## Table of contents:

- Flat universe
- Could our universe be one of the flat forms?
- Spherical shape
- What if our universe is spherical?
- Hyperbolic geometry
- Well, is our Universe hyperbolic?

In our view, the universe is infinite. Today we know that the Earth has the shape of a sphere, but we rarely think about the shape of the Universe. In geometry, there are many three-dimensional shapes as an alternative to the "familiar" infinite space. The authors explain the difference in the most accessible form.

Looking at the night sky, it seems that space goes on forever in all directions. This is how we imagine the Universe - but not the fact that it is true. After all, there was a time when everyone thought that the Earth was flat: the curvature of the earth's surface is imperceptible, and the idea that the Earth is round seemed incomprehensible.

Today we know that the Earth is in 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.

Two questions can be asked about the shape of the universe - separate but interrelated ones. One is about geometry - meticulous calculations of angles and areas. Another is about topology: how separate 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 every direction. Only three geometric shapes correspond to these characteristics - flat, spherical and hyperbolic. Let's take a look at these shapes in turn, some topological considerations and conclusions based on cosmological data.

## Flat universe

In fact, this is school geometry. The angles of a triangle add up to 180 degrees, and the area of a circle is πr2. The simplest example of a flat three-dimensional shape is an ordinary infinite space, mathematicians call it Euclidean, but there are other flat options.

It is not easy to imagine these shapes, but we can connect our 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. Let's say we cut out a rectangular piece of paper and tape the opposite edges of it with tape. If you glue the top edge to the bottom edge, you get a cylinder.

You can also glue the right edge to the left - then we get a donut (mathematicians call this shape a torus).

You will probably object: "Something is not very flat." And you will be right. We were cheating a little about the flat torus. If you really try to make a torus out of a piece of paper in this way, you will run into some difficulties. It is easy to make a cylinder, but it will not work to glue its ends: the paper will crumple along the inner circle of the torus, but it will not be enough for the outer circle. So you have to take some kind of elastic material. But stretching changes the length and angles, and therefore the entire geometry.

It is impossible to construct a real smooth physical torus from a flat material inside an ordinary three-dimensional space without distorting the geometry. It remains to speculate abstractly 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 sheet of paper, all the geometric facts we are used to remain the same - at least on a limited scale: the angles of a triangle add up to 180 degrees, and so on. But with the change in global topology through trimming and gluing, life will change dramatically.

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

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

It's as if, on the original piece of paper, light passed through you, went to the left edge, and then reappeared on the right, like in a video game.

Here's another way to think about it: you (or a ray of light) cross one of the four edges and find yourself in a new room, but in fact it is the same room, only from a different point of view. Wandering through such a universe, you will come across an endless number of copies of the original room.

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

On the torus, each of them corresponds to one or another loop, along 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 other box. We will not be able to depict this space inside an ordinary infinite space - it simply will not fit - but we will be able to abstractly speculate about life inside it.

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

The three-dimensional torus is only one of ten variants of the finite flat world. There are also 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".

## Could our universe be one of the flat forms?

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

Let's say you even saw your own copy, this distant image only shows how you (or your galaxy as a whole) looked in the distant past, since the light has come a long way until it reached you. Maybe we even see our own copies - but changed beyond recognition. Moreover, different copies are at different distances from you, so they are not alike. And besides, so far away that we still won't see anything.

To get around these difficulties, astronomers usually look not for copies of themselves, but for repeating features in the most distant visible phenomenon - the cosmic microwave background radiation, this is a relic of the Big Bang. In practice, this means looking for pairs of circles with matching patterns of hot and cold spots - it is assumed that they are the same, only from different sides.

Astronomers conducted just such a search in 2015 thanks to the Planck Space Telescope. They put together data on the types of coincident circles that we expect to see inside a flat 3D torus or other flat 3D shape - a so-called plate - but they found nothing. This means that if we do live in a torus, then it appears to be so large that any repeating fragments lie outside the observable universe.

## Spherical shape

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

Drawing a three-dimensional sphere is difficult, but it is easy to describe it with a simple analogy. If a two-dimensional sphere is a collection of all points at a fixed distance from some central point in ordinary three-dimensional space, a three-dimensional sphere (or "trisphere") is a collection of all points at a fixed distance from some central point in four-dimensional space.

Life inside a trisphere is very different from life in flat space. To visualize it, imagine that you are a two-dimensional being in a two-dimensional sphere. The 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 travels by the shortest path: in large circles. But these circles seem straight to you.

Now imagine that you and your 2D buddy are hanging out 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 quickly as we are used to. This is because as your visual circle grows, your friend takes up less and less of it.

But as soon as your friend crosses the equator, something strange happens: he begins to increase in size, although in fact he continues to move away. This is because the percentage they occupy in your visual circle is increasing.

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

Having reached the South Pole, it will completely fill your entire visible horizon.

And when there is no one at the South Pole, your visual horizon will be even stranger - it's you. This is because the light you emit will spread throughout the sphere until it comes back.

This directly affects life in the 3D realm. 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 in the background - as if our appearance was superimposed on a balloon, then turned inside out and inflated to the entire horizon.

But even though the trisphere is the foundational model for spherical geometry, it is far from the only possible space. As we built different flat models by cutting and gluing pieces of Euclidean space, so we can build spherical ones by gluing suitable pieces of trisphere. Each of these glued shapes will, like the torus, have the effect of a "room of laughter", only the number of rooms in spherical shapes will be finite.

## What if our universe is spherical?

Even the most narcissistic of us do not see ourselves as the background instead of the night sky. But, as in the case of a 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 larger than the limits of the visible world, and the background is simply not visible.

But unlike a torus, a 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 straight lines in spherical geometry are large circles, the triangles there are "plump" than the Euclidean ones, and the sum of their angles exceeds 180 degrees.

Basically, measuring 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, forming the three sides of the triangle, are known. We can measure the angle formed by the spot in the night sky - and this will be one of the corners of the triangle. We can then check if the combination of the side lengths and the sum of the angles corresponds to planar, spherical, or hyperbolic geometry (where the sum of the angles of the triangle is less than 180 degrees).

Most of these calculations, along with other measurements of curvature, assume that the universe is either completely flat or very close to it. One research team recently suggested that some of the 2018 data from the Planck Space Telescope speaks more in favor of a spherical universe, although other researchers argued that the evidence presented could be attributed to statistical error.

## Hyperbolic geometry

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

From our point of view, the triangles near the boundary circle seem to be much smaller than those near the center, but from the point of view of hyperbolic geometry, all triangles are the same. If we tried to portray these triangles 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 get closer to 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 the radius. A pile of triangles is formed near the boundary of the hyperbolic disk

Because of this feature, mathematicians like to say that it is easy to get lost in hyperbolic space. If your friend moves away from you in normal Euclidean space, he will begin to move away, but rather slowly, because your visual circle is not growing so quickly. In hyperbolic space, your visual circle expands exponentially, so your friend will soon shrink to an infinitely small speck. So, if you have not followed 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 - for example, the sum of the angles of some triangles from the Poincaré disk mosaic is only 165 degrees.

Their sides appear to be indirect, but that's because we're looking at hyperbolic geometry through a distorting lens. For an inhabitant of the Poincaré disc, these curves are actually straight lines, so the fastest way to get from point A to point B (both at the edge) is through a cut to the center.

There is a natural way to make a three-dimensional analogue of the Poincaré disk - take a three-dimensional ball and fill it with three-dimensional shapes, which gradually decrease as they approach the boundary sphere, like triangles on a Poincaré 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. Indeed, as we have already noted, most of the cosmological measurements lean towards 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 was flat - the curvature of the Earth is not visible to the naked eye. The larger the spherical or hyperbolic shape, the flatter each of its parts, therefore, if our Universe has an extremely large spherical or hyperbolic shape, its visible part is so close to flat that its curvature can only be detected with ultra-precise instruments, and we have not yet invented them. …