Mathematics 153
Notes from April 20 - 24, 2020, lecture sessions

In order to catch up, we are going to discuss all three lectures in a single file.
What was covered:
Monday was devoted to Subunit 4X, Wednesday to Subunit 4Y, Friday to Subunit 4Z.
What else is new:
There are two new files in the main directory containing examples of short answer questions and possible responses:
http://math.ucr.edu/~res/math153-2020/short-answers.pdf
http://math.ucr.edu/~res/math153-2020/short-answers2.pdf  
This can be discussed in the lectures following Unit 5 if there is interest in doing so.
http://math.ucr.edu/~res/math153-2020/week0/baloney-detection.pdf   This is an article on critical thinking, a skill that must be developed in order to create and follow mathematical reasoning and is also valuable in nonmathematical settings.  Several years ago I ran a one credit course called Junk Logic which covered the principal points of this article and looked at a few (unfortunately, still) current examples.

Relevant directory files:
Here is a listing with comments on their significance.  These links are all clickable.  There is a common homework file:  
http://math.ucr.edu/~res/math153-2020/week4/unit04/homework04.pdf
Files for Subunit 4X.
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04X.pdf  This is the main file for the subunit 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/week4/unit04/history04a.pdf   Modern proofs (with coordinate geometry and calculus) for some results due to Archimedes and Apollonius (but they used other methods). 
http://math.ucr.edu/~res/math153-2020/week4/unit04/sector-area.pdf  This has a proof of the standard area formula for a circular sector (A  =  r2 q/2) using calculus.
http://math.ucr.edu/~res/math153-2020/week4/unit04/spiralregions.pdf  This is a drawing which shows some regions mentioned in the discussion of the spiral curve.
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.   Archimedes constructed a family of semiregular polyhedra which are closely related to the regular examples.
Files for Subunit 4Y.
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04Y.pdf  This is the main file for the subunit 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/week4/unit04/four%20conic%20sections.jpg  This is a drawing of the classical construction for a conic as the intersection of a cone with a plane. 
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04d.pdf   This file shows how one can show that the classical definition of conic sections as intersections of cones and planes leads to the modern definition as the zero set of a second degree polynomial equation in two variables.
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04b.pdf   This provides a comparison of the definition for tangents to conics in classical terms with the modern definition using calculus.
http://math.ucr.edu/~res/math153-2020/week4/unit04/evolute.pdf    This is a description of the evolute of a curve; the evolute of a conic plays an important role in determining the number of normals which can be drawn to a conic through an external point.  
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.   Archimedes constructed a family of semiregular polyhedra which are closely related to the regular examples.
http://math.ucr.edu/~res/math153-2020/week4/unit04/ellipse-2-and-4-normals.jpg   This file contains drawings of examples where the number of normals from an external point to an ellipse is equal to 2 or 4 (one point of each type).
http://math.ucr.edu/~res/math153-2020/week4/unit04/ellipse-3-normals.jpg   This file contains drawings of examples where the number of normals from an external point to an ellipse is equal to 3.
http://math.ucr.edu/~res/math153-2020/week4/unit04/normals-to-parabola.gif  This is an animated drawing which shows how the number of normals to a parabola changes from 1 to 2 to 3 (and vice versa) as one passes from one side of the evolute curve to another.
http://math.ucr.edu/~res/math153-2020/week4/unit04/more-on-normals.pdf   This is a copy of http://www34.homepage.villanova.edu/robert.jantzen/maple/ellipse/apollonius.pdf .
See also the following online site:
https://math.stackexchange.com/questions/609351/number-of-normals-from-a-point-to-an-ellipse  This file contains a detailed algebraic justification for Apollonius' results in the elliptic case.
http://math.ucr.edu/~res/math153-2020/week4/unit04/epicycles%20-%20Bing%20video.mp4   The YouTube video showing the Ptolemaic view of the solar system, with everything moving around the earth in orbits which are circles modified with epicycles (and nested epicycles and ... ).  It is a matter of personal taste whether the Twilight Zone music in the background adds to or detracts from the video itself.
http://math.ucr.edu/~res/math153-2020/week4/unit04/normals-to-parabola.gif  This is an animated drawing which shows how the number of normals to a parabola changes from 1 to 2 to 3 (and vice versa) as one passes from one side of the evolute curve to another.
http://math.ucr.edu/~res/math153-2020/week4/unit04/ellipse-reflection.pdf  This is a derivation of the reflection property for ellipses which was mentioned in the first document.
Files for Subunit 4Z.
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04Z.pdf  This is the main file for the subunit 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/week4/unit04/confracs.pdf      AND
http://math.ucr.edu/~res/math153-2020/week4/unit04/confracs0.pdf  These two files contain step by step descriptions of a procedure for finding the continued fraction expansion of a rational number.
http://math.ucr.edu/~res/math153-2020/week4/unit04/history04c.pdf   This file contains a proof that the continued fraction expansion of a rational number always terminates after finitely many steps.  It is included mainly for the sake of completeness.  
http://math.ucr.edu/~res/math153-2020/week4/unit04/stereographic.pdf  This file describes a basic one-to-one geometrical correspondence between the set of all points on a sphere with the north pole removed and the Euclidean plane.
http://math.ucr.edu/~res/math153-2020/week4/unit04/projstereo.pdf  This illustrates the flat map obtained by mapping the Northern Hemisphere to the plane by the stereographic projection described in the preceding file.
http://math.ucr.edu/~res/math153-2020/week4/unit04/stereo-conformal.pdf   Proof of a fundamentally important property of stereographic projections, first discovered by Hipparchus and proved rigorously 1800 years later by E. Halley (best know for the comet named after him).
http://math.ucr.edu/~res/math153-2020/week4/unit04/congruence-vs-similarity.pdf  This proves an AAA-area congruence theorem for triangles in Euclidean geometry, the goal being to provide an analogy with the AAA congruence theorem for spherical triangles in spherical geometry.