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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?

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    $\begingroup$ 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… $\endgroup$ Dec 27, 2023 at 2:02
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    $\begingroup$ 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")? $\endgroup$ Dec 27, 2023 at 14:31
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    $\begingroup$ @KarlKnechtel Not quite. The well ordering principle states that every non-empty set of natural numbers has a minimum. The well ordering theorem states that every set can be ordered (this is equivalent to Zorn's lemma and the Axiom of Choice), but that isn't really relevant here, either. The existence of a "farthest point" would require that the universe be finite, plus some continuity assumption on the distance function (to which we can apply the extreme value theorem). $\endgroup$
    – user54348
    Dec 27, 2023 at 14:46
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    $\begingroup$ @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. $\endgroup$ Dec 27, 2023 at 20:45
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    $\begingroup$ 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? $\endgroup$ Dec 28, 2023 at 18:39

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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.

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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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