US, WASHINGTON (ORDO NEWS) — Looking at the night sky, it seems that space continues forever in all directions. So we imagine the universe and imagine – but not the fact that is true.

In the end, there was a time when everyone thought the Earth was flat: the curvature of the Earth’s surface was invisible, and the idea that the Earth was round seemed incomprehensible.

Today we know that the Earth has the shape of a sphere. But we rarely think about the shape of the universe. As the sphere replaced the flat Earth, other three-dimensional forms offer alternatives to the “familiar” infinite space.

As for the shape of the Universe, two questions can be asked – separate, but interconnected. One about geometry – meticulous calculations of angles and area. The other is about topology: how the individual parts merge into a single form.

Cosmological data suggest that the visible part of the Universe is smooth and homogeneous. The local structure of space looks almost the same at every point and in all directions. Only three geometric shapes correspond to these characteristics – flat, spherical and hyperbolic. Let’s take a look at these forms, some topological considerations and conclusions based on cosmological data.

Flat universe

In essence, this is school geometry. The sum of the angles of the triangle is 180 degrees, and the area of the circle is πr2. The simplest example of a planar three-dimensional shape is ordinary infinite space, its mathematicians call it Euclidean, but there are other planar variants.

It is not easy to imagine these forms, but we can connect intuition by thinking in two dimensions instead of three. In addition to the usual Euclidean plane, we can create other flat shapes by cutting out a piece of the plane and gluing its edges. Suppose we cut a rectangular sheet of paper and glued its opposite edges with tape. If you glue the top edge with the bottom, you get a cylinder.

You can also glue the right edge with the left – then we get a bagel (mathematicians call this form a torus).

You probably object: “Something doesn’t turn out very flat.” And you will be right. We deceived a little about the flat torus. If you really try to make a torus out of a sheet of paper in this way, you will encounter some difficulties. It is easy to make a cylinder, but it will not work to glue its ends: the paper will collapse along the inner circle of the torus, but it will not be enough on the outer circle. So you have to take some elastic material. But stretching changes the length and angles, and hence the whole geometry.

It is impossible to build a real smooth physical torus from a flat material inside ordinary three-dimensional space without distorting the geometry. It remains to abstractly speculate about what it is like to live inside a flat torus.

Imagine that you are a two-dimensional being whose universe is a flat torus. Since the shape of this universe is based on a flat piece of paper, all the geometric facts that we are used to remain the same – at least on a limited scale: the sum of the angles of the triangle is 180 degrees and so on. But due to changes in the global topology due to cropping and gluing, life will change dramatically.

To begin with, there are straight lines in the torus that loop and return to the starting point.

On the distorted torus, they look curved, but they seem straight to the inhabitants of the flat torus. And since the light travels in a straight line, then if you look directly in any direction, you will see yourself from behind.

It is as if on the original sheet of paper the light passed through you, reached the left edge, and then appeared again on the right, as if in a video game.

Here is another way to imagine it: you (or a ray of light) cross one of the four edges and appear as if in a new room, but in reality it is the same room, only from a different point of view. Wandering through such a universe, you will find an infinite number of copies of the original room.

This means that you will take away an infinite number of your own copies, wherever you look. This is a kind of mirror effect, only these copies are not quite reflections.

On the torus, each of them corresponds to a particular turn in which the light returns back to you.

In the same way, we get a flat three-dimensional torus by gluing the opposite faces of a cube or another box. We will not be able to depict this space inside an ordinary infinite space – it simply will not fit in – but we can abstractly speculate about life inside it.

If life in a two-dimensional torus is similar to an infinite two-dimensional array of identical rectangular rooms, then life in a three-dimensional torus is similar to an infinite three-dimensional array of identical cubic rooms. You will also see an infinite number of your own copies.

Three-dimensional torus is only one of ten options for a finite flat world. There are infinite flat worlds – for example, a three-dimensional analogue of an infinite cylinder. Each of these worlds will have its own “room of laughter” with “reflections.”

Can our universe be one of the flat forms?

When we look into space, we do not see an infinite number of our own copies. Despite this, eliminating flat shapes is not easy. Firstly, they all have the same local geometry as the Euclidean space, therefore, it will not be possible to distinguish them by local measurements.

Suppose you even saw your own copy, this distant image only shows how you (or your galaxy as a whole) looked in the distant past, because the light has come a long way until it reaches you. Maybe we even see our own copies, but modified beyond recognition. Moreover, different copies are at different distances from you, so they are not alike. And besides, so far that we still will not see anything.

To get around these difficulties, astronomers usually do not look for their copies, but repeated features in the farthest of visible phenomena – cosmic microwave background radiation, this is a relic of the Big Bang. In practice, this means finding pairs of circles with matching patterns of hot and cold spots – it is assumed that this is the same thing, but from different sides.

Just such a search, astronomers conducted in 2015 thanks to the Planck space telescope. They brought together data on the types of matching circles that we expect to see inside a flat three-dimensional torus or another flat three-dimensional shape, the so-called plate, but found nothing. This means that if we really live in a torus, then it is apparently so large that any repeating fragments lie outside the observable Universe.

Spherical shape

We are well acquainted with two-dimensional spheres – this is the surface of a ball, orange or Earth. But what if our universe is a three-dimensional sphere?

It is difficult to depict a three-dimensional sphere, but it is easy to describe it using a simple analogy. If a two-dimensional sphere is a collection of all points at a fixed distance from a certain central point in ordinary three-dimensional space, a three-dimensional sphere (or “trisphere”) is a collection of all points at a fixed distance from a certain central point in four-dimensional space.

Life inside the trisphere is very different from life in a flat space. To imagine her, imagine that you are a two-dimensional being in a two-dimensional sphere. A two-dimensional sphere is the entire Universe, therefore you cannot see the three-dimensional space surrounding you and cannot get into it. In this spherical Universe, light moves in the shortest way: in large circles. But these circles seem straight to you.

Now imagine that you are hanging with a two-dimensional friend at the North Pole, and he went for a walk. Moving away, at first it will gradually decrease in your visual circle – as in the ordinary world, albeit not as fast as we are used to. This is because as your visual circle grows, your friend takes up a smaller percentage.

But as soon as your friend passes the equator, something strange will happen: he will begin to increase in size, although in fact he continues to move away. This is because the percentage that it occupies in your visual circle is growing.

Three meters from the South Pole, your friend will look like he is standing three meters from you.

When he reaches the South Pole, he will completely fill your entire visible horizon.

And when there is nobody at the South Pole, your visual horizon will be even stranger – you yourself. This is because the light emitted by you will spread throughout the sphere until it comes back.

This directly affects life in the three-dimensional sphere. Each point of the trisphere has an opposite, and if there is an object there, we will see it in the whole sky. If there is nothing there, we will see ourselves as the background – as if our appearance was put on a balloon, then turned inside out and inflated into the entire horizon.

But even though the trisphere is the fundamental model for spherical geometry, this is far from the only possible space. As we built different flat models by carving and gluing pieces of Euclidean space, we can also build spherical ones by gluing suitable pieces of trisphere. Each of these glued forms will, like the torus, have the effect of a “room of laughter”, only the number of rooms in spherical forms will be finite.

What if our universe is spherical?

Even the most narcissistic of us do not see ourselves as a backdrop instead of the night sky. But, as in the case of the flat torus, the fact that we do not see something does not mean at all that it does not exist. The boundaries of a spherical universe can be greater than the limits of the visible world, and the background is simply not visible.

But unlike the torus, the spherical universe can be detected using local measurements. Spherical shapes differ from infinite Euclidean space not only in global topology, but also in small geometry. For example, since the straight lines in spherical geometry are large circles, the triangles there are more “puffy” than the Euclidean, and the sum of their angles exceeds 180 degrees.

In fact, the measurement of cosmic triangles is the main way to check how curved the universe is. For each hot or cold spot on the cosmic microwave background, its diameter and distance from the Earth are known, forming three sides of the triangle. We can measure the angle formed by the spot in the night sky – and this will be one of the angles of the triangle. Then we can check whether the combination of the length of the sides and the sum of the angles of the plane, spherical or hyperbolic geometry (where the sum of the angles of the triangle is less than 180 degrees) matches.

Most of these calculations, along with other curvature measurements, suggest that the Universe is either completely flat or very close to this. One research team recently suggested that some of the data from the Planck Space Telescope for 2018 is more likely in favor of a spherical universe, although other researchers have objected that the evidence presented can be attributed to statistical error.

Hyperbolic geometry

Unlike a sphere that closes on itself, hyperbolic geometry or space with negative curvature opens out. This is the geometry of a wide-brimmed hat, coral reef and saddle. The basic model of hyperbolic geometry is infinite space, like flat Euclidean. But since the hyperbolic shape expands outward much faster than the flat one, there is no way to fit even the two-dimensional hyperbolic plane inside the usual Euclidean space if we do not want to distort its geometry. But there is a distorted image of the hyperbolic plane, known as the Poincare disk.

From our point of view, the triangles near the boundary circle seem much smaller than those at the center, but from the point of view of hyperbolic geometry, all triangles are the same. If we tried to depict these triangles of really the same size – perhaps using elastic material and inflating each triangle in turn, moving from the center outward – our disk would resemble a wide-brimmed hat and would bend more and more. And as you approach the border, this curvature would get out of control.

In ordinary Euclidean geometry, the circumference of a circle is directly proportional to its radius, but in hyperbolic geometry, the circle grows exponentially with respect to radius. A pile of triangles forms near the boundary of the hyperbolic disk

Because of this feature, mathematicians like to say that in a hyperbolic space it is easy to get lost. If your friend moves away from you in the usual Euclidean space, he will begin to move away, but rather slowly, because your visual circle does not grow so fast. In hyperbolic space, your visual circle expands exponentially, so your friend will soon shrink to an infinitely small spot. So, if you did not follow his route, you are unlikely to find him later.

Even in hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees – so, the sum of the angles of some triangles from the mosaic of the Poincare disk is only 165 degrees.

Their sides seem indirect, but this is because we look at hyperbolic geometry through a distorting lens. For the inhabitant of the Poincare disk, these curves are actually straight lines, so the quickest way to get from point A to point B (both on the edge) is through a cut to the center.

There is a natural way to make a three-dimensional analogue of a Poincare disk — take a three-dimensional ball and fill it with three-dimensional forms, which gradually decrease as you approach the boundary sphere, like triangles on a Poincare disk. And, as with planes and spheres, we can create a whole host of other three-dimensional hyperbolic spaces by cutting out suitable pieces of a three-dimensional hyperbolic ball and gluing its faces.

Well, is our universe hyperbolic?

Hyperbolic geometry with its narrow triangles and exponentially growing circles is not at all like the space around us. And indeed, as we have already noted, most cosmological dimensions tend to a flat Universe.

But we cannot rule out that we live in a spherical or hyperbolic world, because small fragments of both worlds look almost flat. For example, the sum of the angles of small triangles in spherical geometry is only slightly more than 180 degrees, and in hyperbolic geometry it is only slightly less.

That is why the ancients thought that the Earth is flat – with the naked eye, the curvature of the Earth is not visible. The larger the spherical or hyperbolic shape, the more flat each part of it is, so if our Universe has an extremely large spherical or hyperbolic shape, the visible part is so close to the plane that its curvature can only be detected with ultra-precise tools, but we have not yet invented them.

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*The article is written and prepared by our foreign editors from different countries around the world – material edited and published by Ordo News staff in our US newsroom press.*