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.

posted over 5 years ago

Iter‭

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