Wormhole - How to model it?

Question

I am trying to model wormhole between two points in 3D space, but do not know/understand how to do so.

A concrete example:

1. Think of something like a game where we have a 3D world with a size of 100, i.e., coordinates ranging from $\Big(\begin{smallmatrix} 0\\0\\0 \end{smallmatrix}\Big)$ to $\Big(\begin{smallmatrix} 99\\99\\99\end{smallmatrix}\Big)$
2. We now want to have a worm hole connecting the point $p_1 := \Big(\begin{smallmatrix} 11\\22\\33\end{smallmatrix}\Big)$ with the point $p_2 :=\Big(\begin{smallmatrix} 22\\44\\66\end{smallmatrix}\Big)$.
3. If we walk in a straight line that crosses $p_1$ we would end up "entering" $p_1$ from one side and "exiting" on the other side of $p_2$.

As far as I am understanding the topic wormholes can be modeled in 4-dimensional space. My question is: how?

So, I am searching a function $\mathbb{R}^4 \mapsto \mathbb{R}^3$ that projects the 4D space in such a way to the 3D space that the described worm hole exists.

And a reversed function that allows me to move within the 4D space by using 3D "controls" $(i.e., \Big(\begin{smallmatrix} x\\y\\z\end{smallmatrix}\Big)$ directional vector + moving distance)

This website here may have the solution for the problem, but I do not know how to get from their formulas to what I am searching. At least they have a rendering of what I try to achieve.

Answer 1

Topologically, you are just defining a quotient space: $\mathbb{R}^3/(p_1 \sim p_2)$.

However, I think you want a little more than that. You want that a traveler going towards $p_1$ with direction $\vec{v}$ comes out of $p_2$ with direction vector $\vec{v}$.

If you are just trying to get a computer program to work with this space, then the answer is fairly simple. Just set a threshold that if you are are close enough to $p_1$ you pop out of $p_2$ with same direction and if you are close enough to $p_2$ you pop out of $p_1$ in the direction direction you are going.

Since you are allowing a discrete domain, I am going to use a discrete parameter to describe this system. With that in mind, I think this is the function that you want:

$$f(t+1) = \{x(t) + v(t) \mbox{ if } x(t) + v(t) \ne p_1,p_2,\\ p_1, \mbox{ if } x(t) + v(t) = p_2,\\ p_2, \mbox{ if } x(t) + v(t)=p_1 \}$$

where $x(t)$ is the position at time $t$ and $v(t)$ is the direction vector at time $t.$