Thinking outside the 10-dimensional box
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- Опубліковано 27 вер 2024
- Visualizing high-dimensional spheres to understand a surprising puzzle.
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"The goal here is genuine understanding, not shock" You're the best.
1-dimensional array: a line
2-dimensional array: a square
3-dimensional array: a cube
4-dimensional array: death incarnate
4D=tesseract
@@MattacksRC You mean it has a bunch of cubes. Tesseracts are made up of cubes, not squares.
@@kerseykerman7307 Not just a bunch, infinity ammount of cubes
@@SimonPegasus yeah 2nd dimension is just 1st dimension with a lot more dots
ten grams to snifindor
22:04
he said the correct number of "one"s. now I'm impressed.
I counted them too :D
i counted them
how many
@hanstheexplorer 10
7:59 Why imagine yourself as a bug on that small sphere? A human on a really, really big sphere works too, and you don't even have to imagine that.
@Bruce U stupid bug
Because the Earth is flat. Duh!
*no*
You've definitely triggered a flat Earther... Oh wait, there's no way they understand maths in 4th grade or above.
This is the generic analogy for understanding the idea of manifolds. Classically I have always heard "ant" and it emphasizes you are small and thinking locally.
Those numerical sliders are like Braille for higher dimensions.
Ron Mar Best Comment
this comment poke my all dimension at the same time....
are you rich yet? xD
Braille in a hyper-plane
from now on i will only buy higher dimensional real estate seems more profitable currently
we will build a wall to keep the low dimensions out and make the high dimensions pay for it
Blox117 yeah lets do it. Also is that a reference to Trump
@@pokemoncatch6727 yeah its a joke on his build a wall statements
you should check out what *some* people are doing in selling (or trying to sell) virtual real estate, like in VR worlds
Jonathan Bryant
VR worlds should be free, for humanity!
A mathematician and an engineer attend a talk given by a physicist about string theory. The mathematician is obviously enjoying himself, while the engineer is frustrated and lost, especially when the physicist starts talking about higher dimensions.
Finally, the engineer asks the mathematician: "How can you possibly visualize something in 11-dimensional space!?"
The mathematician replies: "Easy, first visualize it in n-dimensional space, then let n equal 11."
John Chessant Get the fuck outta here, AND CLOSE THE DAMN DOOR ON THE WAY OUT!
John Chessant That wasn't even a joke. It was just *straightforward*
Lol
This supposed "joke" seems like the embodiment of r/iamverysmart. I mean, there''s not even a real punch-line here.
Edvin K It's a joke about mathematicians being detached from reality. It's like the spherical cow joke but without the buildup.
VISUALIZE HIGHER DIMENSIONS WITH THIS 1 WEIRD TRICK!
STEM PROFESSIONALS HATE HIM!
SoopaPop
Haha lol 😂😂
There was a very serious part of me that almost did that for the title :)
The art of clickbaiting.
use sum dank DMT and u'll see the world made out os Calabi-Yau manifolds
DIMENSION 6 WILL SHOCK YOU!!
Hey, im leaving this message to say this video really helped me with my research in economics
It has nothing to do with high dimensional spheres, but something about the investigative way you approached the problem made me remember about the way i used to approach mathematical problems when i was younger, and it gave me insight that might help me with a problem ive been stuck for more than 1 month
There is something about this way of seeing math that is very powerful, and that frequently gets lost when we are too deep in more analytical and formal approaches. It is hard to define what exactly it is, but this channel's videos are very good at inspiring it
It blows my mind how brilliant some peoples' minds are
This is not even a glimpse of how brilliant people like Einstein really are. He invented General relativity, which describes the movement of objects in a 4 dimentsional space which is not even flat 4 dimentional space, but curved space, and it is the matter in that space that is responible for how the space curves, over 100 years ago when geometry in 4 dimentional space was not even knows...
Philippe Durrell: 🤯
To be fair, Einstein did receive some help from mathematician Minkowski
and to blow your mind even more, chances are the smartest people to have ever lived died without ever leaving a trace of their thoughts behind
beepybeetle and unfortunately due to the internet, idiotic fucktards that believe the earth is flat, space is fake, etc. are forever enshrined
Never forget that if someone sells you hyper-oranges in dim-87 by hyper-weight, you'll mainly pay for fruit skin.
i love this
I always get your videos until the halfway mark, after that it all goes over my head.
multiplying information ^^
i am at 16th minute mark and i am down to comment ^_^
same
same happens to me. after half way mark i have to often rewind and rewatch segments multiple times. i think thats ok with a information dense video like this. its tough being 100% alert for 27 minutes.
*It's free real estate*
Yeah but it's mathematical swamp land. You get what you pay for.
It literally says it isn't free, though.
i was thinking of that meme XD
got beat to it
Came to the comment section looking for this comment. I wasn't disappointed.
x: moves a little
y: it's free real estate
dynadude lol
AHAHHAA
I can't understand what real estate means in the video.
@@squealer7235 so basically it's like the point in space that you're supposed to conquer
and they're mutually dependant on each other
imagine a line segment where x+y=1.
you (x) take some part of the land, so obviously the other part belong to (y)
similarly, in the next (2nd) dimension, the curve is a circle defined by x² + y² = 1
so by the relation, as x moves a bit, the length y covers is termed as the "real estate" of your and vice versa
No no no no....you wrongly knowledge
I am indian i am the best of the rest world.i need 5 Nobel prizes
Minimum..
X is a male..gender
Y is female..gender
This is Univesel formula
Don't Changes knowledge. ..
-Quality Content
-Clear Information
-Awesome Animation
-No Ads
-No BS
I need more channels like this
ua-cam.com/channels/KzJFdi57J53Vr_BkTfN3uQ.htmlvideos
Captain Disillusion
Thanks guys, you're both a great help. I subbed to both channels. But I'm still looking for more channels :D
@@ArxxWyvnClaw I don't mind ads though. The man is working his ass off to educate us for free. Least we can do is allow him to survive, even if it means watching a couple ads every video.
@@ArxxWyvnClaw Numberphile?
Thank you! This will come in handy next time I'm stacking my 8-dimensional oranges..
This video helped me a lot on seeing Trump's 4d chess moves from a mile away.
But a mile is so small in 4 dimensions... =D
This helped me a lot when looking through my 6-dimensional walls trying to find where my 9-dimensional pen is, lost it in the 6-dimensional wall
This helped me a lot when looking through my 6-dimensional walls trying to find where my 9-dimensional pen is, lost it in the 6-dimensional wall
8-dimensional oranges? Make sure to use the E8 lattice, it's the best!
Thank you so much for explaining WHY it happens, not just that it does. I never imagined the answer would be so simple. The sides of the cube stay the same but the corners get farther and farther away because more dimensions are contributing to them.
I've typically use a couple of methods to visualize higher dimensions: 1) imagine variation over time, 2) variations of color across the spectrum. The sliders map more easily to the vectors used in analytic methods, and give a better feel for what actually happens at higher dimensions, so I'll definitely be adding this to my "toolkit".
3) adjusting the "w" slider
Grant. youre a deity.
Hi, Jabrils! You are one of my idols! Cool to see ur 25 like comment in a random comment section...
@Danko Kapitan haha
27 minutes!
I'm not watching it!
..Maybe just a couple of minutes out of curiosity
...*watched it all*
I've just watched 27 minutes?!?
the same here hahahaha
and you think anybody cares that you do not watch it?
JAY CURVE and do you think we care about your opinion ?
FlingFlexer 11 minutes in came here to check the comments, f me right
I beg you; please make a series on tensors (contravariant & covariant) , curvature, manifolds etc. Thank you so much for what you are doing for us.
If you truly want beauty please do a video on tessellation and matrice theory. The correlation is astonishing and sublime
Yes differential geometry would be awesome with his clarity and animations
Manifolds and their diffeomorphisms are very obscurely introduced at uni
Dude, that would be so cool..!
Coordinate: *Is close to zero*
3B1B: *It's free real estate*
thats funny
This is some gorgeous animation
Glad at least someone appreciates how much went into all of that.
I really wonder how 20-dimensional beings will think about the 2/3D 'sphere packing' problem. "What do you mean that the inner circle is **smaller** than the outer circle in 2 dimenions?! That has to be impossible! This 2-Dimensional representation is just completely wrong! It has to be wrong!"
20D beings would understand 2/3D efortlessly, just like us, 3D beings naturally understand 1/2D.
Just wait till fractal beings with fractional number of dimensions join the talk...
Or imagine those 3 dimensional beings trying to understand 2 dimensional circles. They’ll be like “the square root of 2!? That can’t be possible it must be 1.8...”
@@thomasbeaumont3668 Now wait a minute here-
Ah so this is how Doctor Who's TARDIS is bigger on the inside than the outside.
OH MY GOD IT WORKS
Time And Relative Dimensions I guess. Doesn’t specify 3 I guess, so it works
Y'know, the classic non-euclidian space.
Ohh yeah!
mind = blown.
How do you expect people to visualize a sphere in higher dimensions when flat earthers can't even visualize a sphere in three!? ;)
earth is not a sphere :D
Hes not talking about earth being a sphere
Hes talking about normal spheres like basket balls or soccerballs, basic stuff
r/woooosh
fact:earth is an oblate spheroid
We exist on the 2 dimensional surface of a black holes event horizon. 3rd dimension is just a hologram, illusion.
I just found out this channel and it is fantastic! Please keep making great videos!
U Wot M8 John Cena dun dun duuuuuuuuuun
Just happened to me. I have an exam today!
Now back to studying
The saying: "Think outside of box", got new dimension.
I had a realisation during this video that really helped my understanding, so I wanted to share it. It involves thinking about the proportion of the n-dimensional cube's surface enclosed by the corner spheres.
In 2D space, all of the surface of the square is enclosed by the corner circles. There is no way to draw a line from the origin through the square without touching one of the corner circles.
When you move up to 3D space, you can already see this change. Focus on one square face of the cube. Notice the diamond shape that's not covered by any of the spheres. There's plenty of room to draw a line from the origin through the box without coming close to any of the spheres. In fact, it is this diamond shape that we filled in the 2D problem!
Things get a bit harder in higher dimensions, but our 3D visualisation can still help in the 4D case. Visualise the solution to the 3D problem. Think about all the space inside of the cube that we filled with the inner sphere, there's actually quite a lot of it. Now, in the same way you can visualise the 2D problem as one of the faces of the 3D problem, think about this solution to the 3D problem as one of the faces of the 4D problem. You can actually see how there's quite a lot of space there that's not covered by the corner spheres!
The visualisation breaks down after this point, but thinking inductively how each solution is a "face" of the problem one dimension higher, and how the space left over creates extra space for the next dimension to use was helpful for me.
This has always pissed me off that I can't visualize in higher dimensions when it's sooooo bloody tempting. But if you think about it, it's not that our brains haven't evolved to see in 4 dimensions or anything like that, it would be physically impossible to do it. You'd have to visualize infinitely many 3-D "slices" simultaneously to perceive anything 4-dimensional.
I would give literally anything to be able to "see" in higher dimensions.
The Flagged Dragon it's easy! Take some LSD ! That is leave your body via OBE , NDE ! Many have. I have.
There's a VR game on Steam called 4D Toys which lets you interact with 4D versions of children's toys by selecting a 3D slice. It can be played with more conventional input and output devices, but I haven't tried it out in either form.
Hallucinogenics just trick you into thinking you've had a profound experience. Colorful shapes and brain-fucking isn't going to shed light on any real truths in the universe.
+Hanniffy Dinn - Nothing about "higher dimension" with LSD... Alternate state of consciousness doesn't "open" any dimension... It can swap perceptions, it can do "post effects" on your 2D projected image, nothing about ">3 D" view...
Or you're getting too poetic here, and words have no more meaning anymore, which would mean that you are actually high on LSD or other deceptive drug... :)
You'll need a four-dimensional creature to take you out of this 3D world, like in the Flatland.
This man has solved the housing problem. Real estate agents hate him!
It's pretty interesting that even though it is so hard to imagine the possibilities of universes in higher spacial dimensions, that the mathematics in those universes will always be the same with our's. It is nearly impossible to predict the properties of these universes, but the language of math will always be universal.
Or you know, multiversal I guess...
@@realityversusfiction9960 what the hell are you talking about
@@peterpemrich6962
😂😂😂😂😂😂😂😂😂😂
Your sliders there actually look like a stack of discs in a row to me with the graduations. Which actually really helps with visualising how points change for multidimensional spheres. Thanks!
This is how I visualise higher dimensions, too
16:51 my brain started to shiver... I think that enough math for today...
This channel is goddamn brilliant. Love your visualizations.
As a computer science student, I've had to deal with some of the weirdnes of high-dimensional spaces up close, when using geometric methods to analyze high-dimensional data. The unit cube alone is so incredibly weird.. mainly because it has an exponential amount of corners. Meaning that if you slice off even a tiny region surrounding each corner (and they are all at east distance 1 apart, so these tiny regions don't overlap, and therefore add up directly), those tiny regions comprise nearly all the volume of the cube.
You can sort of say that properties that can arise from the actions of single dimensions are common, while properties that only arise with the "agreement" of many dimensions is rare. It's even sort of hard to create a large high-dimensional volume, because all of the dimensions have to be large together, if even one of them is small, the volume is small. It also has it's good sides, it means when we do optimization problems, we rarely have to worry about local extrema, since they can only happen if all dimensions curve in the same direction, whih is difficult to arrange.
I'm glad I live in 3 easy dimensions.
Or you could show the 4th dimension using not 4 sliders but 2 2D flats
For those interested: Cubes in this setting would have the rule that at most one slider is not at the edges of the line (+-1).
That doesn't seem right. Even just in three dimensions, the point (1, 0, 0) is on the surface of the 2x2 cube centered at the origin but there's more than one slider that's neither at 1 nor at -1. What you have are the edges of the cube rather than the surface.
I have a simpler explanation for this:
When the circles are replaced by spheres, and so on, the length between the edge of the inner circle and where the outer circles meet each other stay the same, regardless of dimensions. the volume on the other hand, is increasing with each dimension.
Therefore, the space between the shapes grow slower than the volume of the shapes themselves as they enter higher dimensions.
I don't see what's your point!
You can also visualize this, sort of, by first realizing that the long diagonal of the n-dimensional hypercube is increasing without bound. Now, the hypercube in n dimensions will have 2ⁿ long diagonals, and each diagonal will pass through exactly two of the hyperspheres. The radius of each hypersphere remains a constant equal to 1. So, any diagonal is getting longer and longer, with a fixed part of the diagonal inside two of the hyperspheres. The distance along the diagonal between these two hyperspheres has to be increasing without bound.
Higher dimensions: *exists*
3Blue1Brown: *IT'S FREE REAL ESTATE*
well put
Actually, its just cheap real estate. But good meme.
shitty meme, and no, real estate is not free
@@bazzcs yeah the value of that meme has really plummeted over time
Ps3udo that was a long time ago, shush
I wish I could like this video in more than one dimension, because it is the best video ever on helping (at least me) REALLY understand a lot more about how higher dimensions work. And I've watched loads of them. Matt's Numberphile video about "Strange Spheres in Higher Dimensions" was a good companion piece to help me get started, but this video completed the home run (to mix a few metaphors).
I think you should make an updated version with few changes, republish it, and just call it "understanding higher dimensions." Because it does far more than explain just the 10th dimensional riddles.
Whenever I get a new job, I'm becoming a Patron.
I think it's kind of cool that I was able to learn that the Pythagorean Theorem scales nicely with higher dimensions all on my own. I saw a line in 3D and was thinking "... I want to find the length of that.". I projected the line into 2D and found its length there. I brought back its Z data, armed with the knowledge of its length to X and Y and found it's length.
Maybe we should keep every extra dimension constant and change 1 at a time and watch how x y z dimentions change. The pattern we'll see will be characteristic
If you do it with 4D, the 3D sphere will grow and shrink
I've been so confused about this for years and this is genuinely first time I've felt like i understood this. THANK YOU!
I've been trying to wrap my head around higher dimensions for a while, and getting nowhere. This video is the first which has made a dent. Thank you so much!
The reality is, 4 spacial dimensions don't actually exist. When he describes a "hypersphere" its actually fake, and non-sensical. Just like the number 3 can represent a range of things and be used for a bunch of things, a vector simply describes 4 independent "dimensions" where the dimensions are not necessarily spacial. Up to three dimensions sure your vectors can represent physical space, but after that they can only represent things like polynomials up to the fourth degree, solutions to the four dimensional equation like in the above video. These vectors aren't 4 dimensions spacially, its like saying a number is 1 dimensional. Numbers don't have spacial dimensions, neither to vectors
I used to imagine a 4d sphere as a continous collection of 3d spheres in some time perios, first a small sphere that appears at some time and at some point from a dot, grows up to the entire radius at the "middle time", then shrinks until it totally disappears at some moment. Every moment is that "3d slice" of a 4d space, where a 4d sphere is located. Sometimes I feel strange for thinking about such thinkgs, but hey, it's the reason I watch videos like this :D
And I’d guess the sphere grows and decreases at a sin rate
In fact, the thing you have been visualising, was the case of 4d sphere entering the 3d dimension space...for better understanding of what I mean, it's easier to step one dimension down and imagine of regular sphere (3d) going through sheet of paper (2d)...on paper surface, first it would appear as a dot and progress to grow until it's max radius...then getting smaller again :)
@@555artisan if you were a 2 dimensional being, it would look like a line growing then shrinking
In math's your not supposed to relate things to reality, its a complete abstraction away from reality. Four dimensions should be thought of as four independent real numbers. Thats all it is. The hypersphere has no shape we can visualise, we are simply calculating distances based on what we know, except instead of 3 sliders we have 4. Its not actually modelling a 4 dimensional shape at all, thats a 3 dimensional application of 3 real numbers. This is mathematics not physics.
Got that 1st d
The 2nd
3rd
But can't even imagine the fourth
@Evi1M4chine To hell with five dimensions. Who likes to try being a puffer fish?!
but the inner sphere is exactly the same size as the outer spheres!
that really doesn't help at all?
it's so much f***en rounder than 3D spheres...
As always so grateful for your creative work. After much reflection and reviewing of this video here are a few of my key takeaways.
For problems in 2 and 3 dimensional spaces, finding solutions is greatly helped by our ability to move back and forth between analytical and geometrical expressions of the problem. For problems in higher dimensions we do not have access to this back and forth sharing of insights. Many if not most problems are formulated in these higher dimensions.
This video provides an example of a counterintuitive insight that did not emerge at all in 2 or 3 dimensions. Namely: in higher dimensions the radius of the inner hypersphere, tangent to all N boundary defining unit hyperspheres whose centers are one unit from the origin as measured along any one of the N orthogonal axes is greater than the distance from the origin as measured along any single orthogonal axis to the surface of the N hypercube that encompasses all N unit hyperspheres. This is indeed counterintuitive.
After grokking how you beautifully and skillfully led all of us to this insight, I was able to distil the following. To make the counterintuitive point, there is no need to introduce the growing inner hypersphere at all. Even in 2 and 3 dimensions the distance from the origin of any N dimension space to the center of any boundary defining unit sphere is always greater than one and grows as sqrt(N). So already for N=4, the distance from the origin to the center of any boundary unit hypersphere is greater than the distance along any one orthogonal axis from the origin to a surface of the hypercube that contains all N unit hyperspheres.
The final grok (so far): the core insight is about the concept of distance. Not about radii, not about hyperspheres, not about boundary surfaces of all containing hypercubes. Distance from the origin to the centers of unit hyperspheres as you distribute them will always be significantly greater than the distance from the origin along an axis to any one of the containing hypersurfaces as you define them. Except in 2 or 3 dimensions.
In yet another way: the N dimensional distance from the origin to any center of a unit hypersphere as you distribute them is always greater than the distance from the origin measured on an orthogonal axis to a hypersurface defined by your distribution of the centers of the unit hyperspheres. This is true for 2, 3 and higher dimensions. For the hypersurfaces containing all your unit hyperspheres, the distance from the origin measured along an orthogonal axis to any such hypersurface is always less than the distance from the origin to a center of a contained hypersphere except in 2 and 3 dimensions.
Your videos are more accessible than my words. But my struggle to put the insights from you into these words deepens my ability to grasp what you offer. Thank you so much!
Corrections and/or improvements to my text are most welcome!
Something in your explanation/insight (perhaps the 4th paragraph) made it very clear to me:
Each time you add a dimension (and continue to say that a corner is at (1, 1, ..., 1)), you're moving the corner "further away" from the origin. But the sphere anchored there stays the same radius. No matter the number of dimensions, each sphere can always only extend "1" towards the origin from its corner, which is moving ever farther away.
4:47
that's free real estate
Great stuff. The slider illustration is exactly what statisticians are using when they display higher dimensional data with a "parallel coordinates plot". They connect the coordinate dots with lines and one can then find clusters.
Not just Statisticians. Please see "Parallel Coordinates: Visual Multidimensional Geometry" by A. Inselberg. Among others this book was also praised by Stephen Hawking. It is remarkable that it is not referenced.
So basically this is how the TARDIS is bigger on the inside.
Master Marf oh yeah that makes sense now
The visualization at 7:40 REALLY made it clear. This is one of your best videos to date, imho!
Which software do you use to make those incredible 2D animations? They're great!
He wrote his own, actually! He even has the source code posted here (he uses Python):
github.com/3b1b/manim
Though he does say this is not the most user-friendly software ...
This reminds me when we calculated in physics that a high dimensional sphere has almost all of its volume on its infinitely thin surface , this blew my mind and so does this
Wait, why?
This is how I imagine stuff:
4th dimension: The colour of a point in a 3d "space"
5th dimension: The brightness of a point in 3d "space"
6th dimension: Change over time, maybe?
Have you read Matt Parker's book "Things to Make and Do on the 4th Dimension"? It mentions this and he also is a mathematics youtuber [StandUpMaths (you probably know his channel already)].
Scott Goodson I have and it's a ton of fun!
+
Who wouldn't know of the great Parker Square?
I first heard of this in that book. It turns out that spheres are a lot spikier than we tend to think they are.
Nillie Isn't it that the cubes are spiky? If we say that "spikyness" is how much the distance from the center of the object varies, an n-dimensional sphere would be the least spiky object. The cubes, on the other hand, have verticies that get farther from the origin as the dimension increases while the center of each face stays at a fixed distance.
I'm not super great at math, and this was very intuitive for me, and easy to understand. Thank you for creating and sharing this. :)
Michael Jensen mayhe you are good at math and just don't know it because this was really hard for me to pay attention to.
1d: a string
2d:paper
3d: a plane
4d: What 3blue1brown talks about
(x^2 goes down)
y^2: *_it's free real estate_*
10-dimensional cube: exists
Sphere: *i t s f r e e r e a l e s t a t e*
20:22 When the lesson will finish in 5 minutes and teacher hurries:
uuuuuhh. Had this explained before, didn't actually realize how crazy that was. Now my head hurts.
So in infinite dimensions, the corner spheres have zero area, and the inner sphere has infinite? Wow.
AMAZING video by the way, now I get why this works!
Naviron Ghost technically the outer spheres are ALWAYS 1 but the inner sphere approaches infinity (but never reaches it ;) )
I think the best way to understand how the center sphere fits in the 10 dimension example is to create a 2D slice of the setup intersecting the center of the sphere and two opposing edges of the outer bounding box.
This results in a rectangle with the same width as the 10D box, but that is 3 times as high (assuming the x-axis is aligned with one of the basis vectors that defines the 10D box). The surrounding spheres show up with same radius as their 10D counterparts, 2 above and 2 bellow. These circles just fits within the horizontal edges of the rectangle but have plenty of room above and bellow. And the in-between vertical space has just enough room to fit the center circle with it's 10D spheres full radius, while just touching the outer circles and sticking out past the horizontal edges of the rectangle.
"The goal here is genuine understanding; not shock." Liked the video as soon as he said that.
3Blue1Brown: The only content creator that's get's a like on the video BEFORE I watch it.
Excellent video. This makes the counter-intuitive nature of high dimensions make sense. Here are a few facts that were just touched on in this presentation:
An infinite dimensional sphere has no volume, regardless of radius (volume has no meaning in an infinite dimensional space)
Every (multi dimensional) unit sphere contains all the unit spheres of lesser dimensions
In higher dimension spheres, the vast majority of the volume lies near the equator, as does most of the surface area
In higher dimension spheres, the vast majority of the volume lies near the boundary
The most efficient sphere packing in dimension 8 is E(sub 8), and in dimension 24, it is the Leech lattice
Each sphere in the dazzlingly symmetrical packing in the Leech lattice touches 196,560 other spheres
In 8 dimensions, the densest packing fills about 25% of space, and in 24 dimensions, the best packing fills only 0.1% of space
High dimensional hypercubes are spiky - as Dim for the unit hypercube goes up, the longest diagonal is sqrt(Dim)
0:06 tryna ruin my no nut november
Notch likes this
I haven't given any attention to math since grade 11 physics and I am 42 years old and just thought id say that watching this video it gave me an amazing amount of clarity to a topic I haven't addressed in over 20 years. that has to say something for putting things in to visual perspective.
The trick to visualizing other dimensions is impossible because other dimensions are not visual.
They're like frequencies on a radio
Well, yes and no. You can't visualise them as true objects in 2- or 3-dimensional space, but you can _project_ them down to 2 or 3 dimensions and try to _infer_ (deduct) their shape.
That ending quote...
Man, what a mic drop!
I solved this problem. the formula that I found for the radio of the n-circle in the midle of the structure that have this properties is ((n^1/2) - 1)/4 and i deduced this formula uniting the centers of the bigger n-circles to form a n-square that has an edge that is a half of the edge of the original n-square that is in the structure and by finding the diagonal of this new n-square that was formed and subtracting it by the two radios which are summed a half of the edge of the original n-square and dividing that by 2.
Wait ! If, in 4 dimensions, we can have 16 spheres touching themselves (whose centers are the corners of an hypercube of side 2, and whose bounding box is an hypercube of side 4), and if the inner sphere also has radius one, that means we can pack 17 identical spheres inside the larger hypercube ? And is the central sphere special in some way ? I'm puzzled. Very nice video with cute animations and explanations, indeed !
yes, at 4 dimensions we'll have 17 identical hyperspheres. but again, it's impossible to visualize, hence the slider example.
he's showing that, in higher than 4 dimensions, the center n-sphere that touches each surrounding n-sphere exactly once, will be larger in size than those surrounding n-spheres.
it's counter-intuitive because we only know how to visualize 2 and 3 dimensions -- but that's the very point of this video.
To get the idea where does that higher-dimensional inner sphere pokes out of the box: Imagine 3D case with cube, and 8 spheres in its corners. Those spheres cover all the edges of the cube, but they do not cover all the surface of the cube. The centre of the sides is not covered by the corner spheres. And this is the place, where most of the inner sphere "leaks out" of the box in higher dimensions.
In 3 dimensions, the inner sphere cannot reach that areas. However, in higher dimensions, the corners of the cube (and therefore the centers of the corner spheres) are way further away from the centre of the cube, giving more room for inner sphere to grow, eventually reaching to that area.
man, what do you do for living? I wonder who has the time, desire, interest and knowledge to make such a high quality video
So thinking in 5 dimensions, is thinking outside the box?
Why are they called higher dimensional spheres, why not higher dimensional circles?
A circle is a 2D sphere
@@botigamer9011 A circle is a 1D sphere and a sphere is a 2D sphere, the surface of a 4D Ball is a 3D sphere
1d line
2d circle
3d sphere
(n>3) hypersphere
@@therobot1080 No, sphere refers to the surface
a 3-ball has a 2-sphere as its surface, and this surface is what people typically refer to as a "sphere"
@@therobot1080
And a 1-sphere is not a line, it's a circle
a 0-sphere is the pair of endpoints of a line, a line is a 1-ball.
This made it *so simple.* This area of knowledge is not _my major_ in any sense, but I felt enveloped. I take special notice of your narration and how you leave some breathing room (needed!) often enough.
The corners of hypercubes get "pointier" the higher you go in dimension. the corner of a square takes up 1/4 of the circle around it, the corner of a cube takes up 1/8 of the sphere around it, etc. So hypercubes in higher dimensions are SUPER pointy -- the corner of a 10-hypercube only takes up 1/1024 of the volume of the hypersphere around it. So if you think of a cube that's been squished in on the faces so that the edges/corners are extra pointy, that's kinda what's happening in higher dimensions, and that's why the middle sphere can get so big. Because the outer spheres are stuck at the tips of these points, really far away from the middle.
I wonder if there is a way to put the 4D representation of this in some kind of 4D engine like 4D Toys.
omg i'm so excited for marc ten bosch's game miegakure when it finally comes out! i've been waiting almost a decade, and so appreciated him releasing 4D Toys to sate his ravening fan base XD
I just found this channel a week or two ago. I almost watch all of the videos on it.
Why almost D: YOU'RE NOT A REAL FAN! D:< *rage* *rage*
but did u understand it?
Wow... this just blew my mind. I don't know if anyone here is even interested in philosophy or spriituality, but the connection i'm seeing is just too profound.
So here we go:
In spirituality you have those things I like to call the dimensions of conciousness. (Most people would call those the seven chakras.)
Your concoiousness is the inner sphere, your inner universe. The most important aspect of your psyche. (I like to see this as the spiritual realm, you could even call it the "spirit world".)
The outer spheres are the outer world. So everything and everyone else. All the things that are influencing you from outside your own mind. (This is the "material world".)
So as you are climbing up the dimensions of conciousness (or chakras if you want to keep it traditional) your inner world (the inner sphere) keeps growing bigger and bigger.
This means your spiritual capacity is growing. Spiritual capacity is equal to willpower, which is equal to your ability to "manipulate" the outer world. In other words your ability to create things. (Creation is the act of transforming something from one state of beeing to another, which doesn't change what it is made of (the inside) but how it looks like (the outside). On the inside/the smallest level everything consists of the same things (particles and waves).)
The craziest thing about all of this is that the teachings of how you should be able to percieve your reality when you reach a certain dimension (chakra) matches with the math.
Remember that willpower equals your spiritual capacity? The diameter of the spheres equal willpower.
On the lowest 3 dimensions the outer world has more power over you than you have over it. The inner sphere is smaller than the outer ones.
On the 4th dimensions you gain enough willpower to not be an "slave" of the outer world. Your inner world equals the outer world in diameter. You can start to decide your path yourself.
From the 5th dimension onwards your inner world starts to gain more strength than the outer world. You gain the ability to think outside of the box. You now have left the realm of conventional "materialistic" thinking. You will start to see the connections between EVERYTHING.
When you reach the 7th dimension, youre inner world has grown so big that it "swallows up" all of the outer world. The "inner sphere" now contains all of the "outer spheres". Everything is now part of yourself.
This is the point where mind over matter starts to really kick in and the abilities of such people start to seem like magic. (Walking over burning coals without burning your feet for example.)
I hope i made my point somewhat understandable since it is pretty abstract and a bit offtopic. (But still not as abstract as that math. At least for me, lol.)
Sure, but I don't think the inner hypersphere ever swallows the outer hyperspheres in the math.
and the burning coals?
@@fdsfsdfgfdsg1338 From how I understand the OP, I would say your willpower would end up bigger than the ability of the burning coals to control you as you walk over them. They remain a separate but accessible entity, though, just as the outer hyperspheres stay separate from, though still tangential to, the inner hypersphere.
@@pronounjow that’s crazy i see so what about death?
wait hold up. you're saying there isn't a limit to the distance from the center to the corner of higher dimensional cubes of side length 1?!
the way i see it is the space trapped between all of the spheres you placed starts to get bigger in proportion to the size of the spheres every time the dimension increases but after 4 it actually gets so big that there's room for an even bigger sphere than the spheres surounding it.
so essentially:
consider an n-cube where each (n-1) side has a unit length 1. inscribe an n-sphere with a radius of 1 such that the center of the unit n-sphere is at one corner of the n-cube. find the radius of the n-sphere who's center is at the exact opposite corner of the n-cube and is tangent with the unit n-sphere at exactly 1 point.
this is what 3blue1brown did, and explained it step-by-step, noting that if we're talking about more than 4 dimensions then the inner (bounded) n-sphere has a radius greater than 1, and so is BIGGER than the unit n-spheres that "box it in".
essentially, he put this question in plain english and helped us figure out how to solve it utilizing sliders. 🙌🏾🙌🏾🙌🏾
ohhh nice! Which software do you use for this awesome animations?
He, like a boss, created his own:
github.com/3b1b/manim
This is how to teach, amazing explaination. Making the complex simple that is perfection, yes, yes, oh yes. All the comments are wonderful yet mine is of a higher dimension I think.
I caught a glimpse of how the universe is even bigger. Particles appearing and disappearing. Electron energy states. Tunneling effect. Thank you very much!! Great graphics and explanations !
Really well made and outright beautiful video as always!!! ❤️
How do you animate your videos?
It's python or javascript I think.
Beekeeper Honeymoon he programs everything himself in Python, I saw him reply that to this question several times on his other videos
Answer: I create the animations programmatically using a python library named "manim" that I've been building up. If you're curious, you can find it at github.com/3b1b/manim, but you should know that I developed it mainly with my own personal use case in mind. It's not that I want to discourage others from doing similar things, quite the contrary, but often my workflow and development with manim can make it more difficult for an outsider to learn than other better-documented animation tools.
There are aspects of producing videos in a software-driven manner like this that I find quite pleasing, but which are pleasing precisely because it's my own tool. It enforces a uniqueness of style, for example, which is by its very nature a benefit that can't be shared. There's also a certain freedom in being able to tear up the guts of the tool every now and then when I feel a change is in order, since backwards compatibility needs are very limited when you only care about videos moving forward. Not exactly the best practice from a collaborative standpoint.
All that said, if you do want to try it out, never hesitate to ask question, or to let me know about things that can be improved.
-From his website's FAQ section
github.com/3b1b/manim
He wrote his own animation engine, which is available on github (3b1b/manim)
i think the idea of inner circle radius needs a rethinking in over 3 dimensions because of the "point outside of the box" problem, like "it's valid up to 3 dimensions, but not over 3 dimensions"
Me 13 years old watching and pretending i understand something
I'm 13 years old too, I found this channel when I was 11. I used to understand about 20% of what he taught but I used to watch the videos for the sheer joy of learning math. Now I understand a lot more of what he teaches and that makes me want to watch his videos even more.
@@anshum1675 i am triyng to understand too, but its too hard. I really like math and want to learn more than others learn. I am looking for some videos but i dont understand everything because i dont know the formulas
@@anshum1675 and same here, i want more videos from him.
@@hercules2524 I was 11 when I first stumbled across the channel and understood a lot of what he said.
@@user-yg4en5mv2j I mean, i also understand some things But its still hard for me since i haven't been taught many formulas
I don't get it
basically the inner "circle" in higher dimensions is just getting too big to the point it becomes bigger than outer circle and the box which isn't originally in 2d, i'm putting quote around the word circle here because it's technically a circle but in higher dimensions
hieu dang 4 months too late, but ok
Honestly, who does?
This guy mathematicians
Effectively, "inner" and "circle" both have incomprehensible meanings in 4 dimmensions for us 3D folks, except mathematically. When we try to use our typical 3, 2, or 1-D models, they become extremely inept at giving us the intuitive interpretations that we're used to from them.
I think i need to rewatch at least one of your videos per day until i cannot forget them. I love how intuitive you make all this!
4:40 *I T S F R E E R E A L E S T A T E*
GREAT!
You know what 10D's go good with?
plum sauce
What if the closest point of inner sphere to the origin in the higher dimension is not 0.5? What if there are some "fundamental rules" we know in 2-3D are not applicable in the higher dimension (eg. the 0.5 distance I've mentioned, real estate rule of "cheaper and more expensive")analysis?
This is *mathematics* - and these are the fundamental rules for *any* dimensions.
@@EneriGiilaan We all thought so about that "rule of the universe" like the Newtonian law of mechanics, until Einstein came along and told us that there's "new" rule unbeknownst to us. Math too is science, and we can't despise the possible updates to it, including the rules above.
Nope - mathematics is not science - especially in this case it is a type of formal logic. We are free to (actually that is a requirement) to set up the basic assumptions - the axioms - and then study what kind of abstract logical structures rise from those. This is the subject of the video here.
What you are talking about are empirical facts about the reality - physics - and in that case we can indeed find out that the reality does not conform to any particular mathematical model we are currently using to describe it (the physical theory). In that case we need to build a new mathematical model. But the one we have here will still stay as valid as before in its own scope.
@@EneriGiilaan I see
You know it's really fun to learn new things by simply challenging it, but thanks for reminding me that the goal of this video at the first place was to introduce one logical approach to visualize higher dimension, not the one-and-only way to visualize (like physics with their one-and-only formulas for each phenomenon).
About the axiom things, thank you for recalibrating my mindset towards math, damn all this time I've treated math as an empirical science instead of pure logical concept.
No problem.
Actually - whether mathematical concepts and constructs are created by us or discovered by us is a deep philosophical question. In the latter case mathematics can indeed be seen also as a form of 'science' - discovering the facts of the reality.
But from a physicists point of view (I think) mathematics can be seen as a big messy (not to mean that mathematics itself is messy) toolbox where there are plethora vitally necessary logical instruments - mixed with totally mind baffling gadgets we have not found any use (yet).
"Fun high-D sphere phenomena: Most volume is near the equator"
Try thinking about what he means by "equator". We aint talking about no rotation on these spheres, so what does this actually mean? Is any straight line on the surface of the sphere an equator? Is it the same as saying most of the volume is near the surface?
I think it's a generalization to a sort of "circumference" of the sphere. That is, the circumference of a circle which passes through the center of the sphere