Mathematics 133
Notes from December 7, 2020, lecture session

The recording for this lecture is available online as  
http://math.ucr.edu/~res/math133-2020/Dec07.mp4  .  

 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
http://math.ucr.edu/~res/math133-2020/weekTen/geometrynotes05b.f13.pdf   The second part of the fifth chapter of the course notes.   The lecture covered the material in Section V.4 beginning with the subheading The Strong Hyperbolic Parallelism Property and continuing through the end of the section.  The two most important parts are the statement and proof of Theorem V.4.7  ths subheading  Logical consistency of hyperbolic geometry in Section V.6  and the solved problems at the end of the section.  The material in the remaining sections was surveyed and will not be covered on the final examination. 
http://math.ucr.edu/~res/math133-2020/weekTen/geometrynotes05c.f13.pdf   The third part of the fifth chapter of the course notes.   The lecture mainly covered the final Section V.8  on the impact of hyperbolic geometry; more details on some of the points appear in the final two sections of the previous document and the documents listed below.  Section V.7 attempts to explain one reason why hyperbolic geometry plays an important role in contemporary mathematics by means of tilings for the plane by regular polygons.  If nothing else, the pictures are somewhat intriguing, and one example is a print by M, Escher (ESS-kher).   

https://math.ucr.edu/~res/math133-2020/weekTen/history03c.pdf    Reasons for simplifying the axioms for Euclidean and hyperbolic geometry and further references.

https://math.ucr.edu/~res/math133-2020/weekTen/moise-hyperbolic.pdf    Table of contents for relevant sections of Moise's book.

https://math.ucr.edu/~res/math133-2020/weekTen/verifications.pdf    Survey of the proofs that Euclidean and hyperbolic geometry are logically consistent (relative to the consistency question for positive whole numbers).

https://math.ucr.edu/~res/math133-2020/weekTen/irreducibleplanes1.pdf   https://math.ucr.edu/~res/math133-2020/weekTen/irreducibleplanes2.pdf     These files discuss ways of setting up incidence geometry with the Hyperbolic Parallel Postulate.  This material will not be covered in any exams.

https://math.ucr.edu/~res/math133-2020/weekTen/sawyer-excerpt.pdf    An attempt to explain one feature of the Beltrami and Poincare models for the hyperbolic plane; namely, the points lie on a bounded region of the plane or space butthe distances between points in these geometries can be arbitrarily large.

https://math.ucr.edu/~res/math133-2020/weekTen/pseudosphere.pdf    An isometric representation of a piece of the hyperbolic plane on a classical surface of revolution.

https://math.ucr.edu/~res/math133-2020/weekTen/poincare-metric.pdf    An explicit formula for the distance in the Poincare disk model for hyperbolic geometry.

https://math.ucr.edu/~res/math133-2020/weekTen/poincare-trivia.pdf    Relation of Henri Poincare to the early 20th century French political leader Raymond Poincare.

https://math.ucr.edu/~res/math133-2020/weekTen/plausible-impossible.pdf    Link to a classic Walt Disney film discussing the a posteriori nature of physical phenomena; namely, they are not logically inevitable like arithmetic but are instead based upon empirical experience.   One consequence of hyperbolic geometry's consistence is that the view of geometry as logcially inevitable was replaced by a view that geometry is also something which is based upon experience.   The portion of the Disney film shown in the lecture begins at 6:54 and continues for about 60 seconds.
https://math.ucr.edu/~res/math133-2020/weekTen/math133hw05.pdf   Recommended homework assignments for Chapter V.   The previous comments about homework recommendations and examination problems also applies here.