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[Putnam]

"Down" is a direction in the 4th spatial dimension, so you really can't go "down".

...It's not 3rd dimension? I know 2D has forward, back, left and right. I though 3rd was up, down?

I meant the "down" you were referring to, and yes, "kata" is a more proper term.

[BurnedToast]

"Down" is a direction in the 4th spatial dimension, so you really can't go "down".

But why not?

I mean, ok, we can't because we're only 3 dimensional but why does that matter? We can describe it all with mathematics (I think?) so why can't we say the center of the universe is (insert mathematical equation equivalent of a "down" arrow on the 4th dimensional sphere)?

And even if we can't describe it, a hypothetical 4-dimensional person wouldn't have any trouble with it. That we can't understand it does not make it not exist - a two dimensional being would have no concept of how to get to the center of earth, but the center of earth still exists.

"Down" in quotes here refers to the down you'd have to travel to reach the center of a 4-dimensional sphere from its surface. Probably we're looking for kata here.

Yeah, that sounds about right

[tahujdt]

Gentlemen, as an official assigner of reading material (here's my card - and my Artistic License) and Commissar of the Internets, I hereby state that no-one may take part in this tip-top topology topic until they have read 'Flatland: A Romance Of Many Dimensions'.

That is all.

[/facetiousness]

[Putnam]

You're right, though.

[Reelya]

"Down" is a direction in the 4th spatial dimension, so you really can't go "down".

But why not?

I mean, ok, we can't because we're only 3 dimensional but why does that matter? We can describe it all with mathematics (I think?) so why can't we say the center of the universe is (insert mathematical equation equivalent of a "down" arrow on the 4th dimensional sphere)?

And even if we can't describe it, a hypothetical 4-dimensional person wouldn't have any trouble with it. That we can't understand it does not make it not exist - a two dimensional being would have no concept of how to get to the center of earth, but the center of earth still exists.

"Down" in quotes here refers to the down you'd have to travel to reach the center of a 4-dimensional sphere from its surface. Probably we're looking for kata here.

Yeah, that sounds about right

Kind of missing the point. "center of the universe" means something else here than you think. If the universe is 4D and we're in a 3D surface of a hypersphere, there's no "place" you can go that you can say "THERE - that's the center". Look at each lower order example:

1D creature on a circle. They can move backwards and forwards on the circle. Each point looks the same as any other. The 1D movement never takes them to a "center" point.

2D creature on a sphere. They can move backwards and forwards, left and right on the sphere. Each point looks the same as any other. The 2D movement never takes them to a "center" point.

Same with a 3D space that is the surface of a hypersphere. It's a direct extension of the same mathematics. All "places" you can go are equally valid candidates to be the center, and they're all the exact same distance from the hypersphere center when you calculate the 4D distance.

So, as long as you're traversing the surface of any order circle/sphere/hypersphere you are always the same distance from the system-center.

[BurnedToast]

I do understand that. I'm not suggesting the center is a place in the part of the universe we exist in or that we could point at or visit. However, a 2 dimensional being couldn't point to the center of the earth but the center of earth exists anyway. Would a 4 dimensional hypersphere not have a center? I can point to the center of earth, could a 4 dimensional being point at the center of the hypersphere that is the universe?

Is it just that when science™ says there's no center of the universe, they are referring only to the part of the universe we interact with, and not the hypersphere as a whole?

Or is the universe not really a 4(+) dimensional hypersphere and it's just a flawed and confusing analogy?

Or do hyperspheres not have centers (despite the fact lines, circles, and spheres all have centers)?

Or something else?

[Reelya]

You can point to the center of the Earth because we're 3D creatures in a 3D universe. We just happen to be sitting on top of a sphere. A true 2D creature wouldn't have up or down or any way to interpret it's top or bottom.

You could point to the center of the 4D universe if you grew an extra dimension and could point in a direction that's perpendicular to the X, Y and Z axes.

If you look at the 1D circle and 2D sphere models, any possible direction that allowable vectors on those surfaces can point is a tangent to the circle or sphere. Extending that to a 3D surface of a hypersphere, any possible direction you can point that's a combination of X Y and Z vectors is a tangent to the hypersphere.

So, in a sense we can point away from the hypersphere since tangents point "away" from the circle or sphere, but we can never point to a 4D point inside the hypersphere.

[Mech#4]

I found a website at one point that had multi-dimensional mazes. 3D makes perfect sense, 4D meant you had to apparently move between walls in this ::Waves hand vaguely:: direction. Very confusing, I think the 4D maze was portrayed as having an extra way to move onto another layer or a section within the existing 3D maze parts.

[Sheb]

No, but a 2-D creature could determine that the Earth's surface is curved (just like we can tell if the universe is curved or not), and realize he's a 3-D sphere.

[Reelya]

yeah, but as far as I can ascertain, he can't determine what direction it's curved in, just how much it's curved.

[Sheb]

But that means it could determine the sphere's radius. An that it's a sphere. So the center would be "a radius perpendicular to the surface in whatever dimension I can't picture but can totally represent mathematically".

[Reelya]

I already said that pretty much right before you jumped in. So I'm not 100% sure what you're arguing against.

You could point to the center of the 4D universe if you grew an extra dimension and could point in a direction that's perpendicular to the X, Y and Z axes.

You can work out the distance to the center easily on any 1D, 2D etc system. But you can never determine what direction of kata it is.

For example a 1D organism on a circle, could determine that the world loops around, and how big it is, perhaps develop maths to describe it, but it could never determine where on the circle he is, or whether the circle center is to his right or left (which he couldn't really visualize anyway).

The same with a sphere, no matter where you are on a sphere you get the same measurements, hence you can't really tell which higher-dimensional direction that thing is in, just how far away it is (always the same distance). We can really only do that because we reference the points in 3D, with up differentiated from down. A proper 2D organism wouldn't have any reference for up/down or inside/outside the sphere.

[Bauglir]

Yeah, I think everyone gets that at this point. Yes, BurnedToast, when people say there is no center of the universe, it more strictly means that the 3D space we occupy does not contain its own center.

[Il Palazzo]

All this talk erroneously suggests that there must be a centre in higher dimensional space. There is nothing in mathematics that says a space must be embedded in a higher dimensional manifold to have curvature. When there is talk about our universe with its 3 spatial dimensions having curvature, it doesn't imply that it's just a 3D slice of a "true" 4D universe. It could be, but it doesn't have to be, so talking about centre of the universe outside our 3 spatial dimensions is still misleading.

[Sheb]

Now I'm lost. How do you define curvature then?