Concerning Triangles

Triangles, as with lines, vary between types of geometry.  This is merely an overview of their behavior on different planes. A more detailed analysis of triangle similarity and non-Euclidean trigonometry is in store as soon as I figure out how, exactly, non-Euclidean trigonometry works.

Fake Spheres?

   Just as the sphere models elliptic geometry, the "pseudosphere" is the model of hyperbolic geometry. It is, however, a bit less well-known than its elliptic counterpart.

The Line In Its Natural Habitat(s)

   The line is defined, by Euclid, as "that which lies evenly with the points on itself." This is utterly useless as a definition, as it seems complete nonsense unless one already knows what a line is.
   One common alternative way of describing a line is as the shortest distance between two points, or a geodensic. Now, as explained earlier, the shortest distance between two points on a sphere is along a great circle. While lines on flat planes appear straight, on elliptic and hyperbolic planes they are curves. It follows that while parallel lines on flat planes will remain the same distance from each other for their entirety, on curved planes they will curve towards or away from each other.

More Conundrums

The riddle from the last "conundrums" post: a man comes out of his house and walks one league south, then one league east, then one league north, then finds himself back at his front door.
Aaaand, the explanation...

Circles, Great and Small

  Oftentimes the paths that airplanes take to travel long distances don't look straight when drawn on a map. But the fastest path from point A to point B is a straight line, right? The only problem is, that a straight line from, for instance, Chicago to Beijing, would have to go through the center of the earth. 

Conundrums

 Murdered cricketers. Who ever knew there so many, and why on earth are they important enough to have an entire Wikipedia page? That, indeed, is a conundrum.
 Speaking of conundrums, I have one for you. Consider this: a man comes out of his house and walks one league south, then one league east, then one league north, then finds himself back at his front door.
Able to explain? Please do. It is indeed related to non-Euclidean geometry, in case you're wondering.

Elementary, my dear self.

     Non-Euclidean geometry is defined as any geometry in which the parallel postulate is not true, which tends to occur in elliptical and hyperbolic planes (as opposed to flat ones, of course), hence the two main geometries within the non-Euclidean ones (geometry-ception, eh?), aptly named elliptical and hyperbolic geometry. The parallel postulate can refer to Euclid's fifth postulate (If a straight line falls on two straight lines in such a manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles) or Playfair's axiom (Through any point in a plane, there is at most one straight line parallel to a given straight line).