Thinking outside the 10-dimensional box

Visualizing high-dimensional spheres to understand a surprising puzzle.
Help fund future projects: / 3blue1brown
This video was sponsored by Brilliant: brilliant.org/3b1b
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: 3b1b.co/high-d-thanks
Home page: www.3blue1brown.com/
Podcast! www.benbenandblue.com/
Check out Ben Eater's channel: / eaterbc
------------------
Animations largely made using manim, a scrappy open source python library. github.com/3b1b/manim
If you want to check it out, I feel compelled to warn you that it's not the most well-documented tool, and it has many other quirks you might expect in a library someone wrote with only their own use in mind.
Music by Vincent Rubinetti.
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
open.spotify.com/album/1dVyjw...
If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with KZbin, if you want to stay posted on new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: 3b1b.co/recommended
Various social media stuffs:
Website: www.3blue1brown.com
Twitter: / 3blue1brown
Patreon: / 3blue1brown
Facebook: / 3blue1brown
Reddit: / 3blue1brown

[@Blaze6108]

1-dimensional array: a line 2-dimensional array: a square 3-dimensional array: a cube 4-dimensional array: death incarnate

[@slymey_]

4D=tesseract

[@kerseykerman7307]

@@MattacksRC You mean it has a bunch of cubes. Tesseracts are made up of cubes, not squares.

[@SimonPegasus]

@@kerseykerman7307 Not just a bunch, infinity ammount of cubes

[@skytrexz3714]

@@SimonPegasus yeah 2nd dimension is just 1st dimension with a lot more dots

[@237homer238]

ten grams to snifindor

[@abhavishwakarma5035]

"The goal here is genuine understanding, not shock" You're the best.

[@thecrystalmemes5767]

It's nice how the video is over 20 minutes long but he doesn't put any midroll ads. Respect man, respect.

[@Jamberia]

he should get way more praise for that

[@ceff01]

ive reported you for having that stupid bitch picture

[@a.yashwanth]

I can see that picture in almost all comment sections. What what that picture?

[@villerger_27]

Watch out guys, josh thumperluck will report you for a profile picture. Maybe you should get one, it’s better than the lime green default.

[@ceff01]

@@villerger_27 your now reported !!!

[@ArxxWyvnClaw]

-Quality Content -Clear Information -Awesome Animation -No Ads -No BS I need more channels like this

[@portul566]

kzbin.info/door/KzJFdi57J53Vr_BkTfN3uQvideos

[@abaundwal]

Captain Disillusion

[@ArxxWyvnClaw]

Thanks guys, you're both a great help. I subbed to both channels. But I'm still looking for more channels :D

[@SoumilSahu]

@@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.

[@ionisator1]

@@ArxxWyvnClaw Numberphile?

[@Sparkette]

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.

[@cryies]

@Bruce U stupid bug

[@Brawler_1337]

Because the Earth is flat. Duh!

[@AnhThuNguyen-zz2hm]

*no*

[@felixroux]

You've definitely triggered a flat Earther... Oh wait, there's no way they understand maths in 4th grade or above.

[@hassanakhtar7874]

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.

[@johnchessant3012]

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."

[@vivekthomas8]

John Chessant Get the fuck outta here, AND CLOSE THE DAMN DOOR ON THE WAY OUT!

[@leowong8777]

John Chessant That wasn't even a joke. It was just *straightforward*

[@dannydk6]

Lol

[@edvink8766]

This supposed "joke" seems like the embodiment of r/iamverysmart. I mean, there''s not even a real punch-line here.

[@Graknorke]

Edvin K It's a joke about mathematicians being detached from reality. It's like the spherical cow joke but without the buildup.

[@ibieiniid4240]

22:04 he said the correct number of "one"s. now I'm impressed.

[@b.clarenc9517]

I counted them too :D

[@mathiasbrio2105]

i counted them

[@hanstheexplorer]

how many

[@xXMindSoulXx]

@hanstheexplorer 10

[@SSM24_]

That point about how far away the corners of the boxes are in higher dimensions is what really made this click intuitively for me. In higher dimensions, there's effectively more space for the outer spheres to fit into the corners of that outer bounding box, and they're located farther from the origin as a result. This leaves more space for that center sphere to take up while still being adjacent to the outer spheres. At no point, however, does the center sphere overtake the _corners_ of either of the boxes.

[@matthewschad6649]

YES, exactly!

[@RichConnerGMN]

lol hi

[@SoopaPop]

VISUALIZE HIGHER DIMENSIONS WITH THIS 1 WEIRD TRICK! STEM PROFESSIONALS HATE HIM!

[@lucasm4299]

SoopaPop Haha lol 😂😂

[@3blue1brown]

There was a very serious part of me that almost did that for the title :)

[@WarpRulez]

The art of clickbaiting.

[@pe4o243]

use sum dank DMT and u'll see the world made out os Calabi-Yau manifolds

[@oceanliketeacher]

DIMENSION 6 WILL SHOCK YOU!!

[@RonMar]

Those numerical sliders are like Braille for higher dimensions.

[@SamraiCast]

Ron Mar Best Comment

[@mlg-allallong5491]

this comment poke my all dimension at the same time....

[@bauagan]

are you rich yet? xD

[@williamromero-auila7129]

Braille in a hyper-plane

[@evanmeyer1496]

from now on i will only buy higher dimensional real estate seems more profitable currently

[@Blox117]

we will build a wall to keep the low dimensions out and make the high dimensions pay for it

[@pokemoncatch6727]

Blox117 yeah lets do it. Also is that a reference to Trump

[@Blox117]

@@pokemoncatch6727 yeah its a joke on his build a wall statements

[@JBB685]

you should check out what *some* people are doing in selling (or trying to sell) virtual real estate, like in VR worlds

[@LazyRare]

Jonathan Bryant VR worlds should be free, for humanity!

[@legendarypussydestroyer6943]

Higher dimensions: *exists* 3Blue1Brown: *IT'S FREE REAL ESTATE*

[@alexandermatias3938]

well put

[@canaDavid1]

Actually, its just cheap real estate. But good meme.

[@1bazz974]

shitty meme, and no, real estate is not free

[@robinw77]

@@1bazz974 yeah the value of that meme has really plummeted over time

[@realbignoob1886]

Ps3udo that was a long time ago, shush

[@terner1234]

x: moves a little y: it's free real estate

[@realbignoob1886]

dynadude lol

[@pranamyabelvai7488]

AHAHHAA

[@squealer7235]

I can't understand what real estate means in the video.

[@sharqstep]

@@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

[@ramakrishnansubbaiyan3798]

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. ..

[@arnaldo8681]

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

[@feynstein1004]

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.

[@justice576]

Now I understand why I suck at mixing/mastering. Its 10 dimensional circle math lol

[@mirmalchik]

except it's even better because you're not restricted to any surface, in any number of dimensions (wait that's not true, if you only have 10 free variables, you're restricted to some 10-D surface in 11-D space. but still neat)

[@ConspiracySundays]

*It's free real estate*

[@darioinfini]

Yeah but it's mathematical swamp land. You get what you pay for.

[@nemou4985]

It literally says it isn't free, though.

[@taureon_]

i was thinking of that meme XD

[@cordialmanx2334]

got beat to it

[@DehJarlorNoob]

Came to the comment section looking for this comment. I wasn't disappointed.

[@daksh8747]

I always get your videos until the halfway mark, after that it all goes over my head.

[@TimEnjoysGnocchis]

multiplying information ^^

[@99bits46]

i am at 16th minute mark and i am down to comment ^_^

[@mobob7989]

same

[@Acid113377]

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.

[@user-vz3mj5qk4d]

Coordinate: *Is close to zero* 3B1B: *It's free real estate*

[@erikfauser2418]

thats funny

[@timlampers8610]

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!"

[@marekhudac8390]

20D beings would understand 2/3D efortlessly, just like us, 3D beings naturally understand 1/2D.

[@AldorEricsson]

Just wait till fractal beings with fractional number of dimensions join the talk...

[@thomasbeaumont3668]

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...”

[@shadowcween7890]

@@thomasbeaumont3668 Now wait a minute here-

[@danegoodwin3057]

It blows my mind how brilliant some peoples' minds are

[@gruminatorII]

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...

[@MeganBoschen]

Philippe Durrell: 🤯

[@GASPERL1]

To be fair, Einstein did receive some help from mathematician Minkowski

[@hoennnoodle]

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

[@tim_nj_devil176]

beepybeetle and unfortunately due to the internet, idiotic fucktards that believe the earth is flat, space is fake, etc. are forever enshrined

[@MoonDystopia]

Thank you! This will come in handy next time I'm stacking my 8-dimensional oranges..

[@elperronimo]

This video helped me a lot on seeing Trump's 4d chess moves from a mile away.

[@zorgazorga7601]

But a mile is so small in 4 dimensions... =D

[@smacq6000]

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

[@smacq6000]

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

[@computercat8694]

8-dimensional oranges? Make sure to use the E8 lattice, it's the best!

[@MissChanandlerBong1]

Ah so this is how Doctor Who's TARDIS is bigger on the inside than the outside.

[@SinHurr]

OH MY GOD IT WORKS

[@caidenkesler3945]

Time And Relative Dimensions I guess. Doesn’t specify 3 I guess, so it works

[@sonetagu1337]

Y'know, the classic non-euclidian space.

[@MonkOrMan]

Ohh yeah!

[@JoX1231]

mind = blown.

[@box_inabox]

Hi, game designer here, I have been struggling to find a visual way to think about balancing the power of mechanics in games that have multiple contributing factors. After giving up on this problem, this video has got me going back to my white board. Higher dimensional game balancing coming soon to a steam library near you!

[@General12th]

Nice! It's good to remember that coordinate systems and geometric reasoning are good for more than just spatial dimensions. All sorts of properties are orthogonal to each other -- for example, temperature, pressure, and humidity, or health, attack, and cooldown rate -- and that means they can be described with this kind of logic.

[@Wren4123]

I'm like "It's a weekend, take a break from math, enjoy life!" *Sees this video* "Eh, a life without math is no life at all."

[@bwcbiz]

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".

[@nutronstar45]

3) adjusting the "w" slider

[@joshuasusanto6626]

22:42 *I may or may not have used an easy-to-compute but not-totally-accurate curve here due to the surprising difficulty in computing the real proportion :)

[@bwatspro]

The saying: "Think outside of box", got new dimension.

[@IgorDz]

27 minutes! I'm not watching it! ..Maybe just a couple of minutes out of curiosity ...*watched it all*

[@VVV.12345]

I've just watched 27 minutes?!?

[@albertovieira588]

the same here hahahaha

[@Handimedia]

and you think anybody cares that you do not watch it?

[@albertovieira588]

JAY CURVE and do you think we care about your opinion ?

[@bewildo]

FlingFlexer 11 minutes in came here to check the comments, f me right

[@yamansanghavi]

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.

[@thorstambaugh1520]

If you truly want beauty please do a video on tessellation and matrice theory. The correlation is astonishing and sublime

[@Psyle_]

Yes differential geometry would be awesome with his clarity and animations Manifolds and their diffeomorphisms are very obscurely introduced at uni

[@tghuffine6277]

Dude, that would be so cool..!

[@d.m.collins1501]

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.

[@battmarn]

I've been so confused about this for years and this is genuinely first time I've felt like i understood this. THANK YOU!

[@arnavverma4557]

This is some gorgeous animation

[@Smokin438]

Glad at least someone appreciates how much went into all of that.

[@Jabrils]

Grant. youre a deity.

[@enebz3746]

Hi, Jabrils! You are one of my idols! Cool to see ur 25 like comment in a random comment section...

[@iro4201]

Can you make an AI model which operates on multidimensional numbers? ... oh wait-

[@Nucleardoom]

@@iro4201 haha

[@P-nk-m-na]

Grant is the modern incarnation of Thoth

[@toferg.8264]

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!

[@guyfriedman295]

16:51 my brain started to shiver... I think that enough math for today...

[@johno9507]

How do you expect people to visualize a sphere in higher dimensions when flat earthers can't even visualize a sphere in three!? ;)

[@shiftl4]

earth is not a sphere :D

[@elijahheyrosa5398]

Hes not talking about earth being a sphere Hes talking about normal spheres like basket balls or soccerballs, basic stuff

[@benedani9580]

r/woooosh

[@takeyourdamnmeds]

fact:earth is an oblate spheroid

[@pluto8404]

We exist on the 2 dimensional surface of a black holes event horizon. 3rd dimension is just a hologram, illusion.

[@souravzzz]

I just found out this channel and it is fantastic! Please keep making great videos!

[@MatthewSalathe]

U Wot M8 John Cena dun dun duuuuuuuuuun

[@itze_]

Just happened to me. I have an exam today! Now back to studying

[@christheswiss390]

The visualization at 7:40 REALLY made it clear. This is one of your best videos to date, imho!

[@akshaygahoi5034]

This video really gave a new concept of higher dimensional things unlike the other videos that I saw untill now. Worth watching!! Keep making such cool stuff. Really appreciated.

[@mushkamusic]

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!

[@milkwater1204]

This is how I visualise higher dimensions, too

[@jalfire]

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...

[@peterpemrich6962]

@@realityversusfiction9960 what the hell are you talking about

[@goyonman9655]

@@peterpemrich6962 😂😂😂😂😂😂😂😂😂😂

[@jf8442]

Wow. I have just watched the entire video. This is just brilliant. Keep it up! I love watching your videos even if I do not quite understand everything entirely. But I try and I guess 10D is not too easy to get one's head around xD

[@MRF77]

You're the beauty, ma'maaan!! As always, this video just made my day a lot better, even this is an older one! KEEP UP THE AWESOME WORK LIKE ALWAYS!!

[@CarlosMats]

This channel is goddamn brilliant. Love your visualizations.

[@twiggy_witch]

22:41 It says: "I may or may not have used an easy-to-compute but not-totally-accurate curve here, due to the surprising difficulty in computing the real proportion :)"

[@diogocruzdiniz3186]

I love the way you explain things. Instead of throwing math at us you explain the concept in a fun way. I hope you keep doing this!

[@elizabethmaas3907]

Another superb video by 3Blue1Brown clearly explaining the nearly unexplainable. Thank you! PS. I knew of the inner sphere's outcome from the beginning, but hadn't considered it in this manner previously.

[@dwinsemius]

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.

[@alfredinselberg7962]

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.

[@Kabitu1]

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.

[@ffximasterroshi]

Wonderful video, glad that I revisited it. I'm currently doing hobby research on 4d spheres and have a great understanding of 4d hedra and a good understanding of 4d cuboids. The formulas will be great for analyzing. Btw, the ability to slice higher dimensional shapes into sets of sub-dimensional shapes is true for hedra (simplexes), square/cuboids, and circle/spheres. These slices can then be visualized in series to make an analog of the higher dimensional shape.

[@danielheard-illumilands7563]

This is a great way to visualize dimensions! I've personally been trying to translate counter-intuitive results of higher dimensions to more intuitive concepts, so the "Slider" analogy was perfect.

[@theflaggeddragon9472]

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.

[@hanniffydinn6019]

The Flagged Dragon it's easy! Take some LSD ! That is leave your body via OBE , NDE ! Many have. I have.

[@Roxor128]

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.

[@Squideey]

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.

[@garryiglesias4074]

+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... :)

[@redsalmon9966]

You'll need a four-dimensional creature to take you out of this 3D world, like in the Flatland.

[@jasonquidoz3452]

This man has solved the housing problem. Real estate agents hate him!

[@ernstvangelderen9537]

Wow! Thanks! I've seen this in a paper, well sort of the same thing but the paper dealt with how many spheres can be stacked in a box. Funny thing with that, out pops the Euler Beta function! This is such a good introduction in helping you visualize hyper spheres. Wicked!!

[@jamesedward9306]

I've been struggling for a while with what all this math in higher dimensions really means. Watched a number of videos, and like many people I suppose, it starts unravel for me as soon as I get past 3-physical and 1-time dimension model. This is by far THE best video I've seen on this so far. In particular 3:00 to 4:00. That explanation on how to understand this stuff will always be burnt into my brain. Thank you Grant.

[@cheydinal5401]

Or you could show the 4th dimension using not 4 sliders but 2 2D flats

[@user-te2gb1hy9y]

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.

[@RK-hello]

Love your videos... your graphics really help put abstract ideas into perspective.... thanks :)

[@Syzygy-21cm]

This is brilliant! What a clever and instructive way to comprehend multidimensional space. Loved it!

[@RoxanneClimber]

4:47 that's free real estate

[@hans_____]

I think the "real estate" metaphor made it more difficult for me. Now I have to figure out what exactly "real estate" is.

[@melocomanTV]

Real estate is the invariant, the constant set to one in this example, which governs the radius of the spheres. But he uses the metaphor inconsistently when talking about the inner radius

[@matthieudeloget8998]

It's cheap real estate 🙃

[@tiny_toilet]

Yeah, introducing this convoluted term just made things unnecessarily complicated.

[@philippelizotte3760]

Real estate is just a fancy term for land that you can buy.

[@PaladinZaego]

I feel the same. I kinda missed the whole video's point because of that

[@jngf100]

Really fascinating way to think about higher dimensions - thanks for sharing this!

[@mikelavalenzuela9173]

Thank you!!! That slider image was super helpful

[@mattiasselin4955]

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).

[@aaronherman1056]

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.

[@prahas777]

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!

[@fraserpye9667]

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

[@shanewaters9402]

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.

[@iqdx]

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!

[@nez14526]

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.

[@shaihulud4515]

IMHO by far the best YTvideo on this topic! It has mathematical proof, makes plausible assumptions, and comes up with an understandable way of explaining it's concepts. Congratulations to 3Blue1Brown - got yourself a new subscriber :)

[@PasseScience]

Never forget that if someone sells you hyper-oranges in dim-87 by hyper-weight, you'll mainly pay for fruit skin.

[@user-bg6xh7vv7t]

i love this

[@GREGGRCO]

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 !

[@smythee7835]

some people and i'm one of them can only learn on their own. so a very fast paced information delivery without any type of interactions including very long delays with useless questions like you provide is perfect. I always refused to stupidly learn math or to listen to professors forcing their half baked course to students. Because no one has shown to me how to visualize it. I have never passed 2+2 2-2. So i'm grateful to you for uploading these videos and finally change my understand of mathematics. I am convinced that if i've been confronted to a choice to learn in this way I could probably contribute to math by now if I could find a passion to it for example.

[@user-zd5pe7qh5n]

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.

[@andreamarino95]

I don't see what's your point!

[@zanti4132]

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.

[@gorkemvids4839]

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

[@caidenkesler3945]

If you do it with 4D, the 3D sphere will grow and shrink

[@mateowirth1267]

One of my favorite professors ever assigned this as a problem in his probability class. Good memories!

[@spiderrobotheavyduty5028]

This explanation is so detail and clear, really amazing and easy to understand. very much thanks

[@sethapex9670]

there are 2^n boundary N-spheres in N dimensional space. As n grows, each boundary sphere must take up exponentially less and less N dimensional space in a unit 1 N-cube which means the N-sphere which they bound must take up more and more space.

[@InvisiRS]

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.

[@KaliFissure]

Always great videos. Thank you! How convenient that in 4 dimensions that the sphere fits in the gap in the middle of a bunch of spheres the same size as itself. That way every universe can be surrounded by other universes the same size as itself, without them hardly touching at all, kinda.

[@jukokobarinko]

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

[@caidenkesler3945]

And I’d guess the sphere grows and decreases at a sin rate

[@555artisan]

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 :)

[@MB32904]

@@555artisan if you were a 2 dimensional being, it would look like a line growing then shrinking

[@fraserpye9667]

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.

[@EjMacarus]

I am bias toward visuals so finding an established channel that really presents these ideas well is priceless. You are the reason I might actually get a degree. Thank you

[@flamingpi2245]

I mean, there are really no such things as people with different learning styles. The idea is unscientific, people benefit from a combination of all learning styles

[@brownfox651]

Got that 1st d The 2nd 3rd But can't even imagine the fourth

[@AntThinker]

@Evi1M4chine To hell with five dimensions. Who likes to try being a puffer fish?!

[@mirmalchik]

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...

[@jonah1077]

I love when your videos appear in my recommended. I can always have some more math in my life

[@jeffdecker7834]

I am reading Love and Math by Edward Frenkel and early in the book the concept is about symmetries and circles and quarks...this video totally relates and I am intrigued. Thanks.

[@BrainSlugs83]

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. :)

[@jacobgirard7175]

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.

[@p11111]

Visualize the 10D version by drawing the 3D version, but with the corner spheres tiny (and not touching) and the inner sphere huge and reaching outside the box. Then just declare that the corner spheres "touch" in 10D's. Although that last part is hard to visualize, it's easier than imagining a deformed inner sphere somehow poking outside of corner spheres that are actually touching in the model.

[@xanderbackus447]

That's how I envisioned it too

[@ed.puckett]

Your beautiful description made me smarter! Thank you!

[@reynaldocmagana]

Wow!!! This video makes soooo much more sense to me! I don't understand it all, but I feel with time and patience, I'll get there!

[@scottgoodson8295]

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)].

[@wedusk]

Scott Goodson I have and it's a ton of fun!

[@vampyricon7026]

+

[@muizzsiddique]

Who wouldn't know of the great Parker Square?

[@ragnkja]

I first heard of this in that book. It turns out that spheres are a lot spikier than we tend to think they are.

[@xenontesla122]

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.

[@TrueMathSquare]

I just found this channel a week or two ago. I almost watch all of the videos on it.

[@loljustice31]

Why almost D: YOU'RE NOT A REAL FAN! D:< *rage* *rage*

[@nagahumanbeingzooofparticl8836]

but did u understand it?

[@LovedeepDhingra]

You are a beautiful teacher! Thank you so much for making this video!

[@slwtgf]

Humbly grateful this is starting to make sense! Visually, this is mind blowing 🤯 and poetic almost, lol!

[@nadiyayasmeen3928]

"The goal here is genuine understanding; not shock." Liked the video as soon as he said that.

[@samusbros66]

If you put a glass (3D object) under a lamp or something, you'll see that it's shadow makes a circle(a 2D form), imagine how complex a fourth dimensional thing is. Tesseract it's just the three dimensional shadow that we can see in our plane, it's real form it's unbelivable complex

[@gregreeves-smith5578]

Brilliantly visualized! Thank you!

[@jucapeloso]

omg this is amazing! when I first saw this problem I couldn’t accept that a sphere bound by other spheres in the corners of a cube could be the same size and even bigger than the spheres that bound it when you go to higher dimensions. But know it makes so much sense, cause you’re like adding more variables but they are limited by the same statement that their squares must add 1, so the points on the spheres stay closer to its center and the tangent points clearly are getting further and further from the origin, thus the inner sphere is getting bigger and bigger. It’s so obvious now! This real state analogy was very clever congrats

[@thinkingoutloud6741]

Joao Vitor Peloso : ponder on what the center of the 4D sphere looks like. Do you imagine it to be a point? What does a point look like in 4D? If it’s not a simple 3D point, then I’d like to see this same video format analyze the comparative volumes.

[@livintolearn7053]

That ending quote... Man, what a mic drop!

[@matheusbergamo240]

Which software do you use to make those incredible 2D animations? They're great!

[@mike4ty4]

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 ...

[@kairostimeYT]

I like how you explained the concept of derivatives and trigonometric functions, without mentioning them.

[@erikisberg3886]

Outstanding visualization of a difficult concept!