Intermediate Seminar:
Geometry, Projective, Spherical, Non-euclidean

Course Description

This class will focus on topics in synthetic geometry not much covered in high school, such as projective geometry, spherical geometry, and non-euclidean geometry of the hyperbolic plane. We will focus on doing in-class group work, and drawing geometrical constructions. We will also study and write some proofs.

This is a seminar class. Although, I will give some presentations, it will not be based on lectures, but rather on readings and in-class presentations and group discussions.

Objectives

Students who complete this class can expect the following results:

An appreciation of the constructive aspects of geometry

An understanding of mathematical proof in geometry

Practice with drawing geometric diagrams

Practice with working with others on mathematics problems

Practice with mathematics presentations in front of others

Required Texts

There will be two main textbooks from which we will take selections:

Hartshorne, R., 2000, Geometry: Euclid and Beyond (Springer: New York). (Selections can be downloaded below.)

Whittlesey, M.A., 2020, Spherical Geometry and its Applications (CRC Press: Boca Raton). (Selections can be downloaded below.)

Other readings can be downloaded from this website.

Suggested Reading

Brannan, D.A., Esplen, M.F., Gray, J.J., 1999, Geometry (CUP: Cambridge).

Grading

Participation (presentations and group-work)	30%
Weekly assignments	35%
Notebook	35%

General Format

The class meets once a week for a seminar discussion. Students are expected to do the readings and assignments before class, engage in classroom discussions, make in-class presentations of some of the material we study, and work on problems and proofs together in class. Students will keep a notebook for the class that will be graded at the end of the class. Students MUST bring paper (or their notebook) and writing utensils to each class. Students are encouraged to bring a compass and ruler to class for drawing geometric diagrams. (You can buy a cheap geometry set at every combini.)

Classroom Etiquette

Please follow basic norms of decorum – do not sleep, eat, or carry on individual conversations in class. Finally, DO NOT use mobile phones, smart phones, or laptops in class, unless this is specifically required for in-class groupwork. (Unfortunately, a large percentage of students use their laptops to do unrelated things during class, and this distracts both them and everyone around and behind them.) I will be very strict about enforcing the rule about devices and laptops, so if you feel that you must use devices, I encourage you to enroll in a different class.

Assignements

Starting from Week 2, there will be an assignment each week, which must be written out in full and handed in at the end of the class for that week. The assignment can be hand written, or printed out, but you should not work on it in class. The assignment will usually involve preparing a proof or argument from the text, and you should be prepared to present your work in class on the board during the session for each week.

Notebook

Students will be expected to keep a notebook for this class, in which they write out their notes, diagrams, and rough work in listening to the presentations and in reading through the reading assignments. This must be kept on paper, either loose leaf pages, ringed, or bound. This workbook will be marked at the end of the term. You do not need to get everything correct in the notebook, but you should make an effort to record your work as much as possible. The assignments can be put into the notebook after they are returned to you.

Discussion Topics, Readings and Assignments

Week 1: Oct 7

Introduction: Review of basic and solid geometry

Reading: Whittlesey, M.A., “Review of three-dimensional geometry” (focus on sections 1 and 2).

Week 2: Oct 14

Projective geometry: Cross-ratios

Reading: Leonard, I.E., Lewis, J.E., Liu, A.C.F., Liu, Tokarsky, G.W., “Cross Ratio”. Make sure you understand the meaning of cross ratios of segments (p.412) and of pencils of lines (p.417-418). Focus on Theorem 15.1.7 (p.420-422), Theorem 15.2.5 (Desargue’s Theorem), and Theorem 15.2.9 (Pappus’s Theorem).

1st Assignment: Write out the diagram and argument for Pappus’s Theorem, Theorem 15.2.9 (pp.430–430), giving a full justification for each step, explaining what previous propositions (theorems or lemmas) are used, and explaining the thought process as much as you can. You should be prepared to present your work in class on the board.

Week 3: Oct 21

Projective geometry: Circular inversion

Reading: Hartshorne, R., “Circular Inversion”. Make sure you understand the definition of circular inversion (p.334). Focus on Propositions 37.2, 37.3, 37.5, and 37.6.

Supplementary material (optional): Geogebra applets on circular inversion 1 (various), circular inversion 2 (shapes), circular inverstion 3 (a circle).

2nd Assignment: (A) Let there be two points A and B, of which their circular inversions through circle Γ about center O are points A' and B'. Use the property of circular inversion to show that △ABO is similar to △A'B'O. Be prepared to present your work in class on the board. (B) Write out the full argument for Proposition 37.6 (pp.339-341), giving a full justification and explaining as much as you can. Be prepared to present your work in class on the board. In this case, the presentation does not need to be as complete as your written work.

Week 4: Oct 28

Spherical geometry, I (solid geometry)

Reading: Whittlesey, M.A., “The Sphere in Space”. Focus on sections 5 (pp.25-35) and 8 (pp.41-44). Make sure you understand the various definitions and work through the arguments.

3nd Assignment: (A) Do exercise 5.4 (p.35). Namely, use 3d geometry to prove Proposition 5.8, that a small circle is the of all points at a fixed spherical distance from one (or both) of its poles. (B) Do exercise 5.6 (p.35). Namely, use 3d geometry to show that given three points that do not lie on a unique great circle, they must lie on a unique small circle. (C) Write out the full argument for Theorem 8.1 (pp.42-44), giving a full justification and explaining as much as you can. (Remember that in a right triangle, the sin of an angle is the opposite side over the hypotenuse, and the cosine is the adjacent side over the hypotenuse.) Be prepared to present your work in class on the board.

Holiday: Nov 4 (早稲田祭)

No Class

No Reading.

Week 5: Nov 11

Spherical geometry, II (axiomatic)

Reading: Whittlesey, M.A., “Axiomatic Spherical Geometry” (sections 9 and 10, pp.47-71, but SKIP Def.9.38 to the end of section 9, pp.55-59). Focus on the axioms (A-1, A-2, A-3, A-4, A-5, A-6, A-8). Try to understand these axioms and to see how they are useful for doing proofs.

Supplementary Reading (Optional): Sidoli, N., “Constructions and Foundations in the Spherics of Theodosios and Menelaos” (on p.69-72 I discuss how Menelaus proved the equivalent to Axiom 10.18 (A-8), using solid (3D) geometry).

4th Assignment: (A) Read through Prop.10.9 (pp.63-66), and consider the fourth condition (4); namely, that for any point Q on ray BA, there is a point R on ray BC such that P lies on arc QR. Try to draw this configuation on a sphere and compare it to a similar configuration on the plane. Explain why condition (4) fails on the plane. (B) Think about Axiom 10.18 (A-8). Why do you think this has to be taken as an axiom in axiomatic spherical geometry? How might we prove it using 3D geometry?

Week 6: Nov 18

Spherical geometry, III (axiomatic)

Reading: Whittlesey, M.A., “Axiomatic Spherical Geometry” (sections 11 through 14, pp.72-110, but SKIP (or skim) Def.11.20-Def.11.21 (pp.80-82), Def.13.8-Prop.13.12 (pp.97-100), and just scan through all the excercizes. Focus on Def.11.16, Th.11.19 (polar spherical triangles), Th.12.5, Th.13.1, Th.13.2, Th.13.7., Axioms 14.1-3, Th.14.4). Try to understand these definitions and propositions and think about the ways in which spherical triangles are different from plane triangles.

5th Assignment: (A) Read through Prop.11.11 (p.74) and write out the proof with a diagram. Be prepared to present your work in class on the board. (B) Read through Prop.13.3 (pp.94-95) and write out the proof with a diagram. Follow thorugh each of the steps and be sure you understand the role of the polar triangle. Be prepared to present your work in class on the board.

Week 7: Nov 25

Projecting the sphere

Reading: Rosenfeld, B.A., Sergeeva, N.D., Stereographic Projection (Chaps. 1 and 2 (pp.11-23), as well as Chaps. 6 and 7 (pp.40-46)). Work through the various geometrical arguments, and try to understand the 3D geometry being described. See if you can drawn diagrams to help yourself understand.

Supplementary material (optional): Geogebra applets on Stereographic projection 1 (a point), Stereographic projection 2 (circles), Stereographic projection 3 (angles), Stereographic projection 4 (spherical coordinates).

Supplementary reading (optional): Ptolemy's Planisphere, a Arabic translation of a 2nd century Greek work on the stereographic projection of the sphere.

6th Assignment: (A) Read through the “Lemma” on pp.11-12 and write out the proof with a diagram. Be prepared to present your work in class on the board. (B) Read throught the discussion of transformations on p.21. Write out a full discription of the set of “two consecutively performed operations” and the set of “four consecutively performed transformations,” and provide a diagram, or multiple diagrams, to help explain the situation. Be prepared to present your work in class on the board.

Week 8: Dec 2

The ancient analemma

Reading: Sidoli, N., “Mathematical Methods in Ptolemy’s Analemma” (in Section 5, do not worry about all of the mathematical details, just make sure you understand the core argument or the significance of the method described). We will discuss the reading in class.

Supplementary reading (optional): A possible use of the analemma model to compute certain values in spherical astronomy claimed by Hipparchus: Sidoli, N., “Hipparchos and the Ancient Analemma”.

Week 9: Dec 9

Neutral geometry

Reading: Hartshorne, R., “Neutral Geometry”. Focus especially on Theorems 34.5 and 34.7, and Propositions 34.9, 34.10 and 34.11. What do these last three theorems together entail?

7th Assignment: (A) Read through Theorem 34.5 (p.309) and write out the argument with a diagram. (What do you think the paragraph that begins “Finally...” means?) Be prepared to present your work in class on the board. (B) Read through Lemma 34.8 and write out the argument with a diagram. Be prepared to present your work in class on the board. (C) Read excercises 34.9, 34.10 (p.317), make sure you understand the claims, and convince yourself that the claims are sound.

Week 10: Dec 16

Hyperbolic geometry, I

Reading: Hartshorne, R., “Hyperbolic Geometry”. Focus especially on Props. 40.6 and 40.9, and think about how these are different from the situation in the Euclidean plane.

8th Assignment: (A) Read through Corollary 40.3 (p.376) and write out the argument with a diagram. Be prepared to present your work in class on the board. (B) Read through Prop. 40.8 (p.380) and write out the argument with a diagram. Be prepared to present your work in class on the board. (C) Read excercises 40.2 (ALL, p.384), 40.5 (LLL, p.85), make sure you understand the claims. Can you prove these propositions?

Week 11: Dec 23

Axiomatic plane geometry

Reading: “Hilbert's Axioms”. Focus on the axioms themselves, I1-I3 (p.66), B1-B4 (pp.73-74), C1-C6 (p.82, 90-91). Read through the everything, but do not worry about working through the details of the proof.

Holiday: Dec 30

No Class

No Reading.

Week 12: Jan 6

The Poincare disk, I

Reading: “TBA”.

Holiday: Jan 13 (成人の日)

No Class

No Reading.

Week 13: Jan 20

The Poincare disk, II

Reading: “Compass and Straighedge in the Poincaré Disk”. Focus on the constructions in Sections 2 and 3.

Week 14: Jan 27

Conclusion and presentations

No Reading.