system
Given the following post, respond as if you're commenting on Reddit:
human
ELI5: What is non-Euclidean space or geometry?
gpt
So, to consider this properly, we need to consider what Euclidean geometry is.

Euclidean geometry is any framework in which the following five postulates are true:

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. ***This one's the important one!***

The first four postulates can be used to describe an entire geometric system, along with a series of definitions and notions.

Euclid's Fifth Postulate, also known as the *parallel postulate,* has mainly to do with triangles, because that's the shape you get if you construct three lines in that manner.

However, Euclid was never able to prove the fifth as a theorem. In 1823, mathematicians Janos Bolyai and Nicolai Lobachevsky figured out that ***you could ignore the parallel postulate*** and make geometries that made sense!

*Non-Euclidean geometry, then*, is any framework in which the Fifth Postulate does not hold -- that is to say, where the interior angles of a triangle do not sum to 180 degrees. Further, as a consequence, parallel lines don't stay the same distance from each other forever.

Edit for clarity regarding parallel lines: Under Euclidean geometry, the following is true:

* Given a two-dimensional plane P, a line L, and a point A not on L, there is exactly one line that can be drawn through the point A that does not intersect L.

In hyperbolic geometry, there are infinitely many such lines that can be drawn; in elliptic geometry, *any* line drawn through A will intersect L.