An idea...

[The Keeper of Secrets]

Hi,

2-dimentional beings living on a very large 3d sphere (so large that they were not aware of the 3-dimensional curvature) would be very confused by the fact that when they journeyed for long enough they would arrive back at their staring point. They would also percieve their 2d world as infinite, tus adding confusion.

Is it possible that our universe is a 4d shpere (hypersphere)? It would explain the infinte nature or the 3d universe. The 4d sphere is simply so big that we can't detect the 4d curvature. If this were true we could journey for billions of years and come back to our original starting point, confusing us in the same way the 2d being is confused by the large 3d sphere. Is this feasable?

If I have missed something blindingly obvious then go easy, I'm only 17 after all.

Thanks

[Keiji]

Yes, this is certainly possible.

A torus would be more feasible though, as it doesn't suffer from the same singularity problems that spheres do.

[wendy]

One model of the current universe was that it was a giant poincare dodecahedron. This, repeated 120 times, makes the surface of a 4-sphere, it corresponds to the faces of the twelftychoron {5,3,3}. Still, all space, even euclidean, is curved, but the curvature is not "in" some higher space.

[Nick]

The Princeton Plasma Physics Laboratory has lectures on Saturdays that see every week. One lecture discussed this subject. But before I get to that:

You don't have to wrap the 2D world around sphere. Imagine a really long and thin rectangle. If you stretch the bottom edge and shrink the top edge it makes a circle without entering the third dimension, correct? The same is true with a 3D world. Imagine a torus (donut shape); this wraps back in on itself, but doesn't enter any higher spatial dimensions.

Back to the lecture. The speaker used an example with a turtle in a video game. The turtle can move the left or right. If it moves too far to the left (edge of the screen) it comes out the right, and vice versa. So, if you follow the turtle's vision, he will see the back of himself. Actually, he will see an infinite number of himself. They aren't actual turtles, nor are they mirrors of him. They are him. This is true for any direction the turtle can look; he can see himself at all angles. By looking up he sees his legs, by looking down he sees his shell.

He can see something else thats really cool too. Since the speed if light is not infinitely fast, there's a slight delay between himself and the turtle in front of him. If he started moving, it may appear that the turtle in front of him started moving at the same time, but if you look 100,000 turtles down, the 100,000th turtle didn't start yet. It's a reflection of him in the past. If the turtle looks down far enough he can see the day he was born (spontaneously?).

Now imagine if the planet Earth was the only thing in the Universe. We could see reflections of earth in every direction, and we could eventually even see the past, if the space around the earth curved back into itself. We could see the Earth the day it was made, and there wouldn't be any more of this "6000 years" business (actually, there probably still would be). Now here's the kicker: if this was true with the universe, then how big is it really? We may have counted billions of stars, but how many of these stars are unique? What if one of them is actually an image of our sun, billions of years ago as it was being created?

The topic of the lecture was "Is the Universe infinite?", in case you're wondering.

[The Keeper of Secrets]

Surely with a torus shape we can still find the edge of the universe. The point is a sphere provides no visible boundary to a 2d world , so a hypersphere shoud have no visible boundary in our 3d world. The turtle can use distances (if he's that clever) to estimate roughly where the border is. 2d sphere-dwelling beings couldn't find an edge, because from a 2d perspective there is indeed no edge to a sphere. The same should be true as we step everything up a dimension.

[zero]

What you're talking about is true of any finite but unbounded manifold.

[The Keeper of Secrets]

I'm not saying the 4d sphere would be infinite, and certaintly 4d beings could identify its borders. However, from our 3d perpestive, to borders would fill 3d space and sthus we couldn't find them, thus the shape to us is seemingly infinite, even though we know it is not. The same is true with the 2d beings on a sphere.

[zero]

You seem to have a clear understanding that the idea of a border or boundary is dependent on context (and also that appearances can sometimes be deceiving at first glance).

[Eric B]

You don't have to wrap the 2D world around sphere. Imagine a really long and thin rectangle. If you stretch the bottom edge and shrink the top edge it makes a circle without entering the third dimension, correct? The same is true with a 3D world. Imagine a torus (donut shape); this wraps back in on itself, but doesn't enter any higher spatial dimensions.

I didn't get this.
I take it you mean a rectangle that is horizontally ling, and vertically thin? Shrink and stretch the edges how? That would seem to change it from being a rectangle into a long trapezoid. You can wrap it around so that the top edge meets the bottom edge (like a straw), but that is using the third dimension.

Back to the lecture. The speaker used an example with a turtle in a video game. The turtle can move the left or right. If it moves too far to the left (edge of the screen) it comes out the right, and vice versa. So, if you follow the turtle's vision, he will see the back of himself. Actually, he will see an infinite number of himself. They aren't actual turtles, nor are they mirrors of him. They are him. This is true for any direction the turtle can look; he can see himself at all angles. By looking up he sees his legs, by looking down he sees his shell.

He can see something else thats really cool too. Since the speed if light is not infinitely fast, there's a slight delay between himself and the turtle in front of him. If he started moving, it may appear that the turtle in front of him started moving at the same time, but if you look 100,000 turtles down, the 100,000th turtle didn't start yet. It's a reflection of him in the past. If the turtle looks down far enough he can see the day he was born (spontaneously?).

Considering that those light beams have all been circling that universe all that time, it brings to mind the fact that if you are in a little hypershperical universe like that, you might get fried. Like being in a hypersphere with a circumference the size of a room, and turning on any light source. The light keep circling around and around, until it hits you, and more and more light is being emitted. Eventually, I would imagine, the radiation would become lethal. Sort of like a blue sheet, but not blueshifted. It's not like the large universe where it would take so long for light to do a lap around the universe. By the time it came back, the earth and all the stars may be gone. Hey, that's another interesting thought. So as the stars were all burning out, you would hae new (old) images of them arriving to keep it from being totally dark!

Now imagine if the planet Earth was the only thing in the Universe. We could see reflections of earth in every direction, and we could eventually even see the past, if the space around the earth curved back into itself. We could see the Earth the day it was made, and there wouldn't be any more of this "6000 years" business (actually, there probably still would be). Now here's the kicker: if this was true with the universe, then how big is it really? We may have counted billions of stars, but how many of these stars are unique? What if one of them is actually an image of our sun, billions of years ago as it was being created?

The topic of the lecture was "Is the Universe infinite?", in case you're wondering.

But the way the latest theories have gone now, with dark matter absorbing most of the light, and the expansion increasing, and belief in some sort of open ended universe, all of that may not mean much.

[Eric B]

Yes, this is certainly possible.

A torus would be more feasible though, as it doesn't suffer from the same singularity problems that spheres do.

What's a "singularity problem"? When you think of singularity, you think of a black hole, but I don't see what that would have to do with a sphere.
Actually, the truest torus would be one constructed in two more dimensions (d+2), rather than just one more dimension. Like for the 2D torus, you take a square, curl it into a tube (joining opposite sides). Now, to join the circular opposite ends of this tube, you curl it through the fourth dimension, instead of the third. That eliminates the crunching of one side and stretching of the other.
This article, http://cosmos.phy.tufts.edu/~zirbel/ast ... Finite.pdf discusses an unbounded but finite hyperbolic space shaped like a sort of double torus made by doing the same thing with an octagon on a pseudospherical surface. (I like that extended manifold they show. That is the true extension of a saddle).

[wotchadec]

It's believed this is actually how the big bang goes. There is no single point in our 3d universe we can pick out, since it is outside our dimensions. imagine that our universe is a series of stickers on an uninflated balloon. As you blow the balloon up, the 2d stickers get further and further apart. But no matter how hard we try, we will never be able to pinpoint the location of the big bang.

So I would say that your idea is not on pheasable, but probable.

[ParadoxJuice]

The Princeton Plasma Physics Laboratory has lectures on Saturdays that see every week. One lecture discussed this subject. But before I get to that:

You don't have to wrap the 2D world around sphere. Imagine a really long and thin rectangle. If you stretch the bottom edge and shrink the top edge it makes a circle without entering the third dimension, correct? The same is true with a 3D world. Imagine a torus (donut shape); this wraps back in on itself, but doesn't enter any higher spatial dimensions.

Back to the lecture. The speaker used an example with a turtle in a video game. The turtle can move the left or right. If it moves too far to the left (edge of the screen) it comes out the right, and vice versa. So, if you follow the turtle's vision, he will see the back of himself. Actually, he will see an infinite number of himself. They aren't actual turtles, nor are they mirrors of him. They are him. This is true for any direction the turtle can look; he can see himself at all angles. By looking up he sees his legs, by looking down he sees his shell.

He can see something else thats really cool too. Since the speed if light is not infinitely fast, there's a slight delay between himself and the turtle in front of him. If he started moving, it may appear that the turtle in front of him started moving at the same time, but if you look 100,000 turtles down, the 100,000th turtle didn't start yet. It's a reflection of him in the past. If the turtle looks down far enough he can see the day he was born (spontaneously?).

Now imagine if the planet Earth was the only thing in the Universe. We could see reflections of earth in every direction, and we could eventually even see the past, if the space around the earth curved back into itself. We could see the Earth the day it was made, and there wouldn't be any more of this "6000 years" business (actually, there probably still would be). Now here's the kicker: if this was true with the universe, then how big is it really? We may have counted billions of stars, but how many of these stars are unique? What if one of them is actually an image of our sun, billions of years ago as it was being created?

The topic of the lecture was "Is the Universe infinite?", in case you're wondering.

But what if the Turtle surpassed the speed of light and headed towards the point where he saw his own birth? It would create some form of visual distortion, but I simply can't imagine what that would look like.

[wendy]

There are models that suggest that space is finite, being for example, a cell of some repeating tile eg: a face of (5,3,3), octagonny, (3,4,3), (4,3,3) or some hyperbolic tiling like (5,3,5), (3,5,3) or (\3,5,A) or even (4,3,4) or (\3,4,A). or (\4,4,A). All of these pass. However, all of these have a repeatable geometry that is not observed at the scale where the repetitions are supposed to happen.

None the same, it is possible for space to be connected to other bits, and that we are looking more through 'wormholes', rather than having a look at some overall tile repeated over every wall of the tile. [One notes, eg that the cell of the poincare dodecahedron 5,3,3 can be replaced by a pentagonal tegum].