Comments and questions about the February 7 class. See february-7 for the class notes.
Very interesting discussion of Lobachevskian theory. Thank you to whoever put the notes up. I thought it was interesting the way Lobachevsky used the method of making a square to prove his twentieth axiom. I also think Professor Bolker made a good point in critiquing the author's use of the figures for this proof. It would be interesting to know whether Lobachevsky made figures of his own, or whether the author inserted these as a visual aide.
Geometry has never been my favorite field of math, but I have appreciated this introduction to the history of the geometry and getting inside the minds of those who helped to formulate it. It will be interesting to investigate Poincaré's axioms, and to look at the use of the circle and pieces of circles. I am looking forward to the future units as well, especially looking into the mathematicians that helped develop set theory (I believe this is coming up). I was particularly interested in learning about Cantor, as I had investigated some of his methods as part of my math proofs class curriculum.
The study of Lobachevsky's work on Euclidian geometry was very interresting especially the proposition 20 which stipulates that: "If in a rectilineal triangle, the sum of the three angles is equal to two right angles, so is also the case for every other triangle". Professor Bolker guided us throught the proof in class creating two triangles from the one we were given and proving that the sum of the angles of each of them is less or equal to two right angle. The question raised in class was if the two triangle created must be right, and the answer was no. What was also interresting is the expansion of the triangle, building as many triangles as he wants. Again, we wanted to know if the result of the expansion should be a square? And No was the answer.It must be proportional to the original triangle.
We also started talking about Henri Poincare who, working on number theory and complex theory, discovered the way to draw picture of Lobachevsky's hyperbolic geometry. I did not understand the construction in class of the parallel line by Poincare during the last lecture. So I am looking foward that the professor do some review on it.
In class we continued our discussion on Lobechevsky geometry and then went on to talk about Poincare. Poincare was a french mathematician who was most famous for his conjecture. Poincare conjecture in laymens terms is a question about sphere in math. Saying that a 2- sphere is connected, like a donut. The conjecture asks if this is true for a 3-sphere.
- Poincare is indeed famous (at least among mathematicians) for the Poincare conjecture. Bringing it into the discussion is good, but you don't have it quite right, even in lay terms. - please look it up and put down a correct version for ordinary people. It can be as elementary as you like - but not misleading. Ethan
So far we have not discussed anything that concerns 'exterior angles' of a triangle, or any shaped graph. I think it would be interesting when we get to take the exterior angles into our discuss, but, it would be more difficult to look at things, or easier, depends on how you look at it. More difficult, because there are more to 'check'; easier, because there would be more ways to look into graphs or proofs.
Also, there are two links to Poincare conjecture that I looked up for whoever post the last discussion paragraph, and they should be easy to understand, even though I did not completely get it:
- Thanks for the links. I agree that we haven't talked about exterior angles. They would be important if we were studying geometry systematically for its own sake. I don't think we need to go there in order to understand the history of the parallel postulate, which is our main concern. Ethan