February 7
• Answer questions (students can post questions here in advance of class …)
• Work through Lobachevsky proof that if one triangle has angle sum $\pi$ then all triangles do.
• Move on to Poincare
• Think about what happens after Chapter 1. (didn't get to this)

• Who will administer the wiki - so far just the professor
• How to do discussion pages (Separate discussion pages with links on the homepage/notes page was Rob's suggestion) - implemented

Class notes go right here. Discussion questions at february-7-discussion

*Talked about the term paper and that the text book has many refrences.
*Lobachevskian theorem 20.
*Lemma developed during proof: For any triangle ABC, CD divides ABC into two triangle. If the sum of the angles of ABC are two right angles, then for triangle ACD and BCD, the sum of three interior angles are also two right angles

*Proof of Therom 20:
- Assume the sum of three angles of an arbitury triangle is π, then you can create a right triangle(unit block) which has the sum of its three angles π.
- Prove that using the unit block, we can build a larger right triangle which is sum of π
- For any other right triangle, we include it in a biger right triangle which is built by the unit block, then by the lemma, prooved that in any right triagle the sum of three angles equal π.
- The for any triangle, it can be cut into two right triangles, so in it the sum of three angles is π.
*Open question: Did Lobachevsky invent or discover the geometry? Are we living in Eclidean or Lobachevskian space?
*Introduce the definition of Poincare's definition for line, parallel lines.

Mingzhi Liu