February 12

Plan

Questions for class (not sure where to post these because they might get brought up in class):

• Homework - I am honestly somewhat intimidated by these questions. Will we be going over all of the concepts we will need to do the homework questions in class?

Discussion at february-12-discussion

Class notes.

• Lobachevsky: No similar triangles not congruent, can't scale things, area related to triangles.
• Comparison between Geometry and Pseudogeometry
• Since Poicare's unit disk is in Euclidean Space, If get a contracdiction without Pol5, then can get a contradiction with 5
• Stretch factor $\frac{1}{1-r^2}$
• Talked about Pseudo distance and HW2
• Follow one reference from the text book, and study it
• Side point: Poincare might be the only person in history who "conquered" all mathematical fields at one's time. (and he has something to do with the explosion of math after him. Which makes impossible to know every field of math)

-Mingzhi Liu

We also:

• Talked about Saccherri's proof that the top two angles of his quadrilateral are equal; it depended heavily on Euclid's fourth proposition.
• Highlighted the differences between geometry (Euclid) and pseudogeometry (Lobachevsky )
• Eucludean geometry features points, lines and planes and is based on Euclid's first four postulates
• Pseudogeometry features points, pseudolines, which are parts of circles, and open disks in a Euclidean plane as the universe. This geometry is based on Euclid's first four postulates and Lobachevsky's work with the Hypothesis of Acute Angles from Sacherri's quadrilateral. Lines can be described as the shortest distance between two points.

-Angela Rogers

Poincare's Model of Hyperbolic Lobachevskian Geometry

Where the open disk in blue is our universe, and L and M are parallel by Lobachevsky's definition. Note that I eyeballed the angles, so this picture is far from exact, but it gives the general idea. The hardest part was finding a pseudoline perpendicular to L, since by definition it must also be perpendicular to the edge of our universe. You might notice that the dotted line, which is supposed to be the perpendicular, might not actually be a proper pseudoline.

Here is an image which shows everything "outside" of the universe as well:

Note that circle L is inside and tangent to M. If the point P was "inside" the pseudoline L, then M would be inside L.

-Shira