Mathematics 133
Notes from November 23, 2020, pre-recorded lecture session

The recording for this pre-recorded lecture is available online as  
http://math.ucr.edu/~res/math133-2020/Nov23.mp4  .  
Since there was no opportunity for student questions, you are welcome  to submit electronic questions, and the answers will be posted to this subdirectory.

 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
http://math.ucr.edu/~res/math133-2020/week8/geometrynotes05a.f13.pdf   The first part of the fifth chapter of the course notes.   There is a Chapter IV of the course notes, but it will not be covered in the course itself;  a second file in this subdirectory contains links for this omitted chapter.  The lecture surveyed the material in Section V.1.  Here is an explanation for includion of the material from this section (on spherical geometry):   The main subjects of Chapter V are the emergence of Noneuclidean Geometry during the 19th century and some basid results from the latter.   As Carl Friedrich Gauss noted in the early 19th century, The theorems of this geometry appear to be paradoxical and, to the uninitiated, absurd; but calm, steady reflection reveals that they contain nothing at all impossible.   In order to make the main results seem less bizarre, we have included a section on spherical geometry in ordeer to illustrate that (1) there are other important and tangible geometrical systems besides Euclidean plane and solid geometry, (2) knowing some results from spherical geometry might make it easier to develop some intuition for Noneuclidean Geometry.  For our purposes, one of the most significant features of spherical geoemtry is the fact that the angle sum of a spherical triangle (the edges are all great circle arcs) is always greater than 180 degrees, and in fact the excess of the angle sum is proportional to the area of a solid region on the sphere bounded by the spherical triangle. 

https://math.ucr.edu/~res/math133-2020/week8/planetarium.jpg    Photograph of a typical planetarium (a theater with a hemispherical dome, designed for simulating the viewing of the sky from the earth).  To a great extent, both spherical geometry and spherical trigonometry arose from ancient Babylonian and Greek studies of the skies using similar hemispherical models. .  
https://math.ucr.edu/~res/math133-2020/week8/threedim-angles.pdf   Brief description of a geometrical result behind an important principle in spherical geometry:  The shortest curve joining two non-diametrically-opposite points on a sphere is a minor great circle arc.  Recall that a great circle is the intersection of the sphere with a plane through the origin, and a minor great circle arc joining two non-diametrically-opposite points is the shorter of the two great circle arcs joining these points (here the relevant great circle arc is formed by intersecting the sphere with the plane containing the sphere's center and the two given points).
https://math.ucr.edu/~res/math133-2020/week8/Whiteboard.nov18-3.png   Review of an angle-angle-area congruence theorem from Euclidean geometry, which can be viewed as an analog of the angle-angle-angle congruence theorem in spherical geometry and Girard's theorem relating the area of a solid spherical triangular region with the excess of the triangle's angle sum over 180 degrees.  
https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov23-1.png   https://math.ucr.edu/~res/math133-2020/week7/Whiteboard.nov23-2.png   Summary of the main steps in the development of hyperbolic geometry.  Perhaps one should add yet another step on the subsequent impact of these develpments, but in any case these impacts will be covered later.