Announcements

I will put announcements about the class in this space. Please check here periodically as the term progresses.

First Year Seminar:
Geometry, Euclid and Others

Course Description

In this course, we will explore some of the historical contexts of Greek mathematics, read part of Euclid’s Elements of Geometry, some of Lobachevski’s Theory of Parallels, and some of Hilbert’s Foundations of Geometry. We will be focusing on the techniques of proving propositions and will go through the arguments in detail. Students taking this class will develop an understanding of some basic geometry, and appreciation of logically structured arguments, be exposed to some of the ideas of formalized axiomatic geometry, as well as non-Euclidean geometry. Students will develop experience presenting mathematics in front of others.

Required Texts

Euclid: Fitzparick, R., trans., 2008, Euclid’s Elements of Geometry.

Hilbert: Unger, L., trans., 1971, Hilbert’s Foundations of Geometry.

Lobachevski: Halsed, G.B., trans., 1914, Lobachevski’s Theory of Parallels.

Grading

Classroom presentations	40%
Notebook	30%
Active participation	30%

General Format

The class meets once a week for a seminar discussion. Students are expected to do the readings before class, make in-class presentations of the material in the readings, and keep a study notebook about the readings that can be used for their presentations. The student notebooks will be graded at the end of the semester. Students MUST bring paper (or their notebook) and writing utensils to each class. Students are encouraged to use a compass and ruler 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.

Presentations

Starting from Week 3, students will make presentations of the reading material on the white board. Students can use their own written notes during the presentations, but they should not use the readings themselves.

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.

Topics, Readings and Assignments

Week 1: Apr 14

General Introduction

Readings: Sidoli, N., Mathematics education.

Week 2: Apr 21

Introduction to Greek mathematics

Readings: Asper, M., The two cultures of mathematics in ancient Greece.

Week 3: Apr 28

Euclid’s Elements, I

Reading: TBA.

Supplementary material: For Euclidean constructions see, Euclid: The Game! (It sometimes takes a long time to load.)

Holiday: May 5　(子供の日)

No Class

No Reading.

Week 4: May 12

Euclid’s Elements, II

Reading: To be announced (TBA).

Week 5: May 19

Euclid’s Elements, III

Reading: TBA.

Week 6: May 26

Euclid’s Elements, IV

Reading: TBA.

Week 7: Jun 2

Euclid’s Elements, V

Reading: TBA.

Week 8: Jun 9

Euclid’s Elements, VI

Reading: TBA.

Week 9: Jun 16

Euclid’s Elements, VII

Reading: TBA.

Week 10: Jun 23

Non-Euclidean Geometry, I

Reading: Nicholas Lobachevski’s Theory of Parallels pp. 11-16 (up to section 19).

Supplementary Reading: Hartshorne, R., Geometry: Euclid and Beyond Chap. 7) (pp. 373-387).

Supplementary material: See Tibor Marcinek’ss Geogebra website of the Poincaré Disk for a model of the hyperbolic plane.

Week 11: Jun 30

Non-Euclidean Geometry, II

Reading: Nicholas Lobachevski’s Theory of Parallels pp. 16-21 (from section 19).

Supplementary Reading: Hartshorne, R., Geometry: Euclid and Beyond Chap. 7 (pp. 373-387).

Week 12: Jul 7

Hilbert’s Geometry, I

Reading: David Hilbert’s Foundations of Geometry Chap. 1 (pp. 1-7).

Supplementary Reading: Hartshorne, R., Geometry: Euclid and Beyond Chap. 2 (pp. 65-81).

Week 13: Jul 14

Hilbert’s Geometry, II

Reading: David Hilbert’s Foundations of Geometry Chap. 1 (pp. 7-15, 25-26).

Supplementary Reading: Hartshorne, R., Geometry: Euclid and Beyond Chap. 2 (pp. 81-96); Chap. 7 (pp. 295-300).

Week 14: Jul 21

Presentations and discussions

No Reading.