Goals

   As it is almost halfway through the school year, our instructors have decided it is time for us to decide on some concrete goals for what we wish to do with our blogs. Here are my three goals-- one for this blog, one for CWatKPCN (with Jasmine, Nate, and Andy), and one for AFaiG (also with the three aforementioned group members).

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.

People...

   I have not yet contacted any experts on my topic. For research purposes, I honestly prefer to learn from reading than from people. I do, however, have some ideas on people to contact once I have developed a concrete direction to point a potential conversation in.

Sources

 This is a list of the books and sites that I have been using for my research. Most information presented in other blog posts comes from one of the following sources.

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.

It's All Greek at Heart

The first great mathematicians were all Greek. As Leonard Mlodinow said of the Egyptians and Babylonians that came before the Greek, "like our political leaders, they sometimes accomplished astonishing feats with surprisingly little comprehension of what they were doing. Nor did they care." The Greeks, on the other hand, did care. And so while the Egyptians built pyramids with their knowledge, the Greek mathematicians-- Euclid, Pythagoras, Thales, Eratosthenes, Archimedes, Hypatia, and many more-- are immortalized through their work in the creation of the system of mathematics and logic called geometry.

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...

Risk: Takes 2 and 3

Take 2:
     As our Risk board was cleared off since we last played and we didn't take any pictures of it, we had to start over. Jasmine was absent, so I played for her, managing to get the entirety of South America and Africa for her (then having her squander it by putting all her little soldierpeople in Madagascar, but never mind about that). However, we didn't save the board again, so on to take 3.