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!
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.
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 !.
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!
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.
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
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.
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
@@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...
Thank you so much! I’m glad you enjoy how we connect the dots between ideas and videos-that's always fun to put together. Big shoutout to the team for making it all come together! It’s awesome to know it resonates with you!
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.
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
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?
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
If anybody was wondering, the "sliding cubes through the origin" move described in the 5th puzzle is called a central inversion. Another way to describe it is to take any vector and negate all of its coordinates
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
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❤
*Another great example of working in higher dimensions being useful...* The integral from 0 to x of ln(t) can be thought of as the integral of [the integral from 1 to x of 1/y], which can be thought of as the volume of a certain solid, also counting 'negative volume'... That volume can be though of as the difference of two volumes... The integral from 1 to y of x/y minus the integral from 0 to y of y/y. _(The notation I used is a hint, as well.)_
Speaking of intuition, I've always found that one of the most effective ways for me to understand something is to really mess around with math interactively. I would not be here and as interested in math as I am now were it not for Desmos allowing me to play around and develop my intuition. Whenever I tried something that I really thought should work but then it didn't, I'd look into things to figure out where I went wrong. Answering my own questions that were completely constructed by me out of curiosity have led me to esoteric mathematics such as the W Lambert function, which if presented to me in a classroom setting I think I'd struggle to understand the motivation for. Singlehandedly the best development in my understanding of the complex plane was desmos's relatively recent addition of "complex mode," to mess around with complex numbers and whatnot. It has made the complex plane not just intuitive to me, but so utterly obvious that I wonder why it isn't explored earlier on in math. It is so incredibly elegant and I just can't get enough of complex analysis. The unit circle is just split into fourths! If you consider i^θ, θ here corresponds to some angle n/4 around the cricle. You don't have a ton of πs floating around like you do when typically dealing with the unit circle, and it's just... I absolutely love it, and I don't think I'd get anywhere near that amount of intimacy with the complex plane were it not for a dynamic, interactive tool I could play around with.
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.
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.
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.
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.
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.
Hi! Great video :) I often work on tiling problems, always nice to see them pop up! I believe there is another, maybe simpler case to make 4d relevant: here you are tiling with rhombi that have 3 possible edges directions. If now you allow for 4 edge directions, and you are tiling, say, a sort of octagonal region, then (if I remember correctly) you can still relate any two tilings via the same "cube flip" operation. This is due to the fact that the tiling can again be seen as a 2d surface living in R^4, which as been projected onto a plane. The same cube flip "just" amounts to popping up or down a 3d cell in that surface. And similarly, if you allow for d different edge directions, you are merely playing with surfaces in R^d ! Also, this "lifting up a dimension" is relevant in other tilings as well, even for the (more common) domino tilings, where you can define a "height function", and this is used to show that the natural flipping of two dominos is enough to get from any configuration to any other.
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.
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.
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.
This stratagy of looking at a 2d question and making it about three dimensions Has a really nice Hebrew name, מִרְחוּב (mirhuv). 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.
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!
23:45 You can get the same effect by performing a symmetric rotation of the N-1dimentional shape, so that the orientation of the projected-down vectors changes to the furthest outer vertici. In other words, the vertici and the intersection of any 3 cells would translate to to the outside facing vertici of an adjacent cell, and vice versa. If younwant to be extra technical, what youre actually doing is applying a rotation vector equal to N-2 30° rotations for each plane perpendicular to the nth dimention, but thats a less intuitive way of thinking about it for us 3 dimensional creatures. In fact a more complete version of thie problem would be to perform any 2 30° rotations around the x, y, and/or z axes. Which will always be a symetrical rotation for the rhombic dodecahedron while resulting in a new orientation for some or all of the cells that comprise it.
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
I find it utterly wonderful and beautiful to realize how ‘things fit into each other’ like this, especially how things in higher dimensions are more complex generalizations of simpler special cases in fewer dimensions. Thanks for providing such wonder and beauty!
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.
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 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.
I remember your first Numberphile video about a probability puzzle that ended up involving the sum of the volume of each n dimensional ball. Still a mind-blowingly insightful video !
I have to say, your visuals and explanations are phenomenal. Of course this was always the case with your other videos, but I especially appreciate it here since it's hard to be intuitive with 3D geometry
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.
When I saw the "hexagon" puzzle, I immediately started thinking about it as cubes. I feel like the since the video kept calling them hexagons it drove me away from that frame of thinking when it could have been fruitful. I think this may be a good example of how educational videos sometimes can lead viewers to a certain kind of interpretation, even unintentionally, that makes sets up a reveal.
a different intuition for the shapes tiling the 3d space (which also makes it easier to code) if you rotate the shape you get a cube with 4 4-sided pyramids glued to the 6 sides, the base angles are 45 degrees, so the transition between them is seamless, and they are half as high as they are wide now if you put 6 of those pyramids facing inwards, they form that same cube what you do is place the basic cube in a checkerboard pattern: x+y+z=0 mod 2 where x, y and z are whole; the x+y+z=1 mod 2 spaces get covered by the pyramids sticking out of the cube ...and checking the wikipedia page for it, it's well-known, so I guess my comment doesn't matter
This is amazing to get on my feed. I was literally prepraign an internal talk about how granting degrees of freedom to architecture problems truely simplifies our work! Thanks so much!
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
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.
@ 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. That being said I am only an IOL contestant, so my argument might very well be flawed.
My no-so-rigorous proof for the strips-in-a-circle problem: 1. Start with the single, wide strip. 2. To get less than the 2 that is wide, you have to make it smaller, creating a gap. 3. You can either fill that gap with an additional, parallel strip, or you can try to fill it with a rotated one. 4. Due to the shape of a circle, this gap will always be wider than it is tall, so any rotated strip would need to stretch to that width, bringing you above 2. (a combination of multiple strips only makes this worse.) And thus, parallel is the only optimal way there is.
Monges theorem! The 3D proof is quite beautiful but I must point out some fine details. For any two circles there are two centers of similarity, you can think of them as the intersection of the extangents and the intersection of the intangents. Monges theorem doesn't work if you choose any center you like for any pair of circles, however it still does work if the number of intangent intersections is even. And if we are already on the topic, here is an entirely 2D proof of monges theorem that is still very pretty. For some point in the plane we define an homothety to be the scaling of the plane around that point by some nonzero (possibly negative) constant. Now I won't get into the details but it isn't difficult to show that chaining two different homotheties with possibly different centers still results in a homothety, with its center being on the line of the two other centers. Now, think about taking the homothety that takes circle A to circle B via a similarity point and then another from B to C, chaining these results in the homothety from A to C so its center, which is a similarity point, must be on the line of the two other points and you can convince yourself that the number of negative homothety centers must be even (hint: What is the scaling factor of the chain of two homotheties?) which concludes the proof for all cases of intangent intersections as well. This of course works for any shape, not just circles but if the shape happens to be symmetric around a 180 degree rotation then the caveat of even number of negative homothety centers is necessary.
I know a couple of good, related "project into higher dimensions" problems! In mechanics, if you consider the Kepler problem in either 3D configuration space, where the potential is 1/r, it's a classic result that the solutions to trajectories of the motion are conic sections. However, Kustaanheimo and Stiefel showed that if you project the configuration space into 4D, the Kepler problem in 3D (which is nonlinear) is equivalent to the simple harmonic oscillator in 4D (which is linear), thus showing that two of the most notable solvable dynamics equations are in some sense, the same probem. There's a beautiful theory about how the K-S transformation keeps track of the dynamics of a non-unit quaternion too. I should note that the K-S transformation generalizes a previous, analogous result of Levi-Civita showing that the 2-D Kepler problem can be transformed to a complex harmonic oscillator. Beyond the mathematical elegance, This has applications in both celestial and quantum mechanics, that I know of--anywhere that 1/r potential shows up. Thanks for another great video!
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
I've often wondered what would happen if we trained an AI that could see in 4 dimensions, i.e. instead of two 2D visual inputs (pixels), it receives two 3D visual inputs (voxels). Since we don't live in 4D the inputs would have to be all artificially generated of course, but in principle such an AI would be "seeing" the 4th dimension.
2:45 saving my intuitive response here before watching the rest. If we look at the presented picture as one of those 3-d illusions (of small cubes in a cubic space in an isometric projection onto a hexagon) and choose the lower sides of the hexagon to be the "bottom" of the cube, rotating any triplet of rhombi is analogous to adding or removing a smaller cube from the cube stack. So you should be able to get from any one state to another just as you can always add or remove a cube from the stack. The two farthest apart states should be the states of the "empty" stack" and the "full stack", and they should be as far apart as the number of cubes they represent - 0 to 4x4x4 = 64 steps.
25:03 The acute angle on the rhombus needs to be about 70.53 degrees, not 60 degrees. Attempting to make that shape out of 60 degree rhombuses would result in a flattened version of it.
People often describe four-dimensional space as "some direction that we can't quite imagine" and "we struggle to really think about what it looks like" and "to you and me, it's inaccessible" but it seems pretty intuitive to mentally visualize after having thought about it for so many years. The extra perpendicular dimension is sort of like a point at infinity that goes ana or kata, like a sort of "inward" or "outward" direction from itself. Especially after seeing 4D golf and other 4D visualizers all while learning about 4D geometry. It's parallel to itself, which makes sense to me. I mean, we don't have a biological camera to see it, but it's like how if you close your eyes, you still have a 3D sense of spatial awareness and proprioception, a 4D space is still conceptualizable in that same way. In my mind's eye, something like a 4D cube is like added density in two antipodal directions where the corners form new edges and each face being extended through that dimension to make a new cube. I mean, it's hard to put into words, but I have a visual spatial conception of it that works in my head, and I think that was afforded to me by watching so many videos that explained it visually. Like 4D spheres falling onto 4D sloped surfaces that look flat in 3D but are clearly a slope in 4D, so they appear to "sink through" then in 3D space. I'm rambling, but I think that having a sensory intuition of 4D space isn't as impossible as people often say. Even if stereoscopic vision doesn't capture it, a broader spatial sense can, and then that in turn can form a visual. Y'know?
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
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).
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.
For the tetrahedron puzzle; Consider points P2 - P1, P3 - P1, P4 - P1, and use those as where I, j, k land for a 3D matrix. Use the determinant formula (which I definitely do remember) and divide by the ratio of a cube's volume to an "inscribed" right tetrahedron, which I believe is 6 but could be very wrong on
Yeah, you don't need to go up a dimension to solve that problem. I'm curious to see what that solution looks like though, it seems like you get more symmetry that way.
My solution should be similar to yours, if a bit clunky and not at all elegant nor leveraging any 4D insight: - The volume of any pyramid is one third the base area times its height. Say we start from a vertex A, and WLoG pick vertices B and C to be the other points on the base, we can write the (signed) base area as one half AB cross AC, and the height is AC dot the unit vector parallel thereto. Combining the above we have the (signed) volume being one sixth the triple product (AB AC AD). - Pick an origin O and write AB = OB - OA, etc. Expanding the triple product in terms of OA, OB, and OC, the volume V is explicitly equal to ((OB OC OD) - (OA OC OD) + (OA OB OD) - (OA OB OC)) / 6. - Note how we have alternating signs on the triple products, which are reminiscent of determinants. Taking OA, etc. as column vectors, we can thus write $V = \frac{1}{6} \left\|\begin{matrix} 1 & 1 & 1 & 1 \\ OA & OB & OC & OD \end{matrix} ight\|$.
the argument for problem 3 breaks again if you align the circles such that the centers of similarity are all the same point - then the tips of the cones are co-linear and thus only define a line and not a plane. of course, it is trivial that 3 identical points line on any line through that point, and the intersection of the line through the tips with the original plane is the mutual center of similarity, but this case does need to be handled separately.
11:41 : I see a projected 3d scene where 3 spheres of the same size are resting on an infinite plane. The white line is the horizon. You connect pairs of spheres with infinitely long cylinders, and the outlines of each cylinder join at the horizon.
This exact kind of thinking has some applications in physics: to deal with 2D hexagonal crystals like graphene (eg to compute how they conduct current at the quantum level), it is common to use 3 basis vectors (the ones shown in the video resulting from the projection of the 3D ones) although they are not linearly independent, because it preserves the symmetries of the crystal!
Needless to adore your intriguing animations that make us reliable to discern a wide variety of concepts and prone to analytical thinking....yet i had a question, would you mind elaborating the uniqueness of diagonal lines and planes traced by tesseract (ik it could be analysed via flat landers' pov(s) of a square / a cube), the 2D diagrams of 4D object aint aids me up, i guess, which the animation can.?
I solved Monge's theorem by using 3D in a different way. Imagine each circle is the outline of a sphere (or just the same circle in 3D), where all the spheres are the same size, arranged in 3D space and viewed by projecting onto the screen. Then the "center of similarity" of two of these spheres is the focal point of the "tube"/cylinder around both spheres, which is the same as the focal point of the line passing through the centers of the spheres. Since the centers lie on a plane, so do those lines. And so the focal points lie on the horizon corresponding to that plane, which is a straight line. This solution also doesn't care if one is inside another or the specific shape since you can take any point instead of the center, as long as it is in the same place for all shapes. (Similar to the point of the pyramid in the "fixed" proof)
im with you. three identical spheres sitting on a plane, viewed in perspective. a scene like an old raytracing demo. i feel like it takes a little bit more work with this picture to prove this works for arbitrary figures, but the intuition is strong
From what I can find, it seems that the cone argument for Monge's Theorem originates with the engineer John Edson Sweet and was then used by Monge to prove this theorem (which was first proposed by d'Alembert). See page 153-154 of The Penguin Dictionary of Curious and Interesting Geometry by D. Wells. (It cites to the book Ingeneous Mathematical Problems and Methods by L.A. Graham for the quote by Sweet, but I don't have access to this for further verification.)
writing this as grant revealed the 3 dimensional projection of monge's theorem and not having seen the numberphile video because I quickly thought of my own answer and was surprised to see it didnt at all match what is shown in the video; immediately upon seeing the problem, I thought of vanishing points in technical drawing, and so I consider the circles not as 3 circles with different diameters but 3 spheres with the same diameter. this means that the point of intersection of the tangent lines must be a point infinitely far away but colinear with the 2 spheres. since there are 3 spheres, each with it's own centre point, this means they all share a plane. since the cotangent points are colinear with their spheres, they must also share any plane their parents do. so, all 6 points share the same plane. since the cotangent points are also infinitely far away, that means that they lie on the perimeter of said plane. so, how do we prove that the edge of an infinitely large plane is straight? the same reason the horison of earth seems straight, even though it clearly curves as we raise our altitude. the infinite plane is in fact the surface of an infinitely large sphere. even though the surface of the sphere is curved and so is it's horizon, both appear flat as the ratio of the diameter of the sphere relative to our distance from it increases. in fact, since the sphere is of infinite size our distance from it doesn't matter because at it's limit, the surface of any infinitely large sphere appears like an infinite plane and it's horizon appears as a straight line regardless of our distance. the 3 cotangent points lie on the horizon of this infinite sphere, and this horizon is approaching straight so, you can imagine my surprise when the video didn't match my intuition at all. anyway, back to the video. I'm sure the real answer will be similarly elegant.
Another problem in the same vein is as follows: find a 3x3 matrix A =/= I with integer entries such that A^5 = I. Sadly I can’t remember where I first saw this any better than “in a tweet once”
Was feeling particularly masochistic the other day and scrolled through the Wikipedia page on x-ray crystallography. This video hurt my head a good bit less than that. Now wondering how related Demis Hassabis's protein work is. I guess I feel a little bit better knowing that this sort of stuff can make somebody like you sad, too. 🙂 Now definitely have to go back and make myself finish watching the video on quaternions. Simply mind boggling to me that there are apparently tons of humans that "get" all this stuff.
itd be just the greatest thing for you to explore the beauty of the truncated octahedron. optimally fills space, order 4 permutahedron, a slice of a hypercube, unification of hexagon and square, space filling with either squares or hexagons removed to create new space filling shapes, more and more
that was trippy, when that last cube git removed at 5:55 my brain inverted tye empty space and when the next one got added it like warped back to being concave
The three sphere problem - I imagine the three spheres to all be the same size but at different distances from my point of view. That tangent line is where the plane through the centers of the spheres intersects my viewing plane. I think. Something like that, anyway.
Here's another case where thinking about higher dimensions might be useful. It's possible that the universe is curved on itself. However, it's difficult for the human brain to get an idea of what the curvature of 3D space would look like. It's easier to think of the same problem in smaller dimensions. For example, a line (1D) can be curved into a circle (2D). Similarly, a plane (2D) can be transformed into a sphere (3D). Logically, trying to visualize a curved space in 3D would force us to think in 4D and that is why it's so hard.
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!
I don't enjoy doing criticisms (even less to a channel that I love as much as this one), but as no one else is pointing it out, I fear that I must do so: the audio isn't as good equalized as usual (some parts of the narration are a bit too loud). I hope this helps in some capacity and look forward to the next videos ♥
да уж, согласен, это не просто развлечение на вечеринке ботаников. С помощью такого подхода можно решать много сложных задач, где решение не очевидно на первый взгляд. Если была бы нобелевская премия по математике, то этот подход к решениям ее получил бы
I wanna make video to add new perspective about "Dimensions", seems like mine kinda forgotten or maybe long gone forgotten hypothesis. Why not start the dimension from 0D = 1 value of D aka 0=1 and so on, means a singular dot can form a "length", not point A to B and between them imaginary line/dimensional form. From those length produce more same length as diagonal length and picture a cube (you can picture an orb/globe but those body is already finite). Voila you create cube skeleton/ construction, so now what? BEHOLD inside those skeleton, you can fit 6 pyramids! How many dimensions now? 1st. 0; 2nd. Single Length; 3rd. Cube Limits; 4th. Addition of 6 Pyramids; 5th. ??? yeah same rule, add dot(s) as a fixed anchor, put length, put limit, add pyramids = Breaking new dimension! I got my hypothesis of this "new" perspective because isn't so obvious that in "order of operations" you can found after counting -> addition -> multiplication -> exponent -> tetration -> (and beyond) there's likenesses pattern, "I'm too lazy to count so let's create new limit(s)" and BOOM! new realm of mathematical dimension(s).
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
Monge's Theorem should work on any convex hulls, so generalizes easily to any vanishing points and objects. This means the "But actually" argument is simply asking why a trivial or uninteresting drawing has no interesting interpretation.
Its mind blowing to imagine how to view 3D, instead of merely experiencing it, Leave experiencing 4D. Also, the diagonal of a "4D square" would be 2 times the 1D edge, which seems logical as for 3D, It's root3 times, and for 2D, It's root2 times
A year ago I stumbled upon your linear algebra series, knowing barely anything about linear algebra or geometry, but at problem 4 I immediately thought "I know this one from triangles." In my head I think of it as the "shadowpuppet matrix."
18:20 Why wouldn't a 3x3 matrix work? My thinking: Translate shape such that one point is at the origin. Take 3x3 determinant of the 3 remaining points and half it to get the answer. Same is the case in the 2d example.
I wonder why we mostly refer to hypercubes or tesseracts when discussing 4D geometry instead of hyper-tetrahedron To form a 1D line, we need at least 2 dots; for a 2D shape, we need 3 lines; and for a 3D polygon, we require 4 shapes. Therefore, to create a 4D hyper-shape, at least 5 polygons are needed. So why don’t we use a hyper-tetrahedron?
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!
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.
Hi, you can also slide the rhombus in the hexagon.
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 !.
@iota_i_1 this guy trying to answer the meaning of life
did you calculated the det wrong the letter "c" isnt even in the sum.
18:50 "aei-ahf+dbi-dhd+gbf-ged"?????
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!
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.
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
this is something that happens in topology! For instance, computing the homotopy groups for the sphere in all dimensions is possible except for n=4
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.
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
@@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...
@GeorgiiPotapov that is a very interesting question that will haunt me for life now 😂
24:57 Bold of you to assume such a nerdy cocktail party would have people that don't watch 3b1b
Thank you so much! I’m glad you enjoy how we connect the dots between ideas and videos-that's always fun to put together. Big shoutout to the team for making it all come together! It’s awesome to know it resonates with you!
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.
1:05
What I'm told: Three rhombus shapes rotating,
what i see: Stacked cubes, being added/removed, with a rotating fade effect
YES!
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
Wait, he just revealed my way of thinking
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?
seriously I cannot see this as 2D as my brain doesn't allow that
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
4d hogs a lot of the regular shapes though. And it kinda gets samey after that
Can you break the cycle in Hilbert space as an apeironian
Hilbert space beings go brrrr
Just learn to think in nD, and specialize to a particular value of n when necessary.
@@fgvcosmic6752Banach space beings: "oh you are still just a baby."
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.
Yeah, I saw it right away, as well
Although after each rotation, at first an impossible one.
If anybody was wondering, the "sliding cubes through the origin" move described in the 5th puzzle is called a central inversion. Another way to describe it is to take any vector and negate all of its coordinates
17:23 missed opportunity for a pi-ramid
Wow 😂
That's some Parker joke right there
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
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❤
Were you a participant?
*Another great example of working in higher dimensions being useful...*
The integral from 0 to x of ln(t) can be thought of as the integral of [the integral from 1 to x of 1/y], which can be thought of as the volume of a certain solid, also counting 'negative volume'... That volume can be though of as the difference of two volumes... The integral from 1 to y of x/y minus the integral from 0 to y of y/y.
_(The notation I used is a hint, as well.)_
Speaking of intuition, I've always found that one of the most effective ways for me to understand something is to really mess around with math interactively. I would not be here and as interested in math as I am now were it not for Desmos allowing me to play around and develop my intuition. Whenever I tried something that I really thought should work but then it didn't, I'd look into things to figure out where I went wrong. Answering my own questions that were completely constructed by me out of curiosity have led me to esoteric mathematics such as the W Lambert function, which if presented to me in a classroom setting I think I'd struggle to understand the motivation for.
Singlehandedly the best development in my understanding of the complex plane was desmos's relatively recent addition of "complex mode," to mess around with complex numbers and whatnot. It has made the complex plane not just intuitive to me, but so utterly obvious that I wonder why it isn't explored earlier on in math. It is so incredibly elegant and I just can't get enough of complex analysis. The unit circle is just split into fourths! If you consider i^θ, θ here corresponds to some angle n/4 around the cricle. You don't have a ton of πs floating around like you do when typically dealing with the unit circle, and it's just... I absolutely love it, and I don't think I'd get anywhere near that amount of intimacy with the complex plane were it not for a dynamic, interactive tool I could play around with.
This 3D to 4D mapping reminds me of quaternions. Edit: 26:16 Oh lol, I spoke too soon as you bring them up later.
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.
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.
"Pi-ramid shape". I see what you did there.
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.
The 1st puzzle was a cool reminder of the GameCube animation!
This channel still proves I'm a visual learner. Kudos for the animation skills you have.
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.
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.
This happened to me too. The brain is a curious device.
Yes! That starled me a little lol
The best part of your videos is the "connect the dots" to previous ideas and videos. Well done to all you and your team!!!
I have watched the 4π² proof video and remembered why area of strip would be πd
Absolutely loved this video! I really really would like you to make a thematic series/video on Representation Theory or Category Theory!
Hi! Great video :) I often work on tiling problems, always nice to see them pop up! I believe there is another, maybe simpler case to make 4d relevant: here you are tiling with rhombi that have 3 possible edges directions. If now you allow for 4 edge directions, and you are tiling, say, a sort of octagonal region, then (if I remember correctly) you can still relate any two tilings via the same "cube flip" operation. This is due to the fact that the tiling can again be seen as a 2d surface living in R^4, which as been projected onto a plane. The same cube flip "just" amounts to popping up or down a 3d cell in that surface. And similarly, if you allow for d different edge directions, you are merely playing with surfaces in R^d !
Also, this "lifting up a dimension" is relevant in other tilings as well, even for the (more common) domino tilings, where you can define a "height function", and this is used to show that the natural flipping of two dominos is enough to get from any configuration to any other.
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.
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.
25:04 the faces of the rhombic dodecahedron are not 60°-120° rhombi. they have different angles
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?
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.
This stratagy of looking at a 2d question and making it about three dimensions Has a really nice Hebrew name, מִרְחוּב (mirhuv). 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.
Did not expect such a comment, ממש מגניב.
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!
23:45 You can get the same effect by performing a symmetric rotation of the N-1dimentional shape, so that the orientation of the projected-down vectors changes to the furthest outer vertici.
In other words, the vertici and the intersection of any 3 cells would translate to to the outside facing vertici of an adjacent cell, and vice versa.
If younwant to be extra technical, what youre actually doing is applying a rotation vector equal to N-2 30° rotations for each plane perpendicular to the nth dimention, but thats a less intuitive way of thinking about it for us 3 dimensional creatures.
In fact a more complete version of thie problem would be to perform any 2 30° rotations around the x, y, and/or z axes. Which will always be a symetrical rotation for the rhombic dodecahedron while resulting in a new orientation for some or all of the cells that comprise it.
Woah I just learnt about the desargues theorem! It appears very nicely in P3
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
I find it utterly wonderful and beautiful to realize how ‘things fit into each other’ like this, especially how things in higher dimensions are more complex generalizations of simpler special cases in fewer dimensions.
Thanks for providing such wonder and beauty!
Ghank you, nature and mathematics, for making the 2-D shadow of a 3-D cube a hexagon (from the optimal angle).
thank you for all the time youve spent helping us all learn, it has been worth it and we're always excited for the next no matter how long it takes
Thank you for the work you do !!!
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.
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.
In what way is what you are saying contradicting what he said?
@@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.
I remember your first Numberphile video about a probability puzzle that ended up involving the sum of the volume of each n dimensional ball. Still a mind-blowingly insightful video !
love your work ! very much appreciated ! :)
I have to say, your visuals and explanations are phenomenal. Of course this was always the case with your other videos, but I especially appreciate it here since it's hard to be intuitive with 3D geometry
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.
Thanks again for your teaching me adding and improve myself too
When I saw the "hexagon" puzzle, I immediately started thinking about it as cubes. I feel like the since the video kept calling them hexagons it drove me away from that frame of thinking when it could have been fruitful. I think this may be a good example of how educational videos sometimes can lead viewers to a certain kind of interpretation, even unintentionally, that makes sets up a reveal.
the animation for the first puzzle is so satisfying!
a different intuition for the shapes tiling the 3d space (which also makes it easier to code)
if you rotate the shape you get a cube with 4 4-sided pyramids glued to the 6 sides, the base angles are 45 degrees, so the transition between them is seamless, and they are half as high as they are wide
now if you put 6 of those pyramids facing inwards, they form that same cube
what you do is place the basic cube in a checkerboard pattern: x+y+z=0 mod 2 where x, y and z are whole; the x+y+z=1 mod 2 spaces get covered by the pyramids sticking out of the cube
...and checking the wikipedia page for it, it's well-known, so I guess my comment doesn't matter
This is amazing to get on my feed. I was literally prepraign an internal talk about how granting degrees of freedom to architecture problems truely simplifies our work!
Thanks so much!
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
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.
@
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.
That being said I am only an IOL contestant, so my argument might very well be flawed.
My no-so-rigorous proof for the strips-in-a-circle problem:
1. Start with the single, wide strip.
2. To get less than the 2 that is wide, you have to make it smaller, creating a gap.
3. You can either fill that gap with an additional, parallel strip, or you can try to fill it with a rotated one.
4. Due to the shape of a circle, this gap will always be wider than it is tall, so any rotated strip would need to stretch to that width, bringing you above 2. (a combination of multiple strips only makes this worse.)
And thus, parallel is the only optimal way there is.
You are the man! Thank you for doing these videos.
Monges theorem!
The 3D proof is quite beautiful but I must point out some fine details.
For any two circles there are two centers of similarity, you can think of them as the intersection of the extangents and the intersection of the intangents.
Monges theorem doesn't work if you choose any center you like for any pair of circles, however it still does work if the number of intangent intersections is even.
And if we are already on the topic, here is an entirely 2D proof of monges theorem that is still very pretty.
For some point in the plane we define an homothety to be the scaling of the plane around that point by some nonzero (possibly negative) constant.
Now I won't get into the details but it isn't difficult to show that chaining two different homotheties with possibly different centers still results in a homothety, with its center being on the line of the two other centers.
Now, think about taking the homothety that takes circle A to circle B via a similarity point and then another from B to C, chaining these results in the homothety from A to C so its center, which is a similarity point, must be on the line of the two other points and you can convince yourself that the number of negative homothety centers must be even (hint: What is the scaling factor of the chain of two homotheties?) which concludes the proof for all cases of intangent intersections as well.
This of course works for any shape, not just circles but if the shape happens to be symmetric around a 180 degree rotation then the caveat of even number of negative homothety centers is necessary.
I know a couple of good, related "project into higher dimensions" problems!
In mechanics, if you consider the Kepler problem in either 3D configuration space, where the potential is 1/r, it's a classic result that the solutions to trajectories of the motion are conic sections. However, Kustaanheimo and Stiefel showed that if you project the configuration space into 4D, the Kepler problem in 3D (which is nonlinear) is equivalent to the simple harmonic oscillator in 4D (which is linear), thus showing that two of the most notable solvable dynamics equations are in some sense, the same probem. There's a beautiful theory about how the K-S transformation keeps track of the dynamics of a non-unit quaternion too. I should note that the K-S transformation generalizes a previous, analogous result of Levi-Civita showing that the 2-D Kepler problem can be transformed to a complex harmonic oscillator.
Beyond the mathematical elegance, This has applications in both celestial and quantum mechanics, that I know of--anywhere that 1/r potential shows up.
Thanks for another great video!
This was very satisfying to watch, thank you. And yes, I've had this sentiment before, and for me it is a sort of sadness.
You never fail to make me appreciate your existence.
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
I've often wondered what would happen if we trained an AI that could see in 4 dimensions, i.e. instead of two 2D visual inputs (pixels), it receives two 3D visual inputs (voxels). Since we don't live in 4D the inputs would have to be all artificially generated of course, but in principle such an AI would be "seeing" the 4th dimension.
8:30 this fact still scrambles my mind every time I come across it.
2:45 saving my intuitive response here before watching the rest.
If we look at the presented picture as one of those 3-d illusions (of small cubes in a cubic space in an isometric projection onto a hexagon) and choose the lower sides of the hexagon to be the "bottom" of the cube, rotating any triplet of rhombi is analogous to adding or removing a smaller cube from the cube stack. So you should be able to get from any one state to another just as you can always add or remove a cube from the stack. The two farthest apart states should be the states of the "empty" stack" and the "full stack", and they should be as far apart as the number of cubes they represent - 0 to 4x4x4 = 64 steps.
25:03 The acute angle on the rhombus needs to be about 70.53 degrees, not 60 degrees. Attempting to make that shape out of 60 degree rhombuses would result in a flattened version of it.
People often describe four-dimensional space as "some direction that we can't quite imagine" and "we struggle to really think about what it looks like" and "to you and me, it's inaccessible" but it seems pretty intuitive to mentally visualize after having thought about it for so many years. The extra perpendicular dimension is sort of like a point at infinity that goes ana or kata, like a sort of "inward" or "outward" direction from itself. Especially after seeing 4D golf and other 4D visualizers all while learning about 4D geometry. It's parallel to itself, which makes sense to me. I mean, we don't have a biological camera to see it, but it's like how if you close your eyes, you still have a 3D sense of spatial awareness and proprioception, a 4D space is still conceptualizable in that same way. In my mind's eye, something like a 4D cube is like added density in two antipodal directions where the corners form new edges and each face being extended through that dimension to make a new cube. I mean, it's hard to put into words, but I have a visual spatial conception of it that works in my head, and I think that was afforded to me by watching so many videos that explained it visually. Like 4D spheres falling onto 4D sloped surfaces that look flat in 3D but are clearly a slope in 4D, so they appear to "sink through" then in 3D space. I'm rambling, but I think that having a sensory intuition of 4D space isn't as impossible as people often say. Even if stereoscopic vision doesn't capture it, a broader spatial sense can, and then that in turn can form a visual. Y'know?
lovely video, as always
Hyπr cubes
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
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).
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.
For the tetrahedron puzzle;
Consider points P2 - P1, P3 - P1, P4 - P1, and use those as where I, j, k land for a 3D matrix. Use the determinant formula (which I definitely do remember) and divide by the ratio of a cube's volume to an "inscribed" right tetrahedron, which I believe is 6 but could be very wrong on
That's what I was thinking, but it doesn't seem to use any extra dimension
Yeah, you don't need to go up a dimension to solve that problem. I'm curious to see what that solution looks like though, it seems like you get more symmetry that way.
My solution should be similar to yours, if a bit clunky and not at all elegant nor leveraging any 4D insight:
- The volume of any pyramid is one third the base area times its height. Say we start from a vertex A, and WLoG pick vertices B and C to be the other points on the base, we can write the (signed) base area as one half AB cross AC, and the height is AC dot the unit vector parallel thereto. Combining the above we have the (signed) volume being one sixth the triple product (AB AC AD).
- Pick an origin O and write AB = OB - OA, etc. Expanding the triple product in terms of OA, OB, and OC, the volume V is explicitly equal to ((OB OC OD) - (OA OC OD) + (OA OB OD) - (OA OB OC)) / 6.
- Note how we have alternating signs on the triple products, which are reminiscent of determinants. Taking OA, etc. as column vectors, we can thus write $V = \frac{1}{6} \left\|\begin{matrix} 1 & 1 & 1 & 1 \\ OA & OB & OC & OD \end{matrix}
ight\|$.
Thank you very much
the argument for problem 3 breaks again if you align the circles such that the centers of similarity are all the same point - then the tips of the cones are co-linear and thus only define a line and not a plane.
of course, it is trivial that 3 identical points line on any line through that point, and the intersection of the line through the tips with the original plane is the mutual center of similarity, but this case does need to be handled separately.
11:41 : I see a projected 3d scene where 3 spheres of the same size are resting on an infinite plane.
The white line is the horizon.
You connect pairs of spheres with infinitely long cylinders, and the outlines of each cylinder join at the horizon.
This exact kind of thinking has some applications in physics: to deal with 2D hexagonal crystals like graphene (eg to compute how they conduct current at the quantum level), it is common to use 3 basis vectors (the ones shown in the video resulting from the projection of the 3D ones) although they are not linearly independent, because it preserves the symmetries of the crystal!
Needless to adore your intriguing animations that make us reliable to discern a wide variety of concepts and prone to analytical thinking....yet i had a question, would you mind elaborating the uniqueness of diagonal lines and planes traced by tesseract (ik it could be analysed via flat landers' pov(s) of a square / a cube), the 2D diagrams of 4D object aint aids me up, i guess, which the animation can.?
You deeply broke my mind. Thank you 😉.
I solved Monge's theorem by using 3D in a different way.
Imagine each circle is the outline of a sphere (or just the same circle in 3D), where all the spheres are the same size, arranged in 3D space and viewed by projecting onto the screen.
Then the "center of similarity" of two of these spheres is the focal point of the "tube"/cylinder around both spheres, which is the same as the focal point of the line passing through the centers of the spheres.
Since the centers lie on a plane, so do those lines. And so the focal points lie on the horizon corresponding to that plane, which is a straight line.
This solution also doesn't care if one is inside another or the specific shape since you can take any point instead of the center, as long as it is in the same place for all shapes. (Similar to the point of the pyramid in the "fixed" proof)
im with you. three identical spheres sitting on a plane, viewed in perspective. a scene like an old raytracing demo. i feel like it takes a little bit more work with this picture to prove this works for arbitrary figures, but the intuition is strong
From what I can find, it seems that the cone argument for Monge's Theorem originates with the engineer John Edson Sweet and was then used by Monge to prove this theorem (which was first proposed by d'Alembert).
See page 153-154 of The Penguin Dictionary of Curious and Interesting Geometry by D. Wells. (It cites to the book Ingeneous Mathematical Problems and Methods by L.A. Graham for the quote by Sweet, but I don't have access to this for further verification.)
writing this as grant revealed the 3 dimensional projection of monge's theorem and not having seen the numberphile video because I quickly thought of my own answer and was surprised to see it didnt at all match what is shown in the video;
immediately upon seeing the problem, I thought of vanishing points in technical drawing, and so I consider the circles not as 3 circles with different diameters but 3 spheres with the same diameter. this means that the point of intersection of the tangent lines must be a point infinitely far away but colinear with the 2 spheres.
since there are 3 spheres, each with it's own centre point, this means they all share a plane.
since the cotangent points are colinear with their spheres, they must also share any plane their parents do. so, all 6 points share the same plane.
since the cotangent points are also infinitely far away, that means that they lie on the perimeter of said plane.
so, how do we prove that the edge of an infinitely large plane is straight? the same reason the horison of earth seems straight, even though it clearly curves as we raise our altitude.
the infinite plane is in fact the surface of an infinitely large sphere. even though the surface of the sphere is curved and so is it's horizon, both appear flat as the ratio of the diameter of the sphere relative to our distance from it increases.
in fact, since the sphere is of infinite size our distance from it doesn't matter because at it's limit, the surface of any infinitely large sphere appears like an infinite plane and it's horizon appears as a straight line regardless of our distance.
the 3 cotangent points lie on the horizon of this infinite sphere, and this horizon is approaching straight
so, you can imagine my surprise when the video didn't match my intuition at all. anyway, back to the video. I'm sure the real answer will be similarly elegant.
Recently I stumbled upon area of applied mathematics called: Topological Data Analysis. I'd love to see your take on this : )
17:24 That's not a pyramid! That's a pi-ramid!
Another problem in the same vein is as follows: find a 3x3 matrix A =/= I with integer entries such that A^5 = I. Sadly I can’t remember where I first saw this any better than “in a tweet once”
Was feeling particularly masochistic the other day and scrolled through the Wikipedia page on x-ray crystallography. This video hurt my head a good bit less than that. Now wondering how related Demis Hassabis's protein work is. I guess I feel a little bit better knowing that this sort of stuff can make somebody like you sad, too. 🙂 Now definitely have to go back and make myself finish watching the video on quaternions. Simply mind boggling to me that there are apparently tons of humans that "get" all this stuff.
itd be just the greatest thing for you to explore the beauty of the truncated octahedron. optimally fills space, order 4 permutahedron, a slice of a hypercube, unification of hexagon and square, space filling with either squares or hexagons removed to create new space filling shapes, more and more
The truncated octahedron is not a slice of a hypercube. The hypercube only has 8 cells, but the truncated octahedron has 14 faces.
And? The rhombic dodecahedral slice he shows has 12
@@galoomba5559 but i did call it the wrong thing, its actually a slice of a tesseract by a hyperplane, so thank you
I'm happy I woke up and watched your video :)
*@[**11:43**]:* Will the proof for this involve a circle's omni-directional symmetry?...
that was trippy, when that last cube git removed at 5:55 my brain inverted tye empty space and when the next one got added it like warped back to being concave
The three sphere problem - I imagine the three spheres to all be the same size but at different distances from my point of view. That tangent line is where the plane through the centers of the spheres intersects my viewing plane. I think. Something like that, anyway.
Here's another case where thinking about higher dimensions might be useful. It's possible that the universe is curved on itself. However, it's difficult for the human brain to get an idea of what the curvature of 3D space would look like. It's easier to think of the same problem in smaller dimensions. For example, a line (1D) can be curved into a circle (2D). Similarly, a plane (2D) can be transformed into a sphere (3D). Logically, trying to visualize a curved space in 3D would force us to think in 4D and that is why it's so hard.
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!
Can you elaborate more? I am curious
17:25... so, would that be a pi-ramid...? ;)
просто кайф, ты сам додумался или такой способ уже где-то был? 😊
Он этого не сам передумал, но это все ровно очень красиво.
I don't enjoy doing criticisms (even less to a channel that I love as much as this one), but as no one else is pointing it out, I fear that I must do so: the audio isn't as good equalized as usual (some parts of the narration are a bit too loud). I hope this helps in some capacity and look forward to the next videos ♥
да уж, согласен, это не просто развлечение на вечеринке ботаников. С помощью такого подхода можно решать много сложных задач, где решение не очевидно на первый взгляд. Если была бы нобелевская премия по математике, то этот подход к решениям ее получил бы
I wanna make video to add new perspective about "Dimensions", seems like mine kinda forgotten or maybe long gone forgotten hypothesis. Why not start the dimension from 0D = 1 value of D aka 0=1 and so on, means a singular dot can form a "length", not point A to B and between them imaginary line/dimensional form. From those length produce more same length as diagonal length and picture a cube (you can picture an orb/globe but those body is already finite). Voila you create cube skeleton/ construction, so now what?
BEHOLD inside those skeleton, you can fit 6 pyramids! How many dimensions now? 1st. 0; 2nd. Single Length; 3rd. Cube Limits; 4th. Addition of 6 Pyramids; 5th. ??? yeah same rule, add dot(s) as a fixed anchor, put length, put limit, add pyramids = Breaking new dimension!
I got my hypothesis of this "new" perspective because isn't so obvious that in "order of operations" you can found after counting -> addition -> multiplication -> exponent -> tetration -> (and beyond) there's likenesses pattern, "I'm too lazy to count so let's create new limit(s)" and BOOM! new realm of mathematical dimension(s).
I'm really glad the solution to the first puzzle involved 3d cubes, because I couldn't see anything but 3d cubes.
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
Wait, he just revealed my way of thinking
maybe being a gamer isn't so bad after all
Same
Immediately reminded me of that youtuber who beat minecraft in isometric view (ghosttea i think)
Monge's Theorem should work on any convex hulls, so generalizes easily to any vanishing points and objects. This means the "But actually" argument is simply asking why a trivial or uninteresting drawing has no interesting interpretation.
Its mind blowing to imagine how to view 3D, instead of merely experiencing it, Leave experiencing 4D. Also, the diagonal of a "4D square" would be 2 times the 1D edge, which seems logical as for 3D, It's root3 times, and for 2D, It's root2 times
A year ago I stumbled upon your linear algebra series, knowing barely anything about linear algebra or geometry, but at problem 4 I immediately thought "I know this one from triangles." In my head I think of it as the "shadowpuppet matrix."
The tittle of this video becomes haunting you should have dropped this for Halloween. Amazing!!
18:20 Why wouldn't a 3x3 matrix work? My thinking: Translate shape such that one point is at the origin. Take 3x3 determinant of the 3 remaining points and half it to get the answer. Same is the case in the 2d example.
I wonder why we mostly refer to hypercubes or tesseracts when discussing 4D geometry instead of hyper-tetrahedron
To form a 1D line, we need at least 2 dots; for a 2D shape, we need 3 lines; and for a 3D polygon, we require 4 shapes. Therefore, to create a 4D hyper-shape, at least 5 polygons are needed. So why don’t we use a hyper-tetrahedron?