# What 4D shape could best fit a looping world?

I have an idea for a world such that the world itself is like a large loop. The best way I can think to describe it is going step by step through the "loop".

Imagine a planet-like object; a sphere akin to that of Earth. If you dig to the center, the center is actually like an opening to a "fork", such that you can take one of two paths (not including backtracking to go to the surface). Each path goes in the opposite direction of the other, and goes "outward" to then meet together in the sky. From the sky you can fall back down to "Earth".

I have two ways of imagining it currently:

1) is like a torus, except with an extra path through the center. The center of the middle path of the modified torus is the "planet", or ground, or surface. Downward from the center path is the "core" or center of the planet, which branches out to the two paths that can be taken "up" to the "atmosphere" or sky, which is the top of the center path.

2) is like the "planet" is a sphere, and the sky (atmosphere) is another, larger sphere that encompasses the planet. The sky then connects to the center of the planet through one of two "tunnels", neither of which actually pass through the sky or the planet (like how a klein bottle doesn't actually pass through itself).

My first thought is that this could be done with a 4 dimensional world, but that begs the question: would a 3 dimensional creature be able to traverse the world, even through the parts that are "breaking" 3 dimensional rules?

After seeing this video https://youtu.be/kEB11PQ9Eo8, I thought maybe I could use non-Euclidean geometry to accomplish what I'm trying to [EDIT: However, comments have lead me to believe that a 4D shape would be the simpler and more understandable approach]. Unfortunately, I'm not an expert on abstract mathematical geometry, so I was hoping someone could share their insight.

In short: what 4D shape would best fit the world I'm trying to make. (Best meaning "most efficient", or the simplest geometry that meets the requirements). Thank you in advance.

• What does the word "better" mean? Jan 16, 2019 at 2:14
• @Iter What would constitute better? I find your question completley incomprehensible - Through the loop, two paths, fall to "Earth", torus, extra path through centre, centre of the modified middle path... I'm sorry but this is just incomprehensible gibberish. Please figure out what it is you're asking and edit your question. Jan 16, 2019 at 2:16
• More fitting to accomplish the goal desired with the mkst "efficient" solution. So, optimally, being able to make the desired world with, say, a single uniform or regular 4D shape, or rather non-Euclidean geometry that requires the fewest "exceptions" as possible.
– Iter
Jan 16, 2019 at 2:17
• Food for thought: 4D geometry is substantially easier to describe compared to non-Euclidean geometries. If you can do what you want in 4d, you'll find the result is more approachable from a reader's perspective. Jan 16, 2019 at 2:30
• Perhaps I'm taking the wrong tack here, but it's important to note that from an empirical perspective all we can ever say is that our observations of a real object conform to a specific mathematical model, including geometry. Until the 19th century, most non-Euclidean forms of geometry were largely considered a form of heresy and that they had no place in physics. If you're looking for a model that helps the reader understand your structure, Cort is right. If you're looking for one that defines your structure, NE geometry will be too exotic for that I fear. Jan 16, 2019 at 2:54

## 3 Answers

Maybe something like this?

1. Start out at the Red Dot and go upwards with outward-orientation. (Here you're on the "outer surface".)
2. Cross the Horizon and go downwards with inward-orientation. (Here you're on an "inner surface".)
3. Do a moebian flip. (You're still kind of on an inward surface, only flipped.)
4. Go downwards some more with an outward-orientation. (Here you're on the "outer surface" whilst still being on the "inner surface".)
5. Cross the Antihorizon and go outwards with a downward-orientation. (Now in the "air", you're still on an "outer surface" that's an "inner surface".
6. Do a moebian flip. (You're still kind of on an outer surface, only flipped.)
7. Go inwards with downward-orientation.
8. Do a moebian flip.
9. Go upwards with outward-orientation. (You're back on the "surface" again, only not where you started out!

The wiggly bits are the outward-oriented surface and the smoothe bits are the inward-oriented surface.

This is, I think, only sort of half the planet. On account of, if you start out on the downward side (that little cone of wiggly) you'd end up on the inverse side of the upwards-going funnel, which would have its own non-intersecting egresses at the upwards pole of the planet. I only showed the egresses at the downwards pole, accessed from the obverse side of the downwards-going funnel.

Hope that makes some sense!

• Is the Moebian Flip anything like the monster mash? Because that would catch my interest. Jan 16, 2019 at 14:41
• I'm sure it'll catch on in a flash! Jan 16, 2019 at 16:35

I actually only understand 3D geometry, maybe you could use a state in between a normal sphere and an eversed sphere (with maybe some holes), with atmosphere inside and outside ?

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and here is a video of a sphere eversion: https://youtu.be/x7d13SgqUXg?t=497

So, first of all, the center of the world would be a large topological worm hole. What is on the other end of this wormhole? A space that is topologically equivalent to [0,1] x S^2 (the Cartesian product of an interval and a sphere). The spheres at 0 and 1 are topological wormholes that lead to some pair of points above the world.

If this seems hard to visualize, that's because it is. Luckily, there is a nice 2D analogy. Imagine a plane with a disk on it. The disk is the world. Now we do some surgery! First, we cut a hole in the middle of disk, and put a cylinder in it. This is our first worm hole. This cylinder is tube like. It splits into two smaller tubes (like this). Finally, the two end of this tube connect back to the plane somewhere in the atmosphere (we will need to cut two holes so we can make the connection). Now, even though this structure exists in 3D, we can embed 2D creatures in it, who will perceive it as an extremely weird but 2D space. When they go towards the center, they will go along the surface of the tube to the fork, go along the surface of one of the branches, and then pop out somewhere in the atmosphere. Just note that gravity may work in intuitive ways (the tube branches will strongly attract things into them).

Here is what a nest of 2D topological wormholes would look like. For your case, you just need 1 forked wormhole, but in 3D.