Same here. I understand perfectly everything that was said in the video but I still don't understand how the fourth dimension quite works.
However, having only heard of hypercubes and not hyperspheres before I decided to see if there was anything useful about them online and I found this video that I just started watching and already 1 min 50 sec into the video something very interesting was said;
> Everybody knows what the sphere is. I'm thinking of a hollow sphere so like a basketball right. That's a two-dimensional surface living in a three-dimensional space. The hypersphere is generalized one dimension up, so in four dimensions you have this three-dimensional space called the 3-sphere or the hypersphere.
Already this is telling me something that I have not heard before, and which I find much more helpful than talking about what a shape looks like from the perspective of someone living one dimension further down. That being said it was still useful having that explained as well, which is what Flatland: The Movie (2007) was about as well. Just this fact I quoted above was even more useful IMO.
Another quote from the video I linked in parent comment.
6:19
> As complex numbers are to real numbers, quaternions are to complex numbers. It's like a way to build up even further. [...] Real numbers are one-dimensional. Complex numbers are two-dimensional. [...] For three dimensions there is no natural number system, but for four dimensions there is and it looks like this.
That’s really misleading. The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.
In a similar way, the quaternions are the space of transformations (scaling & rotation) of three-dimensional vectors. (It’s a little more complicated because 3-dimensional rotations are not commutative, and must be combined by sandwiching, so there are 2 choices of quaternion corresponding to every scale and orientation in 3-dimensional space. For an introduction see http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf)
>The complex numbers are not a two-dimensional Euclidean space directly, but are a space of transformations (scaling & rotation) on two-dimensional Euclidean vectors, where 1 represents the identity transformation, and i represents a quarter turn anticlockwise.
You might be right, I don't know. Could you explain a bit more how you mean?
Complex numbers have multiplication defined on them. If you multiply two complex numbers, you get another complex number. (They compose via multiplication in exactly the same way as scaling & rotation operators on 2-dimensional vectors.)
If you just have 2-dimensional vectors, there’s no obviously well-defined way to multiply two vectors and get out another vector.
In other words, both 2-dimensional vectors and complex numbers are made up of 2 coordinates, but they don’t have the same mathematical structure.
The rest of the video up until 16:26 has been about projecting down one dimension but it's doing so in different ways instead of just simple cross-section and it's using videos to show rotations and stuff.
However, having only heard of hypercubes and not hyperspheres before I decided to see if there was anything useful about them online and I found this video that I just started watching and already 1 min 50 sec into the video something very interesting was said;
> Everybody knows what the sphere is. I'm thinking of a hollow sphere so like a basketball right. That's a two-dimensional surface living in a three-dimensional space. The hypersphere is generalized one dimension up, so in four dimensions you have this three-dimensional space called the 3-sphere or the hypersphere.
https://www.youtube.com/watch?v=krmV1hDybuU
Already this is telling me something that I have not heard before, and which I find much more helpful than talking about what a shape looks like from the perspective of someone living one dimension further down. That being said it was still useful having that explained as well, which is what Flatland: The Movie (2007) was about as well. Just this fact I quoted above was even more useful IMO.