This thread is obviously inspired by Antichamber; anyway, I get the general gist of what non-euclidean geometry is, sorta...but, uh, not really...

So, it would be great if someone could give a clear, succinct definition of what exactly non-euclidean geometry is - but without getting getting too technical...

You should ask Lovecraft ;)

But also you might find this helpful: http://www.youtube.com/watch?v=zHh9q_nKrbc

Ehhh, I thought it was a fictional concept invented by HP Lovecraft to describe alien cities

Cthulhu's favorite type of architecture ? :P



But seriously, Wikipedia is your best friend:


http://en.wikipedia.org/wiki/Non-Euclidean_geometry

I'm no mathematician, but from what I got, it's geometry on a different space, with a different "topology". Like, trigonometry on a sphere surface, where the sum of a triangle's inner angles won't make 180°. You can have many "surfaces" that have different properties than the flat one. And you can extrapolate this to different 3D spaces, with counter-intuitive topologies.

(Well, 3D or more, that is.)

"Non-Euclidean geometry is a type of geometry. Non-Euclidean geometry only uses some of the "postulates" (assumptions) that Euclidean geometry is based on. In normal geometry, parallel lines can never meet. In non-Euclidean geometry they can meet, either once (elliptic geometry), or infinity many (hyperbolic geometry) times.
An example of Non-Euclidian geometry can be seen by drawing lines on a ball or other round object, straight lines that are parallel at the equator can meet at the poles."

Origin

"It is called "Non-Euclidean" because it is different from Euclidean geometry, which was discovered by an Ancient Greek man named Euclid. The different names for non-Euclidean geometries come from thinking of "straight" lines as curved lines, either curved inwards like an ellipse, or outwards like a hyperbola."

->> http://simple.wikipedia.org/wiki/Non-Euclidean_geometry