A trick to visualizing higher dimensions

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John Chessant : 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."

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

karsaka sdasfa : Love your videos. Will you make a collab with Vsauce?

The Flagged Dragon : 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.

3Blue1Brown : Find the relevant links for the podcast at https://www.benbenandblue.com/ Since it's hot off the presses, it might still be a few days before it's visible by searching in the itunes store. Also, the version on YouTube can be found here: https://www.youtube.com/watch?v=8r5WKpK9-m8 Also, if y'all haven't already seen Ben Eater's YouTube channel, it's full of some really top notch explanations: https://www.youtube.com/user/eaterbc

Error 418: I'm a teapot : This feels like a really complicated trick to vaguely display something that is more intuitive as just numbers. To me this is just over complicating it.

Scott Goodson : 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)].

Alex Blandin : Can we get some math applied to music as higher dimension objects? That'd be pretty cool. Also, would an infinite dimensioned sphere wrap the corner spheres?

Ismail Hossain : It just simply awesome.Thank you sir. Can we have a video on laplace transformation.plz sir. I don't understand it ,where it's came from,and s domain t domain .

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

Will C : Here it is, my proudest fap.

Abdulnaser Sheikh : Why are knowledgeable people atheists?

ProCactus : Is there a purpose to this in the real world ?

DevinDTV : How is the center sphere getting bigger unintuitive at all

An Unearthly Child : All the analogies using more expensive and cheaper make me vomit. Can't escape from fucking capitalism even when watching maths...

ChickenStealer : HOW IS THIS EXAMPLE NOT LOGICAL?!!!! Unless you really really don't know what dimensions are, you should expect that it eventually reaches outside the square!! -Look at a 1 dimensional example: The inner line has size 0. -Then when you add a dimention, the inner circle shifts from the point where it can only be 0 to be slightly bigger. -Add another dimension and the inner sphere has more room because it makes use of the 3rd dimension. -You do not have to think in 5 or more dimensions in your head, but just think of 4D. The inner "4D sphere" will be bigger than the one in the 3D box because it can fit halfway into the extra 4th D space you created between the two sets of 8 3D spheres!! Now keep adding dimensions and the inner sphere will keep growing. Or am I just thinking outside the box?!

Benjamin steffen : if i can slice a 4D sphere into a lot of 3D spheres, would it make sense to think about the 4D sphere as a solid 'Ball'? the x, y and z would move on top of the outer layer of the ball but when i decrease w it would go inside of the 4Dsphere on top of the surface of another 3Dsphere slice...?

aghaanantyab : how large is the inner sphere of 1 million dimensions?

Flamingpaper : What is the 1 dimensional version of a circle? The 0-dimensional is a point, cause everything is. The 2-dimensional is a circle. The 3-dimensional version is a sphere, but what is the 1-dimensional version?

zukodude487987 : How to sort of visualize a 4D sphere. Take a 2D transparent flat surface and have a 3D sphere pass through it. The cross section of the sphere going through the flat plane will look like a dot appearing out of nowhere gradually growing from a point into a circle and as the sphere is half way through it it reverses the order and goes from a circle shrinking back down into a dot and disappearing. Now take that same logic with a 4D hyper sphere. Imagine a 3D cube as a chunk of space the 4D hyper sphere passes through. It starts out with a small dot appearing into existence inside the cube and it grows into a large sphere and after the 4D hyper sphere is half way through the square it slowly starts to shrink again into a small dot and disappearing.

DeathByMischief : I have had this question for over half a year now: is it possible in higher dimensions to have two planes (flat 2d surface that extends infinitely far) that don't intersect, aren't the same or aren't parallel to each other? It intuitively feels like in 4d that should be a thing because in 3d that happens to lines. (Skew lines is what they're called? My mathematics english isn't perfect)

rustyrusky : The "dimensions" of math and the "dimensions" of physics are two different things entirely. While the physicist uses qualifying statements (i.e. direction and orthogonality), math is only concerned with quantity (i.e. ordered triplett). Both are entirely different concepts yet here we are treating n-dimensional "spheres" as "real". They are as real as epicycles, a crutch used to explain away things we don't understand yet.

Beekeeper Honeymoon : How do you animate your videos?

Happy Totter : Estaba viendo este video Me quedé dormido Y soñé con la 5ta dimensión

Staremastershy : Well, calculating the size of the center circle took me like 5 seconds (yeah, sometimes I'm a little slow with that). But then you needed 15 minutes to explain that 'trick' and I wondered what interesting property you are going to show with it and then wtf... I want my 15 minutes back! Really, this was your most boring video ever.

thegustavodag : this video doesnt explain anything dude

Jonathan Castello : I think there's an error in the animation around 16:45. When describing the corners for the 4-dimensional cube, the X coordinate is animated at 0 or 2, where it should only be taking on values at -1 or 1. (I mean, it's technically still a 4-cube -- just, translated.)

Against NAZO! : The inner >5D spheres doesn't reach outside the box for outer spheres! If you project the large radius into two dimensions to visualize it, you make a mistake! It's a distortion, like projecting a 3D-Cube on a plane, the shadow-Edges wouldn't be rectangle any more.

Mario Cantina : I'd love to see a video about quaternions! :)

Kevin Kall : Congratulations! You have just proven at time 6:05 mark why the earth does not go around the sun but the sun moves "around" a fixed non-moving earth.I quote around due to that its not really around but more past the surface. very close past the surface. If the earth was moving around the sun and the sun is 93 million miles away, the sun's movement across the earth's surface at high noon would be slower than at dawn and dusk because the surface area the sun is moving across is more expensive "real estate" and so would move slower. But because the rate the sun is moving across the sky from horizon to horizon, that means that either the earth is flat which is bogus or that the earth is a shell and then sun is moving evenly inside the shell. Before you discount this entry, think about the facts, the science. Something else you never hear is that you have to be in a higher dimension to see the lesser one. So for instance, to understand/see a one dimensional object, you have to be in a 2 or higher dimension. To understand/see a two dimensional object, you have to be in a 3 or higher dimension. We in this dimension see 4, x, y, z, and time which is the fourth dimension. That means we are, *at a minimum*, fifth dimensional beings. Likely higher. Higher than we care to admit or comprehend.

Flare03l : would this work for 1 dimension?

Mike Meyer : Was this video by any chance inspired by the Infinite Series video on honeycombs in higher dimensions?

Hepad : Means that you could probably put helium in metals in a 5D universe. Imagine the properties of this thing

pokechao196 : I had to do a presentation on R-omega (infinite-dimensional space, more or less) for my topology class last semester, and I came up with a similar "slider" visual to help describe some of the properties. Happy to see it show up somewhere else! Also, does the Euclidean metric continue up to infinite-dimensional spaces? Or is distance have to be defined in a different way?

dpcprince : What if the outer spheres are centered on the "faces" or surfaces of a x-dimensional cube? Then, the inner sphere radius can't possibly be bigger than 1. Also, how do you find the centers of the spheres on this problem?

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

Павел Таранов : That's how I see 4th dimention with my puny mind: Imagine earth in space, from the moment of it being glob of magma, then move through time every 1000 years. Those are single slices of 3d in 4d (3d + time dimention). Now emagine every 3d earth that you saw at the same time. Of it flying around the Sun, swirling in Milky way, leaving ghostly image of itself every second, looking as it was at that time, changing from glob of red goo, into single continent planet, into what it is now.

crapvile : we are apes

RIPxBlackHawk : *Teacher when half the class has already left:* Remember this! There's gonna be a test about this next week!

axelasdf : WHAT IS THE SEARCHABLE NAME OF THE PODCAST.

CustardMan : what happens to negative dimensions

Steve Agland : Great video. I wanted to share something interesting I noticed about higher dimensional spheres. Take two points at (uniformly) random locations on the surface and measure the angle between them from the origin. On a high-D hypersphere (say, 50) this angle is likely to be very close to 90 degrees and vanishingly unlikely to be anywhere near 0 (nearby) or 180 (polar opposites). It'd be lonely living on a 50-D planet since everyone else would be half a world away.

Lee Jordan : 15 minutes in. I swear to god, this better not be a complete waste of my time...

Simon Bouchard : 21:45 "bigger than 4" , you mean 2?

Sofía Díaz Esteban : I don't understand what is real estate, can anyone help me?

Griet 'Nicky' Csellak-Claeys : So how does that image of the 4-d sphere work? (That's the mystery that still surrounds me

Brandon Smith : Thank you.

Rígille Scherrer Borges Menezes : Grant, nice video ^^. Do you know the game "4d toys"?

Murvs : Can you see life in 10 dimensions???

Cyan Stickmin : x^2+y^2=1 x=0.6 y=0.8