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

[PeterDonis]

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.

 

[PeterDonis]

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.

 

[greswd]

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

 

[PeterDonis]

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.

 

[greswd]

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

​

 

[greswd]

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.