DoctorKokktor t1_j6nynl9 wrote on January 31, 2023 at 5:57 PM

Reply to comment by NIRPL in ELI5 why time slows down as you go faster by -cool--beans-

Another good question :)

So basically, Euclidean geometry is founded on five posulates, the last of which is called the "parallel postulate". It turns out that the statement that describes this parallel postulate is ambiguous, and so you can have multiple descriptions/variations of this postulate (you can even omit the 5th postulate altogether!), all of which allow you to derive entirely new geometries.

Those 90 degree angles in different shapes are a consequence of the different statements of the parallel postulate. Those figures shown in the article are all examples of parallel lines, which at first seem preposterous but are a natural consequence of accepting a different version of Euclid's parallel postulate.

NIRPL t1_j6oe0q2 wrote on January 31, 2023 at 7:31 PM

I sincerely hope you are an educator. You have a talent for explanations. Thank you for taking the time to respond and teach me something new today.

DoctorKokktor t1_j6om43l wrote on January 31, 2023 at 8:21 PM

Thank you for your kind words :)

Incidentally, Einstein's theory of special and general relativity are also a different kind of geometry! Just like how Euclidean geometry takes place in Euclidean space, special relativity takes place on what's known as "Minkowski" space or spacetime, whereas general relativity takes place on what's known as a pseudo-Riemannian manifold.

These words sound complex but just remember that they are merely names for some kind of foreign geometry that are derived from a completely different set of axioms than the normal Euclidean geometry that we studied in high school. This article is very nice at introducing the history and context behind non-euclidean geometries!

But you don't even have to think about something as abstract as relativity to realize that non-euclidean geometries are everywhere. Even the surface of the earth has a non-euclidean geometry (the geometry that takes place on the surface of the earth is called "spherical geometry"). In this sort of geometry, you can have triangles which can have two right angles!

The reason I am introducing all these different types of geometries is because in the end, relativity theory is a geometric theory of spacetime. If you can understand the context behind some of the different kinds of geometries, then you can understand the context behind relativity theory as well! In this way, relativity won't seem as mysterious anymore. It's still very counter-intuitive, but at least you can understand that relativity is just a consequence of choosing some set of axioms, and drawing conclusions from there, just like any other sort of non-euclidean geometry.

Sorry for all these other links haha. This topic is extremely interesting to me and I wanted to share some resources to hopefully get you excited about it too! :)