[empath-nirvana]

Higher dimensions in general?

An n-dimensional space is just a collection of points, each defined uniquely by a set of n-numbers. The semantic meaning of those numbers doesn't really matter. It might be like actual physical space, but it could just as well be something like "time" and "the price of big macs". We have a bunch of mathematical operations that work well on 2 or 3 dimensional space that correlate nicely with our physical intuitions of 'curvature' and 'holes', and that still work perfectly well in more generalized forms in higher dimensions.

I'm not really sure it's that useful to try and visualize what it means on higher dimensions, to be honest.

[wackycat]

It's not perfect but to get an idea of adding one more dimension on top of the three dimensions we can visualize is thinking of color as the 4th dimension. There's a game called 4D Maze created by a topolgist that's availble in iphone app store. The visualization is 3d but if you can imagine the colors taking up the same space (instead of being right next to each other in 3d space), it kinda works. At least, it's the closest I've ever come to feeling like I could visualize or understand an additional dimension.

[xg15]

Yeah, the "an n-dimensional vector is just a struct with n floats" way of thinking is great - until you actually want to apply geometrical operations in the vector space, such as calculating a distance or performing a rotation. Then you have a problem: You cannot visualise such a space and "pretending" to work in 2D/3D space is convenient but often extremely misleading.

So what kind of intuition could you use instead then? Or what exactly do you mean with "work perfectly well"?

[alistairSH]

“Just a struct” plus “measuring curvature and shapes” is where my mind goes into “must visualize this” mode. How does a struct have curvature/shape? Or is curvature overloaded here (with a technical math definition that is very different than the layman’s “surface of a sphere” mental model).

[monadINtop]

the technical math definition is a rigourous formulation that encapsulates exactly the same thing as what we mean when we say things are curved, but one that also extends far more generally into contexts where our old intuition fails.

The same is true for most mathematics. For example, we are introduced to multiplication as repeated addition: 3x == x + x + x or 2x == x + x and more generally nx == x + x + ... + x, for n number of times. Of course this is only defined over naturals, what would it mean if we instead took n to be fractional, negative, irrational, or even complex? We of can easily generalise multiplication over larger and more complex fields and spaces, but in doing so we must abandon our old intuitive idea that nx is x + x n-times.

[alistairSH]

Yeah, the visualization is where I get hung up. Sounds like I need to stop that.

Given your response, is it fair to say time as the 4th dimension is just a sci-fi concoction?

[gavagai691]

No, that's not exactly a sci-fi concoction. In special and general relativity, there are three dimensions for space and one dimension for time, and this is not something that is of "incidental" importance to special / general relativity, it's a pretty essential shift in perspective to these theories to think of the universe as (curved) four-dimensional spacetime.

But "dimension" is something mathematical. I would say it doesn't quite make sense to say "is the fourth dimension time" in the same way as it wouldn't make sense to say "is the fifth an apple?" The same way that numbers can refer to different things in different contexts (including in the context of different scientific theories), dimensions can correspond to different things in different contexts. For example, statistics and machine learning heavily use "high dimensional" mathematics, but there the "dimensions" would correspond to different variables you are trying to predict or explain. E.g. if you were trying to predict chance of heart attack from 1000 different factors, then you would have 1000+1 total "dimensions," and in that case the "fourth dimension" might be "cigarettes smoked per week" (rather than time).

[markisus]

Contextually of dimension even exists within a specific scientific theory. In relativity, the direction you call time might contain some component of the direction I call space. This implies notions like simultaneity are not well defined in a universal context.

[gary_0]

No, 4D spacetime is a real thing in physics, which explains things like time dilation and the speed of light. But sci-fi does tend to abuse the term "dimension" for other ideas that are not scientific.

[gcr]

yeah, nobody can visualize it. it's something you just get used to after a while.

there's an old joke about a mathematician teaching an engineer about thirteen-dimensional spaces. "What do you think," the mathematician asks. "My head's spinning," the engineer confesses. "How can you develop any intuition for thirteen-dimensional space?"

"Well, it's not so hard. All I do is visualize the situation in arbitrary N-dimensional space and then set N = 13."

[philipswood]

Geoffrey Hinton on visualizing higher dimensions:

"To deal with hyper-planes in a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it."