The study in question: [https://arxiv.org/abs/2106.13205](https://arxiv.org/abs/2106.13205)
The much friendlier Vice Motherboard article I actually read: [https://www.vice.com/en/article/3aqjkn/the-universe-is-a-giant-donut-that-we-live-inside-new-research-suggests](https://www.vice.com/en/article/3aqjkn/the-universe-is-a-giant-donut-that-we-live-inside-new-research-suggests)
The article suggests that scientists currently model the universe as a kind of infinite flat plane. First of all, I don’t even understand how our 3-D world is being understood as a plane. My main question though, is if they’re thinking of it as a plane that wraps around and touches itself, then why is a torus/donut a better model than simply a sphere?
In: Physics
> First of all, I don’t even understand how our 3-D world is being understood as a plane.
It’s not a plane. It refers to the geometry of the universe. A flat universe is the boring universe you always thought you lived in: one where parallel lines never intersect and the angles of a triangle sum to 180 degrees. Scientists have done precise measurements and determined our universe is very very flat.
But that’s not the only possible geometry. We could imagine living in a universe with curvature like a sphere, or a saddle where parallel lines intersect or diverge, and the angles of a triangle sum to something other than 180 degrees.
> My main question though, is if they’re thinking of it as a plane that wraps around and touches itself, then why is a torus/donut a better model than simply a sphere?
When the article talks about a donut shaped universe, they are referring to the topology. How things are connected together. The geometry of a donut is still flat. You can have parallel lines will not intersect. The geometry of a sphere is not flat (think about lines of longitude on a globe).
Having the topology of a torus means that some dimension of the universe might wrap back onto itself. E.g. you looked far enough in one direction, you would see the back of your own head.
As far as we can tell, space is flat. That rules out a lot of possibilities. The sphere will always wrap around itself.
Space is 3 dimensional, so it’s not even a “plane”. The “donut” here is actually a 3-dimensional version of a flat donut, which is already difficult to visualize for human.
For clarification, the donut refer to the torus, which is normally described as the surface of the donut that people know. It’s 2-dimensional because it’s a surface (we don’t care that it lies inside a 3-dimensional space).
However, the picture we normally see of it is a curved surface, because it looks obviously bent. What you need to think of is a flat torus, which means you flatten it up, without cutting it somehow. This is obviously impossible to do to an actual donut, which is why you need to visualize it as a Final Fantasy world map, which is a rectangle that teleport to the other side when you reach the end.
But since space is 3D, you need to visualize this as a box that teleport when you reach the edge.
It helps to think of things in terms of what would happen if you turned the surface of a donut or a sphere into a flat map. Specifically, let’s look at what happens when something goes off of the edge of the map.
You’re already familiar with how a flat map of a sphere works, because that’s what world maps are. Let’s use [this map](https://upload.wikimedia.org/wikipedia/commons/8/83/Equirectangular_projection_SW.jpg) as an example.
If we go off the western edge of that map, we would come back around on the eastern edge, since we’re actually on a sphere, and the eastern and western edges of the map are actually connected. But, it doesn’t work the same going north or south. If we go off of the northern edge, we would come back around on the northern edge but in the opposite hemisphere. This is because the northern edge all comes together at the north pole, so its connected to itself and not to the southern edge. If you go far enough north you get to the north pole and if you keep going straight you’ll start going south from the north pole.
Now let’s imagine for a moment that we lived on a doughnut shaped Earth. [Here is an animation](https://i.imgur.com/7QVgJWi.gif) showing how a flat map of a doughnut’s surface works. So what does going off of the edges of [this map](https://i.imgur.com/OLhCTrl.gif) look like? East and west are still the same – go off of one and wind up on the other. However, the north and south edges work different for a torus. Instead of being each being connected to itself at a pole, now the north and south edges are connected together! That means that if you go off of the northern edge, you come up on the southern edge, just like going east and west.
So far I’ve been talking about a flat 2D map of 3D torii and spheres, because it’s pretty easy to understand. We’re used to living on a sphere and playing video games that take place on doughnuts. Now let’s talk about a 3D projection of 4D shapes.
If the universe was completely flat, then that would mean if we started travelling in a direction we would travel forever and never come back to where we started (as long as we don’t change direction on our own). If the universe is toroidal, that would mean if we travelled far enough in one direction, we would eventually find ourselves back where we started; it would be like if we lived in a cube where going through the top would cause you to come up from the bottom, same as with the 2D maps. Travel far enough and eventually you’ll find yourself behind where you started. Similarly, if the universe is spherical, we could also travel in a straight line and find ourselves back where we started, but depending on the direction we chose, we might find our selves going past our starting point going in the opposite direction, like going over the poles on a map of Earth.
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