# Edwin O'Shea

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## Description, Syllabus, etc for Math 475, Fall 13

• This course is concerned with the foundations of geometry: Euclidean, Coordinate, Projective and Non-Euclidean. We will be reading Euclid's Elements and Stillwell's The Four Pillars of Geometry with occasional interludes from other sources.
• A detailed syllabus is here.
• This beautiful Euclid's Elements in Color by Oliver Byrne was published in 1847. It bankrupted the publisher.
• Here is an online copy of the first few pages of the text, for those of you still waiting for your text in the post.
• Here is an online copy of Elements.
• On the Fifth Pustulate: Heath and Grabiner.

## Homework

Students will be expected to present the propositions on the board. Presenters will be chosen randomly. You may call on the help of other students.
• For 8/27: Memorize all definitions, postulates and common notions from Book I. Read and understand Proposition 1.
• For 8/29: Memorize all definitions, postulates and common notions from Book I. Read and understand Propositions 1--4 inclusive.
• On Book I, Proposition 1 (or [I.1]): State a condition that you think is necessary and sufficient for two circles to intersect.
• On [I.2]: This is sometimes referred to as "the compass has memory." Why do you think this is so?
• On [I.4]: Is it odd that there are no postulates (the geometry axioms) used in this proof, only common notions? Do you recognise this proposition as an old fact from high school geometry?
• For 9/1: Memorize again all definitions, postulates and common notions from Book I. Read and understand Propositions 4--7 inclusive.
• On [I.4]: Is it odd that there are no postulates (the geometry axioms) used in this proof, only common notions? Do you recognise this proposition as an old fact from high school geometry?
• For further discussion on [I.4] and [I.5], read the first two pages of Section 2.2 of Pillars. Why do you think Pillars refers to the "SAS axiom"? Have we encountered a need for a new axion before in Euclid? It also asks if SSA (the angle not between the two sides) is a sufficient criterion for congruence? Is it? If not, provide a counterexample.
• How is [I.6] related to [I.5]? We are encountering a new type of proof here. What is it?
• For 9/3: Read and understand Propositions 5--12 inclusive. As always, talk to classmates about these.
• On [I.7] (and on almost all propositions from here on) it will help a great deal to mark off angles and different triangles in different colors.
• On [I.8], Do you recognise this proposition as an old fact from high school geometry? How does it compare to [I.4]?
• On [I.9-10], can you generalise bisecting (cutting into two equal parts) to cutting into 4 equal parts? Into 3 equal parts (trisecting)? See Exercise 1.3.4 of Pillars.
• For 9/5: Read and understand Propositions 13--17 inclusive. As always, talk to classmates about these. Test 1 will be today and will cover everything from the first two weeks, including this HW for 9/5.
• For 9/8: Read and understand [I.18]--[I.26] inclusive. As always, talk to classmates about these.
• Revise [I.16] again and notice how it is used time and time again in these later propositions.
• How are [I.18] and [I.19] related?
• In what sense could [I.20] and [I.22] said to be converses? How does [I.22] remind you of the additional axiom needed in proving [I.1]?
• Why does [I.23] tell you that Euclid may not have believed in motion, in the sense of [I.4]?
• In [I.26], does the Side need to be ``between'' the two Angles? With this in mind, disprove the other collection of collections of equality in three of the six sides and angles sufficing for congruence of triangles.
• Related to triangle congruence, can you conjecture conditions for four sided figures to be congruent?
• For 9/10: Read and understand [I.27]--[I.34] inclusive. As always, talk to classmates about these.
• Note that we are introducing and using Def. 23 and Post. 5 for the first time.
• Read Sections 2.1 of Pillars. Complete Exercises 2.1.1-2.1.5 of Pillars.
• Read the following very interesting commentary (by Heath) on Postulate 5. What does Ptolemy's "proof" attempt to achieve, and why does it not do so.
• On the topic of the fifth postulate, please read this beautiful exposition, if only for the acknowledgement to the author's high school teacher.
• Per our discussion in class, I wish to amend the following to the tests: They will be every second Wednesday, starting on Wednesday next, 9/17. They will each be for 25-35 mins. There will still be a short test on the Friday before Thanksgiving.
• For 9/12: Read and understand [I.35]--[I.39] and [I.46]--[I.48] inclusive. Understand the statments only of [I.40]--[I.45]. As always, talk to classmates about these.
• Note that in Elements, there is no mention of multiplying the length of one side by another, as we have become accustomed to in computing area. So what are [I.35]--[I.39] really saying about area and, alonmg the same lines, what is [I.47] -- The Pythagorean Theorem -- really saying?
• Read Sections 2.4 and 2.5 of Pillars. Complete Exercises 2.4.1 and 2.5.2--2.5.5 inclusive of Pillars.
• For 9/15: Read and understand the following, making sure to chat with classmates about these:
• Read and understand all propositions and associated exercises assigned on 9/12.
• Read Section 2.3 of Pillars and complete all exercises in that section.
• Read the following commentary and interpretation of this link on [II.12], [II.13] and [II.14].
• Read this link on [II.14]. Why is that the Pythagorean Theorem plays a crucial role in "squaring" a polygon? In what context have you heard the term "squaring the circle" and, in light of this reading what do you think is meant by the expression?
• For 9/17: Test 2 is today, on all material covered since Test 1.
Book III is quite beautiful with a range of results on properties of circles. Read and understand the new definitions at the start of Book III and then study [III.1]--[III.10] inclusive**.
• ** Understand the statements only of [III.7], [III.8],
• How would [III.1] influence you when cutting a cake with a (sharp steel) straight edge? Just a thought: Would you say there is a poetry to its statement, the center being perpendicular to the random?
• How are [III.3] and [III.4] related? And [III.5] and [III.6]?
• Are there statements in the reading that are easier or harder to prove in the usual coordinate/analytic geometry context? For example, [III.10]?
• For 9/19: Read and understand the new definitions at the start of Book III and then study [III.11]--[III.22] inclusive**.
• ** Understand the statements only of [III.14]--[III.16].
• Read and complete the exercises of Section 2.7 of Pillars. Note that the exercises give an another narrative for "squaring" a polygon. The section also allows you to construct right angled triangles with a given hypothenuse. Describe how so.
• If you were designing stained glass windows for a church, who would you be influenced by [III.12]?
• Is [III.12] easier to prove in the coordinate context? Are [III.18] and [III.19] familiar from the calculus? How are they proved there?
• For 9/22: Re-read [III.1]--[III.22] following our discussion today. Read 2.7 of Pillars. Complete exercises 2.7.1--2.7.5 of Pillars. In addition, try to prove the following:
• [III.10] but using coordinate geometry,
• [III.7] and [III.8] in one swoop using the calculus,
• [III.18] using the calculus.
• For 9/24: Read and understand [III.23]--[III.29], [III.32] and [III.37]. Be sure to understand propositions that are used. For example, [III.31] is used in [III.32] so be sure to, at least, understand the statement of that proposition, even if it was not assigned.
• For 9/26: Read and understand [IV.11] and [IV.15] and all propositions used to carry out these constructions. For example, [IV.11] uses [IV.10], which calls upon [IV.5]. You must understand the statements and proofs of [IV.10] and [IV.5]. As motivation, read the exercises of Pillars 1.1.
• For 9/29: Similar triangles and Thales Theorem
• Read Sections 1.3, 1.5, 2.6 and 2.8 of Pillars. Be sure to understand the proof of Thales Theorem (and compare it to [VI. 1 and 2]) and the new proof of the Pythagorean Theorem.
• Do the following exercises from Pillars, and address related questions:
• (Trisecting a line -v- an angle) 1.3.5, 1.3.6. In what sense is bisecting a line and an angle the same?
• (Converse to Thales) 1.4.1, 1.4.2. In addition, after understanding 2.6, interpret [VI.1]--[VI.6] from Euclid.
• (A sketch of an alternative construction of the regular pentagon) 2.8.1, 2.8.2, 2.8.3. How is this construction related to [IV.11]?
• For 10/1: Test 3 is today. Topics are al topics covered in Book III (inlcuding [III.1]--[III.6]), IV and on Thales Theorem.
• For 10/3: π is a constant! Read [XII.1] and [XII.2].
• For 10/6: Re-read [XII.1] and [XII.2] following our discussion today. Read 3.1--3.4 of Pillars. Complete all exercises from those sections, except 3.3.2--3.3.5. In addition, try to prove the following:
• What's the connection between slope and Thales Theorem? (3.2)
• How is 3.3.1 connected to [III.9]? (3.3)
• If trisecting an angle needed the computation a cubed root who would this effect your viewpoint of the possibility of an angle being trisected in the Euclidean context? (3.4)
• For 10/8: Read Sections 3.5(all) and 3.6 (but not glide reflections). Do Exercises 3.5.1--3.5.5 and 3.6.1--3.6.4
It might help to think about how one would define the slope of a line using tan. For the reflections, can you say how a translations on the real line (as opposed to plane) would be a combination of two reflections. Is the rule the same as that for 3.6.3. in the plane?

Note: Test for 10/15 is postponed till 10/17.
• For 10/10: Read remainder of Section 3.6 (that on glide reflections). Do Exercises 3.6.5--3.6.7. In addition, verify the following, connected with glide reflections:
• The only lines preserved by a reflection is the line of reflection itself.
• The only lines preserved by a translation are those lines in the same direction as the translation itself.
• There are no lines preserved by a rotation (unless the rotation is a multiple of π ; which would make such a rotation what?)
• Use the first two parts to classify lines that are preserved by glide reflections.
• For 10/13: Read Section 3.7 and do all exercises in that section.
• For 10/15: Re-read all topics from Sections 3.6 and 3.7.
• For 10/17: Test 4 today, on all topics covered since last test, including Section 3, π being a constant and angle trisection.
• For 10/20: Gentle post-test reading and exercises.
• Revisit Exercises 3.5.3 and 3.5.4 (and finally revisiting Katie's question in class...) by reading just the "Rotation matrices" part of Section 4.7. For these rotation matrices, what is the point being rotated about?
• Describe a matrix A such that A sends (x,y) to its reflection in the y-axis, namely (-x,y). Given that reflections are their own inverses, what can you say about the matrix A and its inverse matrix.
• Thinking in terms of matrix multiplication between this A and the rotation matrices, describe, for any line L that passes through the origin, a matrix B such that B.(x,y) equals the reflection of (x,y) through L. Again, what is the inverse of B?
• Why did we insist that L passes through the origin? Can we create a reflection matrix for lines not passing through the origin? Where does that leave us with matrices for translations?
• Read Section 5.1 and 5.2 and do the exercises. Be sure to play with construzione legittima in 5.1 and the examples of drawing with straight edge alone in 5.2.
• For 10/22: Read Sections 5.3 and 5.5 and do all exercises in Section 5.5. It really helps if you draw lots of example pictures when reading these sections. Note too, in maps like Figure 5.15, you are mapping one line to another by projecting one line to the other from the point of perspective O.
• For 10/24: Read Sections 5.6 and 5.7 and do all exercises in those sections.
• For 10/27: Read Sections 5.8 and 5.9 and do all exercises in Section 5.8.
• For 10/29: Test 5 is today, on all topics covered since last test: Chapter 5 and the discussion of linear transformations and shifts as isometries in the plane.
• For 10/31: Booo! Recall the definition of a group. Read and do all exercises in Sections 7.1 and 7.2. This is mostly recap. Also, attempt once again (as in Test 4) to classify all isometries of the real line. In response to Ex.7.1.3, attempt to classify all reflections in the plane for which r and s such that rs = sr.
Don't forget: There will be a test on Friday 11/21!
• For 11/3: Read and do all exercises in Sections 7.4 and 7.5 and re-read and re-do all of 7.1 and 7.2. Attempt again to classify all isometries of the real line. What are the even isometries? Ignore Section 7.3 for now and ignore question 7.2.4.
• For 11/5: Read and do all exercises in Section 8.1. Re-read this link again Grabiner but with the hard earned viewpoint earned from studying projective geometry. I know you've already read it but the payoff in reading it again will be immense, I promise. Re-read and re-do all exercises in Sections 7.4 and 7.5. On 8.1: The discussion proposes a paradigm in which the fifth postulate does not hold but where the first does. If the second postulate were to hold, then what must be the "length" of a "semicircle?"
• For 11/7: Read and do al exercises from Sections 8.2 and 8.3.
• For 11/10: Read and do all exercises in Sections 8.4 and re-read and re-do all of 8.2 and 8.3.
• For 11/12: Read and do all exercises in Sections 8.5.
• For 11/14: Read and do all exercises in Sections 8.6. In addition, for understanding 8.5 further, confirm that the angle between the two "lines" (semicircles) of radius 2, one centered at 0 the other at 2, is preserved under the generating transformations.
• For 11/17: More on 8.5 and 8.6
• For 11/19: Review.
• For 11/21: Test 6 is today, covering all topics since Test 5.
• For Thanksgiving Week: Read and think about Section 8.7 and 8.8. Review is here.
• For the last week of class: We will have oral exams in pairs this week in preparation for the oral finals in finals week.