Mathematics 153
Notes from April 17, 2020, lecture session

What was covered:
The remaining part of Unit 3.  
 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.
Files for Unit 3.
http://math.ucr.edu/~res/math153-2020/week3/unit03/history03.pdf   This is the main file for the unit and should be read from beginning to end.  After doing so one should read the cited sections of Burton.
http://math.ucr.edu/~res/math153-2020/week3/unit03/history03c.pdf   Modern axiom systems for Euclidean geometry which bridge the gaps in the Elements.  It is especially worthwhile to read this because it describes some fundamentally important properties  which hold in Euclidean geometry.
http://math.ucr.edu/~res/math153-2020/week3/unit03/history03d.pdf   A discussion of regular polyhedra, including drawings of the five Platonic solids and the reasons why there cannot be any other regular polyhedra.  
http://math.ucr.edu/~res/math153-2020/week3/unit03/history03e.pdf   Comments on contemporary questions in Euclidean geometry.   This is optional background material.
http://math.ucr.edu/~res/math153-2020/week3/unit03/isosceles-fallacy.pdf   This file contains drawings for several "cases" in the fallacious proof that every triangle is isosceles.  The purpose is to suggest that the argument does not seem to rely on the choice of drawing.  However, if one takes betweenness considerations into account, then one sees that there are other possibilities as well.  In fact, if one analyzes the argument, it turns out that it yields a valid conclusion about betweenness relations for the collinear triples  {A, B, F}  and  {B, C, G}.  Namely, exactly one of the points F and G is a "between" point.
http://math.ucr.edu/~res/math153-2020/week3/unit03/playfair-proclus.pdf   A drawing with comments to illustrate the postulate and comment on a natural issue associated with it:  How can we be sure there are unique parallels if some lines are supposed to meet at distances that are beyond astronomical?  Is it meaningful to talk about points that we can never reach?
http://math.ucr.edu/~res/math153-2020/week3/unit03/polyhedra.pdf   Paper models for polyhedra.  To form the polyhedra, print these, then cut them out, and finally fold them to get models for basic 3-dimensional figures.
Recommended YouTube video:
The first part of https://www.youtube.com/watch?v=an0dXEImGHM provides an excelllent introduction to non-Euclidean geometry from a viewpoint at a level appropriate for this course.  However, the discussion becomes more complicated at 13:58 when the speaker moves on to discussing the methods that were used in the 19th century to work out the details of the subject, so that time point is a recommended place to stop watching.
Exercises on betweenness, separation and interior regions:
The following material from Math 133 might be helpful for reinforcing or checking one's understanding of the concepts mentioned:
http://math.ucr.edu/~res/math133/math133exercises02.f13.pdf   Exercises II.2.1 - II.2.8, II.2.10, II.3.3 - II.3.5, and II.3.8 - II.3.10
http://math.ucr.edu/~res/math133-2018/math133exercises02a.f13.pdf   All exercises.
All of these are optional but recommended if you are unsure about the listed concepts.
Solutions are posted in the following three files:
http://math.ucr.edu/~res/math133-2018/math133solutions02a.f13.pdf
http://math.ucr.edu/~res/math133-2018/math133solutions02b.f13.pdf
http://math.ucr.edu/~res/math133-2018/math133solutions02ba.f13.pdf