Geometry is a giant field in mathematics and it all begun when Euclid, a greek mathematician, established the five postulates of geometry. He made a type of geometry we use right now and that which you have probably learned about in school – Euclidian geometry. The only problem is, his geometry only deals with flat spaces, but cannot descibe curved space. For example, in a flat space the shortest line between two points is straight but in curved space that would not always be the case. It is very tricky for us to understand curved space as we are not used to it, and many discoveries still need to be made. But to start off we need to understand what rules apply in Euclidian geometry to then extend our knowledge to things that are completely different. Euclid proposed 5 postulates which are basically axioms, of his geometry.

1. It is always possible to draw a straight line form one given point to another.

2. You can extend any line infinitely keeping it a straight line in any direction

3. It will always be possible to draw a circle if you are given a point as a center and a straight line as the radius

4. All right angles are equal

5. Now, this one is trickier, but it also is the most important one - if you have two straight lines that are infinite length and a third straight line going through both, then if the two angles that are created by the lines intersecting(the first two with the third one are less or more than a right angle the two points will eventually meet.

If the sum of angles a and b is less than 180 then the two lines will meet eventually. Another way to say this is – if there is a line and a point is drawn not on the line(anywhere on a plane) then only one line that is parralel to the straight line and goes through the point can be drawn.

This postulate was a last resort for euclid as there was no way to prove it. So during different theorems he didn’t use it much. And in the 1820s two mathematicians, from Russia and Hungary decided that the fifth postulate is not always true. They said that there could be an infinite amount of lines that would go through the point and be parralel. That is how Hyperbolic geometry was born.

Poincare disc (Hyperbolic Geometry)

Euclid started the field of geometry, and for 2000 years the only things we knew were about flat geometry, or Euclidian geometry (same thing) but sooner or later we had to find out that his geometry is not the only type that is true. Still now in schools we learn Euclidian geometry and when you think about Hyperbolic or Spherical geometry ( we will still talk about them) you might be surprised and it may seem as a parallel dimension to you as it is ezxteremely surprising to us, who always think about geometry in 1 way. But if its strange that doesn't mean it's not interesting and many discoveries are yet to be made!

Sources

Picture 1 - Wikipedia, Parallel Postulate

Picture 2 - WIkiwand Hyperbolic Geometry