Do you remember playing Legos, where you had to put similarly shaped pieces together for them to fit properly? It was a basic game to develop shape perception in children. It introduced basic shapes such as Square, Sphere etc to them.
But as we grew up, we learnt about new shapes and polygons, which have numerous shapes all together to form a singular body. So many shapes have we seen in our lives, that the basic question of the shape of any structure doesn’t arise in our minds normally and is now meaningless for us, or maybe not(suggested by the presence of FLAT EARTHERS!).
In this blog, let us try to ponder upon the shape of something bigger, the shape of the universe.
Thinking about the shape of the universe is quite difficult since it is HUMONGOUS. It is so big that it contains more than a million galaxies and each galaxy has more than hundreds of earth. Normally, for determining the shape of something, we observe it from the outside. But we can’t do the same for the Universe. So how do we find its shape from the inside? We can try to start finding our answer by observing how geometry behaves in cosmos.
Let us select 2 points anywhere in the universe. Using a bit of common sense, we can say that the distance(Note that this is not the same as used in physics!!) between them can be measured. Such a system is known as a Metric System, in which 2 points can be selected randomly and the distance between them can be measured. But let us try to think about the distance. Distance can be thought of as the minimum separation between 2 points(can be equal to a straight line in simple cases). Let’s take an example to understand the above line- say you have to go to a music concert alone from your house. So even if you chose the longest way possible to be out for as long as possible, the distance will be equal to the straight path from your house to the venue. If you had to go with your friend, you have to pick her from her house and then go to the venue. Even in this case, the distance is the same as the previous one- it will be equal to the straight path from your house to the venue. BUT if the straight path to the venue is closed due to some construction work, the distance changes as you cannot walk through the worksite. You must remember how we defined distance – minimum separation- so the distance, in this case, must be equal to the curve nearest to the construction side available for moving. From this example, we can generalise that distance is equal to the smallest separation possible under the constraints given. This generalisation in mathematical terms is known as Geodesic.
Now, let’s apply the concept we just learnt! For simplicity, we start with a 2D world. Here the distance is the straight line segment that starts at the first point and ends at the second. What if we do the same thing on something 3D like a sphere, such as the Earth? Here, to find the distance, we must keep in mind the constraint given, that is, all curves should be on the surface of the sphere.
As you can see, the shortest distance between A and B is a curved line this time, which follows the sphere surface. We can conclude that the features of distance are related to the shape in which the points are chosen. This follows in the 3D world and hence will be applicable in the Universe. If the space is curved, the distance will be a curve but if it is flat, the distance shall be a straight line. But this presents to us a new problem, how will we know if the line is straight or curved if we are itself inside the Universe? If we were outside it, we could have easily observed the line and told whether it’s straight or not.
To study the curvature of a line, we can try to take some help from Euclidean Geometry, which is the Geometry of Plane Objects in mathematics. Hence, if Euclidean Geometry holds in-universe, then the universe is flat, as the distance is a straight line otherwise it must be curved.
An easy way of checking this is to draw a triangle and check the sum of internal angles. Doing this in a plane, the sum always is found to be equal to 180. But if we do the same on sphere-like earth or convex side of a spherical mirror, the sum is always greater than 180(known as positive curvature), and when we do it for hyperboloid-like space, like the concave surface of a spherical mirror, then it is always less than 180(or negative curvature).
Now, some questions arise in our stupid brain. First, how do we draw a triangle on the surface of the universe, and secondly, what triangle is better to study? The answer to the first question would be Electromagnetic waves(like light). Since light is faster than anything in the world, the path light follows will be the distance and there will exist no shorter way than travelled by light to transverse through a line segment.
Which triangle should we choose? If we choose a small triangle, then we will only study the shape of that specific space in the cosmos and not the universe. Furthermore, we also know that near something as huge in mass as Earth, the Universe can be curved and our results can get distorted. Hence, we must choose a very very big triangle.
To satisfy the above conditions, Astrophysicists chose a triangle that has one vertex here on earth and the other vertices and the end of the OBSERVABLE universe, i.e. from where we can see a Cosmic microwave background radiation, which is a remnant from an early stage of the universe, and look something like a thermal scan showing red as hot places and blue as cooler ones. By knowing the dimensions of these blue and red stains of CMB radiation, the scientists defined a triangle of precise dimensions, that allows us to calculate the internal angles. The last step is to add all these angles and the results here are very astonishing for some people.
The Universe is FLAT!! This however does not mean it is like a slice of Pizza, it only means that Universe has no curvature and Euclidean Geometry is well followed in it.
You win this round, Flat Earthers!
The above piece is a work of Shivansh Srivastava, all credit goes to him. CHEERS!