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Comment by CT4u8798 | Hacker News Reader

I have no clue about maths beyond extremely basic stuff, but am fascinated by this sort of thing, and I need pictures to understand stuff like this. What an excellent video. During it, when they introduced how you can map the 2D to 3 dimensions, my initial thought was "I wonder if this is how you could map 3D into the 4th dimension?". Then later they mentioned 4 dimensions. This is something I cannot visualise or really understand.

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> I have no clue about maths beyond extremely basic stuff, but am fascinated by this sort of thing, and I need pictures to understand stuff like this.

Fascination is all you need. I find many people have a lot of self-limiting beliefs around math. There’s many reasons for them to develop, but I firmly believe that many people are legitimately interested in mathematics and have the capability despite their beliefs.

One of the problems with math, like a lot of things, is that even though you may find it deeply interesting and fascinating and you may even see great utility in it, becoming an expert is very difficult and is fraught with a lot of failure which many people can't, or won't, stomach.

I guess that's true for most things. Say, learning to play an instrument can be similarly difficult at first.

Motivation is vaning, you need discipline to actually stick to something and get better at it. But even getting better day-by-day by only a tiny percentage will result in huge gains over long periods.

I believe the most important aspect for learning math is being comfortable with not understanding and being able to smack your head into the wall repeatedly. It takes some stubborn determination to push past

I gave up on trying to visualize 4 dimensions. I don't know if it's possible. Instead I just try to think of 4D as more of ideas and less geometry: rules, consequences, capabilities, etc. We can do the same thing in 3 dimensions by saying things like "two objects can't exist at the same place and at the same time" or "parallel lines meet at infinity" or "parallel lines never meet" or something. We usually don't do that for 3 dimensions because we have visualizations and intuitions which we can use instead of breaking everything down formally all the time.

Time is nature's forth dimension, so I think considering the various stages of a slice moving through a four dimensional object at once counts as a visualization.

Donnie Darko style.

Time is not a dimension of the same kind as spatial dimensions. It has a different metric and you can’t move freely back and forth on it. When you rotate on the XT plane, it doesn’t mean the same thing as rotating on the XY plane. It is not a good candidate for the sort of fourth dimension we’re interested in.

The Euclidian group in four dimensions is a fourth dimension, but the Lorentz/Poincare group is the fourth dimension. ;)

My understanding is that time can be a 4th dimension, but n-dimensional spaces themselves are simply a very basic mathematical structure, where a point can be described by n numbers (you can actually be abstract even in that, no need to stick to rational numbers, I believe).

As long as you can map time to a number line, it's a valid representation. We just happen to have hardware acceleration for 3-dimensions, and the 4th is just completely unintuitive to us.

If we're only talking about simple vector spaces, your understanding is accurate, but when we're talking about visualizing shapes in 4 dimensions, we typically want something more. We are doing geometry then, and so we want a metric space that defines a concept of distance (which vector space don't have).

When it comes to geometry and not just vector spaces, time dimensions have a different definition of distance than do space dimensions. There's a minus in the formula where you would usually have a plus. And this means that shapes in this space behave very differently than what we're after when imagining a hypercube or hypersphere, for example.

We want to think of a 4 dimensional space where all the dimensions are indistinguishable, but the minus sign in the metric distinctly identifies the time dimension. For this reason, physicists typically call this kind of space a 3+1 dimensional space rather than a 4 dimensional one.

https://www.youtube.com/watch?v=GkCWywO93b8

There’s a video from the same channel on visualizing quaternions as a projection into 3d that was really fun for this. Only a restricted section of a 4d space, but i feel like the principle generalizes a little because of the idea of, like, imagining one 3d space thats finite as equivalent to an infinite 3d space, just stretched

I've always wanted to make a 4d space in VR. That way it's only one dimension higher, technically. Could help to visualize it in a way that hasn't been done yet

There's a 4D mini golf VR game you might be interested in checking out. It's called, uh, 4D Golf. Creative! I've not played it myself, but it's on my list. I hear it's pretty cool!

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