<p>Given a point <span class="math-container">$p_0$</span> and a metric <span class="math-container">$d$</span>, we can define the antipode <span class="math-container">$p_1$</span> as the argmax of <span class="math-container">$d(p_0,p)$</span> as <span class="math-container">$p$</span> varies across all points. There are things that could happen that would make this definition go wrong:</p>
<ol>
<li><p>There is no upper bound on <span class="math-container">$d(p_0,p)$</span>. Note that even if the <em>volume</em> of the universe is finite, it's theoretically possible that the <em>diameter</em> is infinite.</p>
</li>
<li><p>The upper bound is not achieved. There is some distance <span class="math-container">$r$</span> such that given any <span class="math-container">$\epsilon$</span>, there is a point such that its distance from <span class="math-container">$p_0$</span> is greater than <span class="math-container">$r-\epsilon$</span>, but there is no point whose distance from <span class="math-container">$p_0$</span> is exactly <span class="math-container">$r$</span>. This suggests that there is some point &quot;missing&quot; from the space.</p>
</li>
<li><p>The point is not unique. That is, there is some <span class="math-container">$r$</span> such that  there are multiple points that are distance <span class="math-container">$r$</span> and no point has a distance greater than <span class="math-container">$r$</span>.</p>
</li>
</ol>
<p>There is the further question of whether we're using some three-dimensional distance, in which case we have to fix some coordinate system, or we're using proper distance.</p>

CC BY-SA 4.0
2023-12-27T21:11:24.033Z
2023-12-27T21:11:24.033Z
Acccumulation
<p>Yes, it's conceivable. If the universe is spatially hyperspherical, then there would be a most distant location from every location in the universe, much like there is on the surface of a sphere.</p>
<p>There is no evidence that the universe is hyperspherical, but it's also not ruled out. What is ruled out is a hyperspherical universe small enough that we can see all the way around it, so if there is a most distant galaxy from the Milky Way, it's probably invisible to us.</p>

CC BY-SA 4.0
2023-12-27T19:08:38.433Z
For fun, [most remote land on Earth](https://en.wikipedia.org/wiki/Extremes_on_Earth#Other_places_considered_the_most_remote)
chux
CC BY-SA 4.0
2023-12-27T02:53:45.853Z
2023-12-27T02:53:45.853Z
benrg
<p>Some time ago, I wondered where the center of the universe could be, and the answers I found were pretty much in agreement that there was no center of the universe, as paradoxical as that might seem.</p>
<p>So I started wondering, just as the surface of the Earth has no true center, yet it has antipodes, could the Universe itself -- having no true center -- also have antipodes?</p>
<p>(The term &quot;antipodes&quot; usually refers to two points on the surface of a sphere that are opposite each other, and are therefore mutually the farthest points from each other.)</p>
<p>I know that at some time in the past, when educated people started discussing the Earth as a sphere, no longer did they talk about a specific point on the Earth's surface as its true center, but instead pondered the mysteries and consequences of its antipodes, such as &quot;What's on the other side of here?  Is it reachable?  And could it be populated by people?&quot;</p>
<p>Following the same logic, now that I've learned that the universe has no true center, I've started wondering if it makes sense that the universe has antipodes.  For example, could there conceivably be a point (or area) of the universe that is the farthest away from Earth, our solar system, or the Milky Way galaxy?</p>

CC BY-SA 4.0
2023-12-27T02:02:29.880Z
Strictly speaking, “antipodes” means the point opposite your feet, so the point diametrically opposite to where you are on the Earth. So, of course, the Earth has antipodes, as you don’t need a specific centre point for it to have antipodes. Now, for the Universe, the question is pointless if using “antipodes” in that sense, as there is no “other side” to it…
Pierre Paquette
CC BY-SA 4.0
2023-12-27T14:31:26.760Z
Doesn't the existence of "a point that is the farthest away from Earth" naturally follow from the universe being finite plus the well-ordering principle (applied to the function "distance from Earth to each point")?
Karl Knechtel
CC BY-SA 4.0
2023-12-27T20:45:29.363Z
@PierrePaquette The term "antipode" can be defined in the context of $S^2$ as a two-dimensional manifold without any reference to the embedding space.
Acccumulation
CC BY-SA 4.0
2023-12-27T20:48:35.443Z
@KarlKnechtel The well-ordering theorem says that for every set, we could come up with an ordering is a well-ordering. It says nothing about whether the *existing* ordering is a well-ordering. For instance, the rational numbers are isomorphic to the natural numbers, and that isomorphism can be used to induce a well-ordering on the rational numbers, but the normal order on rational numbers is not a well-ordering.
Acccumulation
CC BY-SA 4.0
2023-12-27T23:32:37.977Z
My point is that there should be a farthest point, because if there weren't then for any given point you could find a farther point, but that should imply infinite extent.
Karl Knechtel
CC BY-SA 4.0
2023-12-28T18:30:29.993Z
How could that ever work, please?

What meaning could the Question have, until you first explained the obviously vital  meaning of 'antipodes' here?
Robbie Goodwin
CC BY-SA 4.0
2023-12-28T18:39:08.100Z
I assume OP means if *in principle* you could travel in a "straight line" near-infinitely fast, would you after some time T end up back at your starting position? And if so, does that imply that at time T/2 you were at the "antipodes" of Earth?
FumbleFingers
<universe>
If the universe has no true center, could it have antipodes?
J-L
# If the universe has no true center, could it have antipodes?

(Asked by: username_0 on 2023-12-27)

Some time ago, I wondered where the center of the universe could be, and the answers I found were pretty much in agreement that there was no center of the universe, as paradoxical as that might seem.

So I started wondering, just as the surface of the Earth has no true center, yet it has antipodes, could the Universe itself -- having no true center -- also have antipodes?

(The term "antipodes" usually refers to two points on the surface of a sphere that are opposite each other, and are therefore mutually the farthest points from each other.)

I know that at some time in the past, when educated people started discussing the Earth as a sphere, no longer did they talk about a specific point on the Earth's surface as its true center, but instead pondered the mysteries and consequences of its antipodes, such as "What's on the other side of here? Is it reachable? And could it be populated by people?"

Following the same logic, now that I've learned that the universe has no true center, I've started wondering if it makes sense that the universe has antipodes. For example, could there conceivably be a point (or area) of the universe that is the farthest away from Earth, our solar system, or the Milky Way galaxy?

**Tags:** universe
> **Comments:**
> **username_1**: Strictly speaking, “antipodes” means the point opposite your feet, so the point diametrically opposite to where you are on the Earth. So, of course, the Earth has antipodes, as you don’t need a specific centre point for it to have antipodes. Now, for the Universe, the question is pointless if using “antipodes” in that sense, as there is no “other side” to it…  
> **username_2**: Doesn't the existence of "a point that is the farthest away from Earth" naturally follow from the universe being finite plus the well-ordering principle (applied to the function "distance from Earth to each point")?  
> **username_3**: @PierrePaquette The term "antipode" can be defined in the context of $S^2$ as a two-dimensional manifold without any reference to the embedding space.  
> **username_3**: @KarlKnechtel The well-ordering theorem says that for every set, we could come up with an ordering is a well-ordering. It says nothing about whether the *existing* ordering is a well-ordering. For instance, the rational numbers are isomorphic to the natural numbers, and that isomorphism can be used to induce a well-ordering on the rational numbers, but the normal order on rational numbers is not a well-ordering.  
> **username_2**: My point is that there should be a farthest point, because if there weren't then for any given point you could find a farther point, but that should imply infinite extent.  
> **username_4**: How could that ever work, please?  What meaning could the Question have, until you first explained the obviously vital  meaning of 'antipodes' here?  
> **username_5**: I assume OP means if *in principle* you could travel in a "straight line" near-infinitely fast, would you after some time T end up back at your starting position? And if so, does that imply that at time T/2 you were at the "antipodes" of Earth?  

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## 2 answers

### Answer by username_6 on 2023-12-27. Score: 26 | Accepted answer

Yes, it's conceivable. If the universe is spatially hyperspherical, then there would be a most distant location from every location in the universe, much like there is on the surface of a sphere.

There is no evidence that the universe is hyperspherical, but it's also not ruled out. What is ruled out is a hyperspherical universe small enough that we can see all the way around it, so if there is a most distant galaxy from the Milky Way, it's probably invisible to us.

> **Comments:**
> **username_7**: For fun, [most remote land on Earth](https://en.wikipedia.org/wiki/Extremes_on_Earth#Other_places_considered_the_most_remote)  

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### Answer by username_3 on 2023-12-27. Score: 2

Given a point $p\_0$ and a metric $d$, we can define the antipode $p\_1$ as the argmax of $d(p\_0,p)$ as $p$ varies across all points. There are things that could happen that would make this definition go wrong:

1. There is no upper bound on $d(p\_0,p)$. Note that even if the *volume* of the universe is finite, it's theoretically possible that the *diameter* is infinite.
2. The upper bound is not achieved. There is some distance $r$ such that given any $\epsilon$, there is a point such that its distance from $p\_0$ is greater than $r-\epsilon$, but there is no point whose distance from $p\_0$ is exactly $r$. This suggests that there is some point "missing" from the space.
3. The point is not unique. That is, there is some $r$ such that there are multiple points that are distance $r$ and no point has a distance greater than $r$.

There is the further question of whether we're using some three-dimensional distance, in which case we have to fix some coordinate system, or we're using proper distance.


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