Thinking visually about 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."

Will C : Here it is, my proudest fap.

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

3Blue1Brown : Find the relevant links for the podcast at 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: Also, if y'all haven't already seen Ben Eater's YouTube channel, it's full of some really top notch explanations:

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


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)].

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.

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?

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?

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...?

Martin FTW : 22:04 But the point 11111111111111111111111111111111111111111111111 WHAT XDDDDDDDDDD

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

Beekeeper Honeymoon : How do you animate your videos?

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.

Seth Apex : 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.

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

drz : Can't we all think in 4 dimensions? Our minds are not restricted to 3 dimensions like our eye balls are. I thought 4D thinking was normal.

Eric Espinoza : Hey how often do you study? I want to become as smart as you and solve real world problems. However, I'm not quite as educated in math as you are. How often do you immerse yourself into new knowledge?

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

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.

Trystan Hooper : i feel really dumb for asking this, but it does pythagoras's theorem have a logarithmic element. like instead of using it twice in two separate operations. can u just do h = root(X^2 + y^2 + z^2...) for the diagonal distance of a cuboid?

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

Caduceus : If the area of a circle is always irrational, how can a circle ever truly exist in a finite state?

Sid Chou : how are the animation made??? i would imagin this to be a night mare if its on after effect

Max Pheby : But surely if you add a dimension to a sphere its no longer a sphere it may look like a sphere but it isnt. This is the problem with maths fundamentally while it can explain geometric shapes it is not those shapes its a representation of that shape in a numeric form they are not the same thing.

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

Oskar Krogsgård : I am not a native english speaker an this has really been annoying me because i have heard it everywhere. Can someone plz explain to me what the term "real estate" means

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

Ben Ward : I love doing meth at higher dimensions.

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

Саша Леськів : You can't really call it a 'sphere' when talking about not 3-dimension spaces :/

Byron Lowry : Distance is /always/ the square root of the sum of squares of the coordinates he says! :-p

Alex Tritt : Wow, I just realised that every state in quantum mechanics is just a point on a unit sphere (although the values of the coordinates can be complex). Because of course S<ψ|x>dx = 1. And I love how near the end where you see the 10 dimensional sphere you could imagine joining the dots to make a function - it really shows how functions can be vectors.

Wolf Edmunds : I wish to be reincarnated into a 4D universe.

Divergent Evolution : If it is common knowledge that a 2d entity cannot perceive 3d space, then why do we assume we can see past 3d space as 3d entities?

Crogon Grey : Nice idea, but I never understood the problem some people have with visualizing >3 dimensions. It seems so trivial and this seems very over-elaborate. Just imagine a cube that is within itself infinitely large but looks finite from the outside and now put an infinite amount of them next to each other in a row. When you then move them in your imagination to the same spot so that they overlap you have an infinite amount of infinitely large 3d spaces overlapping each other and that's basically a 4d space. You can do the same with a plane layed out with cubes or a cube of cubes and then collapse them onto each other for higher dimensions. If you imagine a structure with a somewhat flat surface that spans 4 dimensions you imagine one 3-dimensional part in one cube and another one in the next cube. Sort of like individual frames to a 3d video (as in a video that actually contains a 3d space in each frame and not just an image).

Psychotic Ape : I think I thought of a way to see how the inner sphere touches the sides of the 4D hypercube. Imagine the 4th dimension as pointing inward from the vertices of the 3-cube, and if the 3-cube has side length 2, the the length pointed in ward will be 2, now connect all the points such that you construct an "inner cube". The inner cube is not actually smaller it's just further away into the 4th dimension. Now in order to imagine the centre of the hyper cube, take the point halfway between each vertice between the outer and inner cube. After connecting all those points and forming a third cube between the outer and inner you simply take the centre of this new cube. Now any point on this new cube will be and equal distance away from the corresponding points on both the inner and outer cubes, including the centre point, therefore this must be the centre. Now set up a hypersphere centered at this point. If the radius is 1, the sphere will contact the middle cube.

Kootness : I'm still struggling to understand what a 4th (or higher) dimension could "be." Supposedly physicists talk about time being a 4th dimension, but it is clearly different from the spatial dimensions, which share properties, esp. units of measure. Assigning four variables to something to describe its properties is well and good, but I get lost as soon as I start to think about geometry in this space. What does it mean for space to form an angle with time? What is a vector with four dimensions? Etc.

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.

ronan stephens : I need a clarification. In N dimensions we have an N dimensional cube, but are the spheres we are placing at the corners all 3-spheres or N-spheres?

Flare03l : would this work for 1 dimension?

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 .

Blender Dumbass : How much inspirations for algorythms

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

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?

texasdee slinglead : what would be the application for using 5d or more . I get x, y ,z axis and time , velocity ect , but I don't see geometry /trig needing this for anything currently. would this be applicable to quantum computing.

Jonathan Alcaraz : Should have done 9 dimensional space so that the inner circle radius is exactly 2. Lol. Also you can make that inner radius any positive integer. Interesting.

Silas Ifeanyi : Just so you know how much of a nerd I am for your channel. I am currently dressing up for a party that I have to drive to right now but I couldn't resist clicking on this video because understanding multiple dimensions (beyond 3-d) spatially is something I have always wanted to do. So, I will contain my excitement and wait because this video will still be here but I have to leave right now lol