What happens at the end of a static, flat and finite universe?

[ryan]

We live in a flat and finite universe, but it is growing too fast for us to ever get to its "wall" of existence.

My question is, what if the universe were the size of, say, the Earth plus only 10 feet of atmosphere, and you jumped on a trampoline with an acceleration that would get you 11 feet high?

In a closed universe of uniform curvature, you are always at the center of the "universe" even if you explore every inch of it. But what about an open and static universe? What do you think happens at the edge?

 

[Lion IRC]

You press your face up against the boundary of an adjacent universe.

 

[fast]

You breach the hull which suggests you enter into a space outside the universe which toys with the very assumption that it's finite.

 

[bilby]

We live in a flat and finite universe, but it is growing too fast for us to ever get to its "wall" of existence.

My question is, what if the universe were the size of, say, the Earth plus only 10 feet of atmosphere, and you jumped on a trampoline with an acceleration that would get you 11 feet high?

In a closed universe of uniform curvature, you are always at the center of the "universe" even if you explore every inch of it. But what about an open and static universe? What do you think happens at the edge?

According to Pacman, you disappear, and reappear at the opposite edge.

 

[beero1000]

Beware, there be math below...

We need to be careful with our assumptions here. There are potential flat, finite, and static universes that are possible that don't have any 'edges'.

There are 18 candidate topologies for flat universes without edges (the number of Euclidean 3-manifolds) and we don't really know which one we live in. Of the 18, eight are non-orientable. While it's possible the universe is non-orientable that would have super weird consequences that we don't see in our universe (we've checked).

So there are 10 candidate orientable Euclidean 3-manifolds. Of these, 4 are non-compact (essentially, infinite in at least one direction) and 6 are compact (essentially, finite in every direction). I'm going to try and describe them, but they are a challenging exercise in 3D visualization and I'm too lazy to post figures, so bear with me.

The 4 non-compact, orientable, Euclidean 3-manifolds are:

1. \(\mathbb{R}^3\). This is our intuitive, standard, 3D Euclidean space.
2. \(\mathbb{R}^2 \times S^1\). This is the 3D analog of a cylinder
3. \(\mathbb{R} \times S^1 \times S^1\). This space has one 'normal' infinite dimension while the other two form Pac-man's space (a torus)
4. The easiest way to describe this space is as the previous one but with a half twist (e.g. Pac-man goes through one side and comes out the other, not directly across but rather flipped about the infinite axis.

The 6 compact, orientable, Euclidean 3-manifolds are:

1. \(S^1 \times S^1 \times S^1\). This is the 3-torus, the 3D analog of Pac-man space.
2. 3D Pac-man space but with a quarter twist of one dimension (e.g. you go through that square wall and come out on the opposite wall rotated 90 degrees).
3. 3D Pac-man space but with a half twist of one dimension.
4. Think of this one as 3D Pac-man, but where one pair of walls is split in two and you teleport diagonally across the split pairs.
5. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/6 twist in the hexagonal dimension.
6. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/3 twist in the hexagonal dimension.

From what I understand, most cosmologists think that we live in one of the latter 6 (the compact, orientable, Euclidean 3-manifolds). But which? We don't currently know, but studies in cosmic microwave background might find out in the future.

To answer your question about jumping off a trampoline, it would depend on the shape of the universe and the direction you were jumping. Pick and we can work through what would happen...

 

[ryan]

Beware, there be math below...

We need to be careful with our assumptions here. There are potential flat, finite, and static universes that are possible that don't have any 'edges'.

There are 18 candidate topologies for flat universes without edges (the number of Euclidean 3-manifolds) and we don't really know which one we live in. Of the 18, eight are non-orientable. While it's possible the universe is non-orientable that would have super weird consequences that we don't see in our universe (we've checked).

So there are 10 candidate orientable Euclidean 3-manifolds. Of these, 4 are non-compact (essentially, infinite in at least one direction) and 6 are compact (essentially, finite in every direction). I'm going to try and describe them, but they are a challenging exercise in 3D visualization and I'm too lazy to post figures, so bear with me.

The 4 non-compact, orientable, Euclidean 3-manifolds are:

1. \(\mathbb{R}^3\). This is our intuitive, standard, 3D Euclidean space.
2. \(\mathbb{R}^2 \times S^1\). This is the 3D analog of a cylinder
3. \(\mathbb{R} \times S^1 \times S^1\). This space has one 'normal' infinite dimension while the other two form Pac-man's space (a torus)
4. The easiest way to describe this space is as the previous one but with a half twist (e.g. Pac-man goes through one side and comes out the other, not directly across but rather flipped about the infinite axis.

The 6 compact, orientable, Euclidean 3-manifolds are:

1. \(S^1 \times S^1 \times S^1\). This is the 3-torus, the 3D analog of Pac-man space.
2. 3D Pac-man space but with a quarter twist of one dimension (e.g. you go through that square wall and come out on the opposite wall rotated 90 degrees).
3. 3D Pac-man space but with a half twist of one dimension.
4. Think of this one as 3D Pac-man, but where one pair of walls is split in two and you teleport diagonally across the split pairs.
5. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/6 twist in the hexagonal dimension.
6. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/3 twist in the hexagonal dimension.

From what I understand, most cosmologists think that we live in one of the latter 6 (the compact, orientable, Euclidean 3-manifolds). But which? We don't currently know, but studies in cosmic microwave background might find out in the future.

To answer your question about jumping off a trampoline, it would depend on the shape of the universe and the direction you were jumping. Pick and we can work through what would happen...

Okay, the jump will be vertical. The entire universe would be the Earth plus an extra 10 feet of space/atmosphere, so the radius of the universe is the Earth's radius plus 10 feet. Assume a perfect sphere. And let's assume only 4 dimensions (3 space + 1 time).

 

[beero1000]

Beware, there be math below...

We need to be careful with our assumptions here. There are potential flat, finite, and static universes that are possible that don't have any 'edges'.

There are 18 candidate topologies for flat universes without edges (the number of Euclidean 3-manifolds) and we don't really know which one we live in. Of the 18, eight are non-orientable. While it's possible the universe is non-orientable that would have super weird consequences that we don't see in our universe (we've checked).

So there are 10 candidate orientable Euclidean 3-manifolds. Of these, 4 are non-compact (essentially, infinite in at least one direction) and 6 are compact (essentially, finite in every direction). I'm going to try and describe them, but they are a challenging exercise in 3D visualization and I'm too lazy to post figures, so bear with me.

The 4 non-compact, orientable, Euclidean 3-manifolds are:

1. \(\mathbb{R}^3\). This is our intuitive, standard, 3D Euclidean space.
2. \(\mathbb{R}^2 \times S^1\). This is the 3D analog of a cylinder
3. \(\mathbb{R} \times S^1 \times S^1\). This space has one 'normal' infinite dimension while the other two form Pac-man's space (a torus)
4. The easiest way to describe this space is as the previous one but with a half twist (e.g. Pac-man goes through one side and comes out the other, not directly across but rather flipped about the infinite axis.

The 6 compact, orientable, Euclidean 3-manifolds are:

1. \(S^1 \times S^1 \times S^1\). This is the 3-torus, the 3D analog of Pac-man space.
2. 3D Pac-man space but with a quarter twist of one dimension (e.g. you go through that square wall and come out on the opposite wall rotated 90 degrees).
3. 3D Pac-man space but with a half twist of one dimension.
4. Think of this one as 3D Pac-man, but where one pair of walls is split in two and you teleport diagonally across the split pairs.
5. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/6 twist in the hexagonal dimension.
6. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/3 twist in the hexagonal dimension.

From what I understand, most cosmologists think that we live in one of the latter 6 (the compact, orientable, Euclidean 3-manifolds). But which? We don't currently know, but studies in cosmic microwave background might find out in the future.

To answer your question about jumping off a trampoline, it would depend on the shape of the universe and the direction you were jumping. Pick and we can work through what would happen...

Okay, the jump will be vertical. The entire universe would be the Earth plus an extra 10 feet of space/atmosphere, so the radius of the universe is the Earth's radius plus 10 feet. Assume a perfect sphere. And let's assume only 4 dimensions (3 space + 1 time).

But I wrote... all of that... and...

 

[LordKiran]

Beware, there be math below...

We need to be careful with our assumptions here. There are potential flat, finite, and static universes that are possible that don't have any 'edges'.

There are 18 candidate topologies for flat universes without edges (the number of Euclidean 3-manifolds) and we don't really know which one we live in. Of the 18, eight are non-orientable. While it's possible the universe is non-orientable that would have super weird consequences that we don't see in our universe (we've checked).

So there are 10 candidate orientable Euclidean 3-manifolds. Of these, 4 are non-compact (essentially, infinite in at least one direction) and 6 are compact (essentially, finite in every direction). I'm going to try and describe them, but they are a challenging exercise in 3D visualization and I'm too lazy to post figures, so bear with me.

The 4 non-compact, orientable, Euclidean 3-manifolds are:

1. \(\mathbb{R}^3\). This is our intuitive, standard, 3D Euclidean space.
2. \(\mathbb{R}^2 \times S^1\). This is the 3D analog of a cylinder
3. \(\mathbb{R} \times S^1 \times S^1\). This space has one 'normal' infinite dimension while the other two form Pac-man's space (a torus)
4. The easiest way to describe this space is as the previous one but with a half twist (e.g. Pac-man goes through one side and comes out the other, not directly across but rather flipped about the infinite axis.

The 6 compact, orientable, Euclidean 3-manifolds are:

1. \(S^1 \times S^1 \times S^1\). This is the 3-torus, the 3D analog of Pac-man space.
2. 3D Pac-man space but with a quarter twist of one dimension (e.g. you go through that square wall and come out on the opposite wall rotated 90 degrees).
3. 3D Pac-man space but with a half twist of one dimension.
4. Think of this one as 3D Pac-man, but where one pair of walls is split in two and you teleport diagonally across the split pairs.
5. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/6 twist in the hexagonal dimension.
6. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/3 twist in the hexagonal dimension.

From what I understand, most cosmologists think that we live in one of the latter 6 (the compact, orientable, Euclidean 3-manifolds). But which? We don't currently know, but studies in cosmic microwave background might find out in the future.

To answer your question about jumping off a trampoline, it would depend on the shape of the universe and the direction you were jumping. Pick and we can work through what would happen...

How can something be infinite in only one direction? Wouldn't the logical consequence of that be something that is infinite in at least two directions?

 

[beero1000]

Beware, there be math below...

We need to be careful with our assumptions here. There are potential flat, finite, and static universes that are possible that don't have any 'edges'.

There are 18 candidate topologies for flat universes without edges (the number of Euclidean 3-manifolds) and we don't really know which one we live in. Of the 18, eight are non-orientable. While it's possible the universe is non-orientable that would have super weird consequences that we don't see in our universe (we've checked).

So there are 10 candidate orientable Euclidean 3-manifolds. Of these, 4 are non-compact (essentially, infinite in at least one direction) and 6 are compact (essentially, finite in every direction). I'm going to try and describe them, but they are a challenging exercise in 3D visualization and I'm too lazy to post figures, so bear with me.

The 4 non-compact, orientable, Euclidean 3-manifolds are:

1. \(\mathbb{R}^3\). This is our intuitive, standard, 3D Euclidean space.
2. \(\mathbb{R}^2 \times S^1\). This is the 3D analog of a cylinder
3. \(\mathbb{R} \times S^1 \times S^1\). This space has one 'normal' infinite dimension while the other two form Pac-man's space (a torus)
4. The easiest way to describe this space is as the previous one but with a half twist (e.g. Pac-man goes through one side and comes out the other, not directly across but rather flipped about the infinite axis.

The 6 compact, orientable, Euclidean 3-manifolds are:

1. \(S^1 \times S^1 \times S^1\). This is the 3-torus, the 3D analog of Pac-man space.
2. 3D Pac-man space but with a quarter twist of one dimension (e.g. you go through that square wall and come out on the opposite wall rotated 90 degrees).
3. 3D Pac-man space but with a half twist of one dimension.
4. Think of this one as 3D Pac-man, but where one pair of walls is split in two and you teleport diagonally across the split pairs.
5. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/6 twist in the hexagonal dimension.
6. Think of this one as 3D Pac-man on a hexagonal prism, but with a 1/3 twist in the hexagonal dimension.

From what I understand, most cosmologists think that we live in one of the latter 6 (the compact, orientable, Euclidean 3-manifolds). But which? We don't currently know, but studies in cosmic microwave background might find out in the future.

To answer your question about jumping off a trampoline, it would depend on the shape of the universe and the direction you were jumping. Pick and we can work through what would happen...

How can something be infinite in only one direction? Wouldn't the logical consequence of that be something that is infinite in at least two directions?

Not necessarily, what about a ray?

 

[Malintent]

I am reminded of a Next Generation episode. People start disappearing from the ship. Not just disappearing, though, they are being eliminated from reality... as if they never existed. The Dr. is the only person that remembers the "missing" people. By the end of the episode, it is just the captain, and the Dr. The Captain thinks everything is status quo, and still the Dr. is the only one that realizes something is wrong. finally, once the Captain disappears, and the Dr. is the only person in existence on the ship, hull breaches start happening. When she asks the ship's computer what is going on, it reports "hull breach on multiple decks". The Dr. asks the cause. The ship reports "design flaw". The Dr. pulls up a picture of the ship, and it is encircled in a collapsing bubble that is intersecting the ship. Here is the cool part:

The Dr. says, "here's a question you shouldn't be able to answer.." and asks the ship, "what is the nature of the universe".
The ship responds, "The Universe is a spheroid, approximately 500 meters in diameter".

My head spun a little when I heard that for the first time... cool as hell. great show.

The Dr. was trapped in a quantum bubble that created a small, but unstable, universe... and as it collapsed upon itself, those things that fell outside the universe ceased to ever have existed.

 

[ryan]

Okay, the jump will be vertical. The entire universe would be the Earth plus an extra 10 feet of space/atmosphere, so the radius of the universe is the Earth's radius plus 10 feet. Assume a perfect sphere. And let's assume only 4 dimensions (3 space + 1 time).

But I wrote... all of that... and...

I think my question is somehow impossible because everything I have been finding on flat universes seem to suggest that a flat or open universe must mean it's infinite in size. I don't know why that is. Do you know? Like why can't I have my finite flat universe?

And if we can have a finite and flat universe (no twists, no torus, no hexagons, no net curvature) what would happen if you tried to go through the edge? Is there even an edge?

 

[beero1000]

But I wrote... all of that... and...

I think my question is somehow impossible because everything I have been finding on flat universes seem to suggest that a flat or open universe must mean it's infinite in size. I don't know why that is. Do you know? Like why can't I have my finite flat universe?

And if we can have a finite and flat universe (no twists, no torus, no hexagons, no net curvature) what would happen if you tried to go through the edge? Is there even an edge?

No, that's wrong. All 18 of the topologies I talked about are flat - they have zero curvature, and 10 of them are finite. None of them have edges.

 

[ryan]

How can something be infinite in only one direction? Wouldn't the logical consequence of that be something that is infinite in at least two directions?

Interesting, but I think you can have one direction of infinity. Imagine you are at the beginning of an infinitely long string. No matter how many meters you move along the string, you will never get to the end and you will always be a finite distance from the beginning. You won't even make any gain from a reference of guy infinitely far from you who can see the string in its entirety. He will never see your gain any distance other than not being at 0. He will never even find a number to represent your position.

So it seems completely possible that one direction is infinite and the other is not (since we are assuming infinities in space-time exist anyways).

 

[ryan]

I think my question is somehow impossible because everything I have been finding on flat universes seem to suggest that a flat or open universe must mean it's infinite in size. I don't know why that is. Do you know? Like why can't I have my finite flat universe?

And if we can have a finite and flat universe (no twists, no torus, no hexagons, no net curvature) what would happen if you tried to go through the edge? Is there even an edge?

No, that's wrong. All 18 of the topologies I talked about are flat - they have zero curvature, and 10 of them are finite. None of them have edges.

I must be missing something then. Are these closed universes? How can you have a Pac-man situation without a loop? My mind is feeling pain.

 

[Shadowy Man]

How can something be infinite in only one direction? Wouldn't the logical consequence of that be something that is infinite in at least two directions?

An infinitely long cylinder has an infinite length but a finite circumference.

 

[beero1000]

No, that's wrong. All 18 of the topologies I talked about are flat - they have zero curvature, and 10 of them are finite. None of them have edges.

I must be missing something then. Are these closed universes? How can you have a Pac-man situation without a loop? My mind is feeling pain.

You can have a loop without being intrinsically curved. Pac-man's universe isn't intrinsically curved - if Pac-man draws a triangle, the angles add up to 180 degrees, even though he lives on a (topological) torus. It's only when you try to embed that shape into \(\mathbb{R}^3\) that you force extrinsic curvature, which is why you need to be careful not to implicitly embed the universe into another space. And you need to clarify what you mean by closed - the compact Euclidean manifolds I listed are all examples of a  closed manifold.

 

[beero1000]

Found this image to help visualize my attempted explanations. These are the 6 compact, orientable Euclidean 3-manifolds (the ones that most cosmologists consider the probable candidates for our universe):



The unmarked faces are mapped to those directly across from themselves (like in Pac-man), with the doors as markers used to demonstrate the potential twists. They're in almost the same order as in my post, but I have the Hantzsche-Wendt manifold (my 4, their f) with a cubic fundamental domain, while theirs is the rhombic dodecahedron version.

(a-f it's Pac-man, quarter-twist Pac-man, half-twist Pac-man, sixth-twist hexagonal Pac-man, third-twist hexagonal Pac-man, Hantzshe-Wendt).

 

[ryan]

Okay, I think I get it now. They are trying to say that you don't need a closed universe to come out the other side. I always thought only a closed universe allows that.

But I would like to know why they think we would come out the other side, or see the back of a door in a universe that is small enough.

Can we not just have a 2d ant on a finite 2d space the size of a dinner plate? Why not? If we can, then what happens to the ant if he tries to go through the boundary?

 

[bilby]

Okay, I think I get it now. They are trying to say that you don't need a closed universe to come out the other side. I always thought only a closed universe allows that.

But I would like to know why they think we would come out the other side, or see the back of a door in a universe that is small enough.

Can we not just have a 2d ant on a finite 2d space the size of a dinner plate? Why not? If we can, then what happens to the ant if he tries to go through the boundary?

It's a thought experiment; we can have whatever we can imagine. But what happens at the boundary is determined by what you imagine happening at the boundary - so if you don't know, we can't tell you.

Amongst the options are that the ant ceases to exist; or that he re-enters the universe in one of the ways described by beero1000; or something else happens, that arises due to one or more of the infinite number of attributes of the universe that are left undefined in your thought experiment.

 

[ryan]

Okay, I think I get it now. They are trying to say that you don't need a closed universe to come out the other side. I always thought only a closed universe allows that.

But I would like to know why they think we would come out the other side, or see the back of a door in a universe that is small enough.

Can we not just have a 2d ant on a finite 2d space the size of a dinner plate? Why not? If we can, then what happens to the ant if he tries to go through the boundary?

It's a thought experiment; we can have whatever we can imagine. But what happens at the boundary is determined by what you imagine happening at the boundary - so if you don't know, we can't tell you.

Nobody can tell me or you can't tell me?

Amongst the options are that the ant ceases to exist; or that he re-enters the universe in one of the ways described by beero1000; or something else happens, that arises due to one or more of the infinite number of attributes of the universe that are left undefined in your thought experiment.

I would like to know why they think that a finite and flat universe "loops" back in on itself. Do you know? I read some places that it does loop back, but I also read that a flat universe implies an infinitely large universe, neither explaining how they get to that conclusion.