brown  12 дней назад +6

      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
      @aloosh1375 12 дней назад +4

      @
      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
      @Noname-67 12 дней назад +1

      ​@@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_
      @Thiefy_ 12 дней назад

      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.

  • @piyushchoudhury3709
    @piyushchoudhury3709 10 дней назад +1

    This video was a delight to watch and please make follow up videos like reed Solomon code for hamming code and beta distribution for binomial distribution videos

  • @4g4m3n0n
    @4g4m3n0n 12 дней назад +5

    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
      @cameron7374 12 дней назад

      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
      @karolakkolo123 12 дней назад

      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

  • @briton1509
    @briton1509 12 дней назад

    Your video's are hard to watch properly, but are deeply enjoyable because of that fact.

  • @taukakao
    @taukakao 11 дней назад +3

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

  • @sirgog
    @sirgog 11 дней назад

    So glad to hear of the Slovakian IMO team again. At my second IMO (1999) the Slovakian team were the most fun people around. They studied hard and partied hard. Great memories of having water balloon fights both with and against them in that period after we'd sat the exams and were waiting for adjudication. I remember them being nervous after the exams and ecstatic with their results.

  • @Player-Leli
    @Player-Leli 12 дней назад +5

    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
      @ZeroPlayerGame 12 дней назад +6

      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
      @-tera-3345 12 дней назад +5

      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.

  • @jaredkelnhofer
    @jaredkelnhofer 8 дней назад

    That proof with the 3 spheres is beautiful

  • @PauLtus_B
    @PauLtus_B 12 дней назад +11

    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__
      @-hc__ 12 дней назад +3

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

    • @CollinWilliams-by5cs
      @CollinWilliams-by5cs 12 дней назад +1

      Yes, when he started putting them back 😮

    • @PauLtus_B
      @PauLtus_B 7 дней назад

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

    • @PauLtus_B
      @PauLtus_B 7 дней назад

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

  • @silvesteronono4157
    @silvesteronono4157 8 дней назад

    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.

  • @TheSmegPod
    @TheSmegPod 12 дней назад +3

    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
      @geoffreysimms2520 12 дней назад

      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
      @amagicmuffin1191 12 дней назад +2

      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
      @danielyuan9862 12 дней назад

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

  • @abuzarov
    @abuzarov 11 дней назад

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

  • @douglasheld
    @douglasheld 12 дней назад +3

    6:17 I disagree. Because we are not really talking about cubes but projections of cubes, to go from "empty" to "full" only requires flipping of the three "walls" in our view. I.e. wall1 of 16, + wall2 of 12 remaining, + wall3 of the final 9 = 37 flips.

    • @nonbreapelido3549
      @nonbreapelido3549 12 дней назад

      Agreed, was just about to comment. The "fullness" of the cube doesnt alter the pattern.

    • @riggs234
      @riggs234 11 дней назад

      Yeah wondered if anyone picked up on this too

    • @samgibson6408
      @samgibson6408 9 дней назад +1

      I also thought of this, but I do think N^3 is still necessary. It is true from the perspective of stacking that we could just stack the outer walls. But the "stacking" is actually somewhat constrained.
      For example, consider the spot at the bottom (the bottom tip, opposite to the back corner) when we start at empty. We will not immediately be able to stack a cube there, since the stack is really just a rotation of the 3 rhombus pattern, and we do not yet have that pattern at that spot. This is to say that in order to stack that spot, we need to stack/rotate the entire bottom layer so that we can finally validly do a rotation which corresponds to a stack at that spot. A similar argument should show that we need N^3 rotations to get the final tiling.
      I think the analogy to stacking is actually somewhat limited, since how we think about stacking in the 3-dimensional sense may correspond to invalid stacks in this problem.

  • @cutewavelets
    @cutewavelets 12 дней назад

    YOOO IS THE TOP PUZZLE OF BONUS VIDEOS FROM MATHCAMP I RECOGNIZE THAT

  • @JP_Hatecrew
    @JP_Hatecrew 12 дней назад +13

    1:20 I'm just seeing cubes

  • @sheikhalsumaiya7230
    @sheikhalsumaiya7230 12 дней назад +1

    Thanks for making my weekend better ❤

  • @Inspirator_AG112
    @Inspirator_AG112 12 дней назад +56

    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
      @issholland 12 дней назад +10

      Speedrunners already use trigonometry for an added edge so yeah

    • @eagle32349
      @eagle32349 12 дней назад +4

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

    • @issholland
      @issholland 12 дней назад +18

      @@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
      @eagle32349 12 дней назад +3

      @@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
      @issholland 12 дней назад +11

      @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.

  • @Thiefy_
    @Thiefy_ 12 дней назад +1

    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.

  • @Living_Murphys_Law
    @Living_Murphys_Law 12 дней назад +5

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

    • @null-0x
      @null-0x 12 дней назад +1

      after*

  • @C134B
    @C134B 11 дней назад

    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

  • @emre_ez
    @emre_ez 12 дней назад +7

    0:51 hexagons are the bestagons

  • @maths.visualization
    @maths.visualization 6 дней назад +1

    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! 😊

  • @signordoge8721
    @signordoge8721 12 дней назад +20

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

    • @-YELDAH
      @-YELDAH 11 дней назад +1

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

  • @thechocopi2146
    @thechocopi2146 12 дней назад +1

    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

  • @Nevermind445
    @Nevermind445 9 дней назад +2

    *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

  • @thatapollo7773
    @thatapollo7773 8 дней назад

    One small extension of Monge's theorem is due to the fact that circles have 2 points of similarity, the intersection of external tangents AND that of internal tangents. The extension is that the 2 points of internal similarity and 1 of external are collinear. It can be proven by making one of the cones have its vertex on the opposite side of the plane of the circles.

  • @TheMichaelmorad
    @TheMichaelmorad 12 дней назад +8

    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
      @Fun_maths 12 дней назад +2

      Did not expect such a comment, ממש מגניב.

    • @angeldude101
      @angeldude101 12 дней назад +1

      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
      @TheMichaelmorad 12 дней назад

      @@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
      @angeldude101 12 дней назад

      @@TheMichaelmorad Ah. Thank you.

    • @horrorspirit
      @horrorspirit 10 дней назад

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