Thinking visually about higher dimensions

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U Wot M8 : I just found out this channel and it is fantastic! Please keep making great videos!

Pwn Tech : i would of thought this video would of been so boring, but it turned out keeping my attention and very educational, as i am a visual learner.

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

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

dartagnanx1 : GREAT!

VELOCITY FPV : I don't get it

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

Arnav Verma : This is some gorgeous animation

Higgins2001 : I'm just clever enough to understand the basics and get excited about implications, and just dumb enough to give up thinking past this video after 2 minutes... (might explain why I only minored in math) However, can someone please help me: I'd love to know at which dimension the VOLUME of the inner sphere exceeds that of the bounding spheres... or if this even happens at all?

maisto : If my math teacher was like this man!

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

BEISisICE : I think this guy watches Rick & Morty

Ivan Žuti : Thank you for this video. Now I can't stop thinking if this is somehow related to the expansion of the universe. What if the universe is expanding into a higher dimensional space and we observe it as the expansion? Also what if the big bang was the expansion from a plane into a 3d universe? Mind is melting right now.

Michael Jensen : 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. :)

Yaman Sanghavi : 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.

Jonathan groberg : For some reason, my brain is trying to connect neural networks to higher dimensions problems. Your use of sliders seems very similar to the function of weights determining the value of a neuron, so much so that it seems they must be connected in some way. Also, your video and GoldPlatedGoof's video about higher dimension problems make me wonder, can backpropagation be viewed as a higher dimension problem? Is my brain onto something or do I simply love machine learning so much it is clouding my vision.

Dane Goodwin : It blows my mind how brilliant some peoples' minds are

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.

Ciaran : It's Free Real Estate.

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

Nilay Patel : Awesome man

Nilay Patel : Seariously awesome

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

Shai Hulud : 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 :)

SuccMyEgg : *It's free real estate*

Naviron Ghost : 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!


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...

David Winsemius : 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.

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?!

Josef Wöss : ohhh nice! Which software do you use for this awesome animations?

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)].

Connor Stegner : Wow. Are you an educator? This was awesome.

Amulaya Bhatia : Came here from HFS this is awesome

TheXalkk : Pretty freakin hard for my little brain) But its interesting)

FuriousBean : For calculating such plots as at 22:41 you can use Monte-Carlo simulation. It won't take long to calculate the proportion with the accuracy needed to show it here.

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 .

focus : How much time does it take to make such awesome looking yet helpful visualizations?

Carlucio Leite : I'm blown away. Again.

OGF : wow what a simple trick!

Christina Andwena : Ts;dw Too smart; didn't watch

Dercio Silveira : Sorry.

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

Chaos : Amazing videos mate. The way you illustrate these mathematical theories with geometry made it all click for the first time in my life.

jakeinator21 : This video felt like the quickest 27 minutes of my life.

MD HAMID USMAN : Way Beyond my level !

iOSMinecraft120 : Woah 3blue1brown also uses momentum woo

Kevin Stoffel : One of my favorite explainers from this channel thus far. Great job!

Ryan Santa : To me, this makes me think that in higher dimensions you can fit more and more stuff in the same space- as in there are possibly invisible data/stuff within everything we see in 3D (as in the bigger center sphere). Does this make sense to anyone else?

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?