Didn’t Chronos claim to be able to do it? Perhaps he could explain how he did it? If not, then I guess I am stuck in visualizing only three dimensions. 
Oh, and doing the same thing with a cube would not give you visualization through time.
One misconception you continue to make is that there are “THE” dimensions versus “A” dimensions.
What do I mean by that?
A dimension is just that - a dimension of measure.
If you have a point on a plane, you only need 2 numbers to describe it’s position. If you can restrict it to a line, you can describe it’s position with just 1 number.
Since the Earth’s surface is more or less 2 dimensional, we can describe our positions with 2 numbers (x/y or lat/long or whatever). Sure, we might have some elevation above the ground, but that’s introducing the 3rd dimension.
“A” 4th dimension is just that. It can even be spatial. We don’t think our universe has 4 spatial dimensions, but hey, we could be wrong. There might be dozens of spatial dimensions we are unaware of. If so, our spatial coordinates would be described with dozens of numbers.
Forget temporal dimensions and spacetime.
First, realize that you can posit any arbitrary number of spatial dimensions you please.
This is what you are actually doing with your current visualization. You are really trying to visualize a 4th spatial dimension and just labeling it “time”.
When you ask how this extends to 5 dimensions or beyond, you are still thinking of spatial dimensions.
In this thread? Nope. Nobody else has, either. I know I can’t visualize four dimensions, either. I can get ‘close-ish’ via slicing, but it’s different than a true visualization. Kind of like staring at a Picasso.
We can understand additional dimensions mathematically, though.
It was in another thread.
Honestly, I’m really rusty at it. Ah, to have the flexible brain of a teenager, again! But for some relatively simple objects like a hypercube or hypersphere, using time as the fourth dimension, it’s doable. For instance, for a cube, picture a cube appearing for an instant, then everything but the vertices disappearing, with the vertices persisting for a little while, then the entire cube appearing again for an instant, then everything disappearing. For a hypersphere, picture a sphere expanding from a point, very rapidly at first, then slowing down, stopping, and then contracting at increasing speed until it vanishes back into a point. Note that your visualizations are almost certainly not to scale, since you’re probably imagining something much longer along the time axis than along the spatial axes.
And please, I implore you, completely ignore that “10 dimensions” YouTube video. It’s ludicrously wrong, across the board. It’s much easier to try to understand n dimensions for an arbitrary value of n, and then set n to some number, than it is to try to understand some specific number of dimensions like 10.
Well, how will I know if I am visualizing it correctly when viewing a rotating tesseract?
Really? Ludicrously? Google defines this word as, “So foolish, unreasonable, or out of place as to be amusing.”
So is the video really that bad? :eek:
Yes.
A dimension is basically just an axis on a graph. There’s nothing magical about them. In machine learning (and statistical modelling in general) some problems can be reduced to boggling number of dimensions depending on what you’re considering (I’m talking about dozens, hundreds, or thousands, though above a certain number it usually means your problem is overconstrained or needlessly complex in some way and you’ll get bad results). Of course, the unfortunate tradeoff of this is it makes the graph impossible to draw or visualize, which can make interpretation of the results very hard.
I’m not entirely sure exactly what the rigorous definition of “dimension” is in a physics context where you get your 3+1 spatial+temporal dimension, but it’s essentially just an extension of the concept that we’re measuring four things, so we need four “axes” on our “graph” to represent it.
In quantum mechanics the state space is infinite dimensional!
The best definition of the spacetime dimension is the manifold dimension, so spacetime has 4 dimensions as it is locally homeomorphic to E[sup]4[/sup] (i.e as long as you take a small enough patch, it will always topologically look like 4-D Euclidean space). 3+1 = 4, but stating the dimension as “3+1” as well as telling you the dimension, also carries additional information about the signature of the metric (more rigourously it tells you how many 1s and how many -1s there are in the set of diagonal components of the metric in an orthonormal basis).
A friend of mine had a research project which consisted, basically, of reconstructing an image, given some data. The image he was trying to reconstruct had some large number of pixels (millions, at least), and there was of course a value associated with each pixel. So each of the trial images he was considering could be described as a single point in a million-dimensional space. In his case, this meant that his problem was severely under-constrained, since he didn’t actually have enough data to give him that many pixels, so he had to use a bunch of heuristics to pick out the “best” image out of the myriad that could fit the data.
In my much better universe dimensions go to 11.
I’m sorry… I could not resist