Mathematics 133
Notes from October 14, 2020, lecture session

The recordings for this lecture are available online as 
 http://math.ucr.edu/~res/math133-2020/Oct14.mp4 
http://math.ucr.edu/~res/math133-2020/Oct14a.mp4  .
The first file covers the first part of the class, before Internet service was interrupted, and the second covers the part after Internet service was restored.

 Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable
http://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/geometrynotes02a.f13.pdf   The first part of the second chapter of the course notes.   The lecture covered Section II.1.
http://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/geometrynotes00.f20.pdf   Material under the subheading Viewpoint of this course was covered; the remaining portions were covered at the beginning of the course.  
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/two-circles.pdf  This file describes the logical gap in the first proposition from the first book of Euclid's Elements. 
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/foundations01.pdf  This file describes the basic setting for the analytic approach to geometry (using coordinates).
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/foundations02.pdf   This file describes the basic setting for the synthetic approach to geometry (a modernized formulation for the classical Greek approach).
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/foundations03.pdf    A side by side comparison of the ways the synthetic and analytic pproaches are related. 
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/axioms4geometry.pdf  A chart which summarizes the discussion in the preceding three documents.
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/verifications.pdf   A sumary of the work needed to show that the coordinate models for geometry satisfy the axioms for synhetic geometry.  This is included for the sake of completeness; it is not necessary to understand the details, but you should understand why it is necessary to do these verificaations.
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/Whiteboard.oct14-1.png Edited version of a whiteboard created during the October 14 lecture.  This file gives a drawing for the proof of Proposition II.1.2.
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/Whiteboard.oct14-2.png Edited version of a whiteboard created during the October 14 lecture.  This file gives a drawing for the proof of Proposition II.1.3.
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/Whiteboard.oct14-3.png Edited version of a whiteboard created during the October 14 lecture.  This file gives examples of the axioms for 4 – dimensional incidence geometry.
https://math.ucr.edu/~res/math133-2020/week2/sections_2.1-2.2/Whiteboard.oct14-4.png Edited version of a whiteboard created during the October 14 lecture.  This file displays the formula formula from linear algebra which relates the dimensions of the sum and intersection of two subspaces with the dimensions of these subspaces.   Special cases provide the motivation for the 4 – dimensional incidence properties appearing in the preceding document (the dimension formulas yield the assertions in the previous document if we are dealing with planex and hperplanes which pass through the origin).
https://math.ucr.edu/~res/math133-2020/pg-all.pdf  Excerpts from these notes provide more explicit details on setting up incidence geometry in higher dimensions.   See Sections II.2 and II.3 for the generalization of incidence geometry to higher dimensional coordinate spaces and Section II.5 for an approach to generalizing the abstract incidence axioms (the latter is not really explicit in the mathematical literature; the standard mathematical treatments are somewhat more general).