- #26

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Interesting!@robphy linked these models the other day

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- #26

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Interesting!@robphy linked these models the other day

- #27

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

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.I don't see what you are trying to achieve. Why not just use a Mercator projection, or stereographic projection, or whatever?

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.

- #28

Ibix

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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?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 fitmorecubes radially than you would expect based on tangential distances, whereas in a 3-sphere you can fit fewer.

- #29

Ibix

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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.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.

- #30

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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.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.

- #31

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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:On a sphere you should see more cubes radially than naive analysis of the circumference would suggest, surely?

$$

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

- #32

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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.from #20, I think the map of space should continue outwards indefinitely

- #33

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yup, a finite play space, though in #20 I mentioned infinite looping: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.

and in #27 to Ibix repetitionsA 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.

with the infinite map representing repeated loopsAnd 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.

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- #34

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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.with the infinite map representing repeated loops

- #35

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I was thinking in terms of something similar to Pac-Man on a torus.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.

- #36

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@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.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.

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.

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