# Possible illogicalness of a 3-sphere shape of the Universe's space?

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PeterDonis
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And that's why we apply non-Euclidean geometry - to provide a mathematical description of something we cannot visualise.
Though I was trying to express how a flatlander's perception, or "reality", is still very much within a 2-D planar framework.

I don't see what you are trying to achieve. Why not just use a Mercator projection, or stereographic projection, or whatever?
I'm trying to expand the map of space outwards from our flatlander's location. From his POV, which is limited to two dimensions of space.

For a stereographic projection, locations further away would have their distances stretched greatly, until the opposite pole is at infinity, which does not conform to the actual distance.

And as our flatlander can also perceive that he is moving in one direction only in a straight line while he is actually looping around, the map of space could go on infinitely, but with repetitions.

Ibix
Note, however, that the issue with a black hole spacetime is different from the issue with a 3-sphere. In a black hole spacetime, you can fit more cubes radially than you would expect based on tangential distances, whereas in a 3-sphere you can fit fewer.
Wait - do you have that the right way round? On a sphere you should see more cubes radially than naive analysis of the circumference would suggest, surely?

Ibix
For a stereographic projection, locations further away would have their distances stretched greatly, until the opposite pole is at infinity, which does not conform to the actual distance.
You can't draw a sphere on a plane without stretching or tearing it. It's not possible. You could certainly rescale a stereographic projection so that the pole is at finite distance, but each circle of constant latitude must be larger than its predecessor, which isn't the case on the sphere.

You can't draw a sphere on a plane without stretching or tearing it. It's not possible. You could certainly rescale a stereographic projection so that the pole is at finite distance, but each circle of constant latitude must be larger than its predecessor, which isn't the case on the sphere.
so from #20, I think the map of space should continue outwards indefinitely, but as mentioned, unclear how to proceed after inadvertently reproducing Schlegel diagrams.

PeterDonis
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On a sphere you should see more cubes radially than naive analysis of the circumference would suggest, surely?
Yes, you're right, I misstated the difference between the black hole spacetime (more precisely, a single spacelike slice of constant Schwarzschild coordinate time in a black hole spacetime) and a 3-sphere. Giving the math will make it clearer: the 3-sphere metric, in "Schwarzschild-like" coordinates where ##r## is the areal radius and we have picked any point we like as the origin, is:

$$ds^2 = \frac{1}{1 - k r^2} dr^2 + r^2 d\Omega^2$$

where ##k## is a constant related to the "Radius of curvature" of the 3-sphere.

The metric of the Flamm paraboloid, which is the geometry of a constant Schwarzschild time spacelike slice of a black hole spacetime, is:

$$ds^2 = \frac{1}{1 - \frac{2M}{r}} dr^2 + r^2 d\Omega^2$$

where ##M## is the mass of the hole.

Both of these metrics have the coefficient of ##dr^2## larger than 1, so you are right that they both allow you to fit more cubes radially than you would expect based on tangential distances. But the dependence on ##r## of how many more cubes you can fit radially is very different.

Ibix
PeterDonis
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from #20, I think the map of space should continue outwards indefinitely
Obviously it can't, since the area of a 2-sphere is finite. Similarly, it is impossible to continue any "cube" construction outward indefinitely in a 3-sphere space, since the volume of the space is finite.

Obviously it can't, since the area of a 2-sphere is finite. Similarly, it is impossible to continue any "cube" construction outward indefinitely in a 3-sphere space, since the volume of the space is finite.
yup, a finite play space, though in #20 I mentioned infinite looping:
A flatlander has no conception of the 3rd dimension of space, so to him the 2-sphere is a flat plane in which he can move about infinitely in any direction, though he might end up looping around back to where he started.
and in #27 to Ibix repetitions
And as our flatlander can also perceive that he is moving in one direction only in a straight line while he is actually looping around, the map of space could go on infinitely, but with repetitions.
with the infinite map representing repeated loops

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PeterDonis
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with the infinite map representing repeated loops
As long as you allow multiple squares or cubes to occupy the same space, which I suppose is OK if they're just abstract, but doesn't work if they are actual objects.

As long as you allow multiple squares or cubes to occupy the same space, which I suppose is OK if they're just abstract, but doesn't work if they are actual objects.
I was thinking in terms of something similar to Pac-Man on a torus.

The infinite map would look like this:

And the infinite map includes Pac-Man himself repeated infinitely, as its like the 2007 physicsy video game Portal in which you can look at your own back:

There's also a video, though it shows the environment moving around the torus instead of Pac-Man

so from #20, I think the map of space should continue outwards indefinitely, but as mentioned, unclear how to proceed after inadvertently reproducing Schlegel diagrams.
@Ibix as I'm unclear on how to proceed, I'm looking at another example which has a nice infinite map, that of the torus.
above I've posted the infinite map for Pac-Man on a torus. Its a modified Pac-Man, while the 1979 game Asteroids is originally toroidal.