📷 🤶🏽 📈 What is the geometry of the universe? 🧚🏿 👨‍👩‍👦‍👦 🧓

Cloud solutions are good because they allow you to create projects of any complexity, up to a virtual data center. If you try to visualize these structures, you get a sort of mini-universe. Let's play with geometry by trying to visualize different models of our universe.

In our minds, the universe seems infinite. But with the help of geometry, we can consider various three-dimensional shapes that offer an alternative to “ordinary” infinite space.

When you look at the night sky, it seems as if space is expanding in all directions. This is our mental model of the universe, but it is not always true. In the end, there was a time when everyone thought that the Earth was flat, because the bends of our planet were extremely difficult to notice, and they did not even think about the spherical shape of the Earth.

Today we know that the Earth has the shape of a sphere. But few people think about the shape of the universe. Just as a sphere has become an alternative to a flat Earth, other three-dimensional shapes offer an alternative to “ordinary” infinite space.

We can ask two different, but still closely related questions about the shape of the universe. One of them relates to its geometry: fine-grained local measurements of elements such as angles and regions. Another is about topology: how these local parts are sewn together into a common shape.

Cosmological evidence suggests that the part of the universe that we can see is smooth and homogeneous, at least approximately. The local fabric of space looks the same at every point and in all directions. Only three geometric shapes fit this description: flat, spherical and hyperbolic. Let's look at these models, some topological assumptions, and also what cosmological data say about the shapes that best describe our universe.

Flat geometry (planimetry)

This is the geometry we studied at school. The angles of the triangle are 180 degrees, and the area of the circle is πr2. The simplest example of a planar three-dimensional shape is the usual infinite space - what mathematicians call Euclidean space - but there are other flat forms that also need to be taken into account.

These forms are more difficult to visualize, but we can try to fantasize by thinking in two dimensions rather than three. In addition to the usual Euclidean plane, we can create other flat shapes by cutting out part of the plane and holding its edges together. For example, suppose we cut a rectangular sheet of paper and fasten it with opposite edges. Gluing the upper and lower faces gives us a cylinder:

Then we can glue the right and left edges to get a donut (what mathematicians call a torus):

Now you probably think: “but it doesn't seem flat to me.” And you will be right. We cheated a little, describing how the flat torus works. If you really tried to make a torus out of a piece of paper in this way, you would run into certain difficulties. It would be easy to make a cylinder, but you would not be able to glue the ends of the cylinder: The paper would wrinkle along the inner circle of the torus and not stretch far enough along the outer circle. Instead of paper, some stretch material would have to be used. But this stretching distorts the lengths and angles, changing the geometry.

Inside an ordinary three-dimensional space, it is impossible to build a real, smooth physical torus from a flat material without distorting its geometry. But we can speculate abstractly about how it feels to live inside a flat torus.

Imagine you are a two-dimensional creature whose universe is a flat torus. Since the geometry of this universe comes from a flat sheet of paper, all the geometrical facts we are used to are the same, only on a small scale: the angles in the triangle add up to 180 degrees and so on. But the changes that we have made to the global topology by cutting and pasting, mean that the experience of staying in the torus will be very different from what we are used to.

To begin with, there are direct paths on the torus that bend and return to where they started:

These paths look curved on a distorted torus, but they seem straight to the inhabitants of the flat torus. And since light travels in straight paths, then if you look right, you can see yourself from behind:

On a sheet of paper light, you see, was held back until it reaches the left edge, then reappeared on the right, like in a video game:

You can imagine it is different. For example, you (or a ray of light) cross one of four borders, appearing in what appears to be a new “room." But actually it is the same room, only seen from a new perspective.

This means that you can also see an infinite number of different copies of yourself, looking in different directions. This is a kind of Mirror Corridor effect, except that copies of you are not reflections:

On the donut, they correspond to many different rings along which the light can move from you to you:

In the same way, we can build a flat three-dimensional torus by gluing the opposite sides of the cube. It will not work to visualize this space as an object inside an ordinary infinite space, but we can abstractly talk about life inside it.

Just as life in a two-dimensional torus was similar to life in an infinite two-dimensional array of identical rectangular rooms, life in a three-dimensional torus was similar to life in an infinite three-dimensional array of identical cubic rooms. You will see infinitely many copies of yourself:

Three-dimensional torus is just one of 10 different flat finite worlds. There are also flat infinite worlds, such as a three-dimensional analogue of an infinite cylinder. In each of these worlds, there is a different set of mirror rooms.

Is our universe one of these flat forms?

When we look into space, we do not see infinitely many copies of ourselves. However, it is surprisingly difficult to exclude these flat shapes. Firstly, they all have the same local geometry as Euclidean space, so no local dimension can distinguish between them.

And if you saw a copy of yourself, this distant image would show how you (or your galaxy, for example) looked in the distant past, since the light had to travel a long time to reach you. Maybe we see unrecognizable copies of ourselves there. Worse, different copies of yourself tend to be at different distances from you, so most of them will look different. And perhaps they are still too far away for us to see.

To get around these difficulties, astronomers, as a rule, do not look for copies of themselves, but for repeating features in the farthest of what we can see: cosmic microwave background radiation (CMB) left after the Big Bang. In practice, this means finding pairs of circles in the CMB that have matching patterns of hot and cold spots, which suggests that this is really the same circle that we see from two different points.

In 2015, astronomers conducted just such an analysis using data from the Planck space telescope. They combed data on the types of coincident circles that we expected to see inside a flat three-dimensional torus or other flat three-dimensional shape, called a plate, but they could not find them.

This means that if we really live in a torus, then it is probably so large that any repeating patterns lie outside the observable universe.

Spherical geometry

We are all familiar with two-dimensional spheres - the surface of a ball, orange, Earth. But what would it mean for our universe to be a three-dimensional sphere?

It is difficult to imagine a three-dimensional sphere, but it is easy to describe using a simple analogy. Just as 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 “three-sphere”) is a collection of all points at a fixed distance from a certain central point in four-dimensional space.

Life in three areas is very different from life in a flat space. To feel this, imagine that you are a two-dimensional being living in a two-dimensional sphere. A two-dimensional sphere is the entire Universe - you cannot see and cannot access any of the surrounding three-dimensional spaces. Inside this spherical universe, light moves along the shortest paths: in large circles. For you, these big circles seem to be straight lines.

Now imagine that you and your two-dimensional friend hang out at the North Pole, and your friend goes for a walk. While your friend is walking, at first he will become less and less in your visual space, as well as in our ordinary world (although he will not decrease as quickly as we are used to). This is due to the fact that while your visual space will increase, your friend will take up less and less space in it:

But as soon as a friend passes the equator, something strange happens: he begins to seem more and more, the farther he goes . This is because the percentage that it occupies in your visual space grows:

When your friend is three meters from the South Pole, he will look as big as three meters from you:

And when it reaches the South Pole, it can be seen in all directions, so it will fill your entire visual horizon:

If there is nobody at the South Pole, then your visual horizon is something even stranger: you yourself. This is because the light emanating from you will travel throughout the sphere until it returns to you.

This can be correlated with life in the three-dimensional sphere. Each point on the three-sphere has an opposite point, and if there is an object there, we will see it as a background, as if it is the sky. If there is nothing there, then instead we will see ourselves as a background - as if our exterior was superimposed on a balloon, then turned inside out and inflated to become a whole horizon.

Three-sphere is a fundamental model of spherical geometry, but this is not the only such space. Just as we built flat spaces by cutting a piece from Euclidean space and gluing it together, we can build spherical spaces by gluing a suitable piece from three spheres. Each of these glued forms, as in the torus, will have the effect of a “labyrinth of reflections”, but in these spherical forms there is only a limited number of rooms through which you can go.

Can our universe be spherical?

Even the most narcissistic people cannot imagine themselves as the backdrop of the whole night sky. But, as in the case of the flat torus, the fact that we do not see any phenomenon does not mean that it cannot exist. The circumference of a spherical universe can be larger than the size of the observable universe, which makes the background too distant to be seen.

But unlike the torus, the spherical universe can be detected using purely local measurements. Spherical shapes differ from infinite Euclidean space not only in the global topology, but also in the finest geometry. For example, due to the fact that the straight lines in spherical geometry are large circles, the triangles are more puffy than their Euclidean counterparts, and the sum of the angles is more than 180 degrees:

In fact, the measurement of cosmic triangles is the main way cosmologists verify whether the universe is curved. For each hot or cold spot on the cosmic microwave background, its horizontal diameter and distance from the Earth are known, which forms three sides of the triangle. We can measure the angle at which a spot is hiding in the night sky - one of the three angles of a triangle. Then check whether a combination of the length of the sides and the measured angle is suitable for flat, spherical or hyperbolic geometry (in which the sum of the angles of the triangle is more than 180 degrees).

Most of these studies, along with other measurements of curvature, indicate that the Universe is either flat or very close to flat. But one research team recently stated that some of the data obtained with the Planck space telescope in 2018 indicates the existence of a spherical universe. Other researchers object to this statement, believing that this is most likely a statistical accident.

Hyperbolic geometry

Unlike a sphere that bends by itself, hyperbolic geometry unfolds outward. This is the geometry of flexible hats, coral reefs and saddles. The basic model of hyperbolic geometry is infinite space, like a flat Euclidean space. But since hyperbolic geometry propagates outward much faster than flat, there is no way to place even a two-dimensional hyperbolic plane inside an ordinary Euclidean space, unless we want to distort its geometry. Here, for example, the notion of a hyperbolic plane known as the Poincare disk is distorted:

From our point of view, the triangles near the boundary circle look much smaller than near the center, but from the point of view of hyperbolic geometry, all triangles are the same size. If we tried to make triangles of the same size - for example, using stretching material for our disk and increasing each triangle in turn, going out from the center - our disk would look like a flexible hat and bend more and more as we made our way out. As we approach the border, this bend would become more and more uncontrollable.

From the point of view of hyperbolic geometry, the boundary circle is infinitely far from any internal point, since for this you need to intersect an infinitely many triangles. Thus, the hyperbolic plane extends to infinity in all directions, just like the Euclidean plane. But from the point of view of local geometry, life in the hyperbolic plane is very different from what we are used to.

In simple Euclidean geometry, a circle is directly proportional to its radius, but in hyperbolic geometry, the circle grows exponentially compared to the radius. We can see an exponential cluster in the masses of triangles near the boundary of a hyperbolic disk.

Because of this feature, mathematicians like to say that in a hyperbolic space it is easy to get lost. If your friend leaves you in the usual Euclidean space, he will begin to look smaller, but this will happen slowly, because your visual circle is not growing so fast. In hyperbolic space, your visual circle grows exponentially, so that soon your friend will look compressed to an exponentially shallow point. If you have not carefully tracked his route, it will be almost impossible to find a way to him.

And in hyperbolic geometry, the sum of the angles of a triangle is less than 180 degrees - for example, the triangles in our Poincare disk tile have angles of 165 degrees:

The sides of these triangles do not look straight, but this is only because we look at the hyperbolic geometry through a distorted lens. For a resident of the Poincare disk, these curves are straight lines, because the fastest way to get from point A to point B is to cut the path to the center:

There is a completely natural way to make a three-dimensional analogue of the Poincare disk - just make a three-dimensional ball and fill it with three-dimensional shapes that become less as you approach the boundary zone, like triangles in the Poincare disk. And just like in plane and spherical geometry, we can make a number of other three-dimensional hyperbolic spaces by cutting out a suitable piece of a three-dimensional hyperbolic ball and gluing its faces.

Can our universe be hyperbolic?

Hyperbolic geometry, with its narrow triangles and exponentially growing circles, is not like the geometry of the space around us. Indeed, as we have already seen, most cosmological measurements point to a flat universe.

But at the same time, the possibility that we live either in a spherical or in a hyperbolic world is not excluded, since small pieces of both of these worlds look almost flat. For example, small triangles in spherical geometry have angles that are only slightly more than 180 degrees, and small triangles in hyperbolic geometry have angles that are only slightly less than 180 degrees.

It is no accident that the ancient people believed that the Earth was flat - the curvature of the Earth was too small to be detectable. The larger the spherical or hyperbolic shape, the flatter each small part. Therefore, if our Universe has an extremely large spherical or hyperbolic shape, then the part that we can observe can be so close to flat that its curvature can be detected only with the help of ultra-precise instruments that we have yet to invent.

What else is useful to read on the Cloud4Y blog

→ Computer brands of the 90s, part 3, final
→ Can a ship be hacked?
→ Easter eggs on topographic maps of Switzerland
→How the hacker’s mother entered the prison and infected the boss’s computer
→ How the bank “broke”

Subscribe to our Telegram channel so as not to miss another article. We write no more than twice a week and only on business.