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Krebert: It's Lobachevsky's Hyperbolic Geometry. The "It's just in your imagination" part is referring to it being non-Euclidian geometry.

That means that one of Euclid's Postulates is changed and then the rest of the geometry system is constructed around these new premises.

The word "Postulate" is important here, referring to an "assumption" or a "given". That means that it's one of the few rules that we "take to be true" before deriving the rest of geometry based on those assumptions. All math systems need a set of "startup rules" called axioms (or postulates) that are simply taken as "true" and then, if those axioms are complete enough, you can derive a math system from it which may or may not be useful, but *will* be internally consistent.

If you want a math system to be useful in the real world, you would obviously want the postulates or axioms upon which you build the math system to correspond to the real world, right? Therefore, in Euclidean geometry, the axioms/postulates are things that seem "obvious", but have to be "assumed" or "given" rather than "proved" or "derived".

For example: the Parallel Postulate: For any given line R and point P not on R, there is exactly one line which passes through P and is parallel to R.

If you draw it out on paper, this seems obvious, but it turns out that it's not possible to "prove" this using just the other axioms of Euclidean geometry. It must be "given". That means that it's possible for an internally consistent geometry system to exist in which the parallel postulate is *not* true or is different.

"But isn't that just playing around with your imagination?" you ask. If you take a piece of paper, draw a line, then draw a dot next to the line, it's quite obvious that you can only draw one parallel line through that dot. So how would a geometry in which you can draw at least two parallel lines through that one dot be possible?

In reality, it *seems* obvious that you can't just change the basic rules of the world and expect the resulting math system to be useful, but it turns out that there are in fact situations in which alternate geometries are not just useful, but essential.

Relativity Theory, for example, is highly dependent upon non-Euclidean geometries. The reason for that is that Euclidean geometry is for "flat" space, like paper on a desk, whereas real space is actually not flat (Euclidean), but "hyperbolic", while space-time is described by yet another geometry.

Alternate geometries as a concept is important because is leads us toward "Minkowski space", a four-dimensional "manifold" instead of three-dimensional Euclidean space, which is the foundation for Special Relativity and the concept of a unified space-time with inertial reference frames.

Throne of Magical Arcana · C420
1 week ago
Anyone else find the current ZF personality and actions strange and different than his usual personality. He seems indecisive relatively weak without strategy or tactic and overall just a loser. View More
King of Gods · C1367
2 months ago

Fataki: I remenber that lith is a grown up man in his past life , I find it weird how he indulge himself with those kids .
I started to dislike how the story became in the last few chapters especially solus she doesn't make sense at all. the story became more about lith and his friends , we don't see a solitary lith he never does anything alone anymore , and the difference in strength between them make it even more bad . He could have gone to the forest alone and made it back easily but now he has to make the babysitter of 4 kids. Don't misunderstand me , I don't mean he shouldn't have friends but this is too much for someone who is not social and has many secrets and who especially loves being alone.

Supreme Magus · C235
4 months ago
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