(photos source: http://www.varini.org)

## Geometry (course 430)

### Time: Tuesdays, Thursdays at 2pm. Room: Mathematics Building , room B0427

Teacher: Y.A. Rubinstein. Office hours: Tuesday, Thursday 1-2pm, and by appointment.

Course plan:
There are many routes to Geometry. We will start by reviewing some high-school geometry going back to Euclid's original book, emphasizing coordinates and complex numbers as we go along. We will explore some beautiful theorems in Euclidean geometry. Then we will move on to explore the fascinating realms of spherical and hyperbolic geometry.

Main references:

J. Stillwell, The four pillars of geometry, Springer, 2005.

V. V. Prasolov, V. M. Tikhomirov, Geometry, American Mathematical Society, 2001.

J. Stillwell, Geometry of surfaces, Springer, 1992.

J. Stillwell, Sources of hyperbolic geometry, American Mathematical Society, 1996.

Additional references: TBA as we go along.

Assignments:
Homeworks on a weekly basis. Homework from the previous week due on the next Thursday in-class. No late homework will be accepted out of consideration to the grader. Each student's worst homework grade will be omitted for the computation of the final grade. There will be two midterms (October 9th, and November 20th, in class) and a final. Each of these and the homework will contribute a quarter of the final grade.

Schedule:

• August 30
Overview/syllabus/references. Initial quiz.

• September 4
Review of some high-school geometry. Compass and straightedge constructions. Descartes' approach.

• September 6
Trisecting angles, Cardano's formula, and complex numbers.

HW1.

• September 11
Complex numbers and Euclidean geometry, commensurability and continued fractions.

• September 13
Cosmic distances, the golden ratio.

HW2.

• September 18
Isometries of the Euclidean plane: group structure, reflections, rotations, translations, the Three Reflections Theorem.

• September 20
The Three Reflection Theorem continued, the subgroup of rotations and translations.

HW3 .

• September 25
The sphere: geodesics, great circles as lines, isometries.

• September 27
The sphere - continued. Stereographic projection.

HW4 .

• October 2
The sphere - inversions, reflections, conformality.

• October 4
The sphere - inversions, reflections, conformality (continued).

• October 9
Midterm 1: Euclidean and spherical geometry discussed in class and in HWs 1,2,3. This includes (but not limited to): Stillwell FPG Chapters 1-3, Stillwell GOS Chapter 1 and subsections 3.1-3.4 of Chapter 3. In grading, emphasis will be given to rigorous proofs.

Models of the Platonic solids created by the students.

Close-ups of some of the Platonic solids created by the students.

• October 11
Spherical triangles.

HW5.

• October 16
Writing rigorous proofs via a review of midterm 1.

• October 18
Reflections, inversions, and matrices. Stereographic projection continued.

HW6.

• October 23
Characterizing isometries of the sphere in terms of isometries of Euclidean space.

• October 25
The n-sphere. Isometries of the n-sphere and orthogonal matrices. Clifford translations.

HW7.

• October 30 (cancelled due to University closing)

• November 1
The sphere is not locally isometric to Euclidean space. Isometries viewed infinitesimally.

• November 6
Existence of a flat torus, locally isometric to the Euclidean plane, inside the 3-sphere. The notion of curvature. Hyperbolic space - motivation and definitions. Beltrami's pseudosphere.

HW8.

• November 8
The distance function on the hyperbolic line. The Beltrami-Klein disk model of hyperbolic space.

• November 13
Other models of hyperbolic space. Isometries of hyperbolic space.

• November 15
Geodesics and triangles in hyperbolic geometry. Tilings of the hyperbolic disk.

• November 15, 5pm, Mathematics Building room 1313 (note special time AND room)
Review for midterm 2, part 1.

• November 19, 4:30pm, Mathematics Building room 1313 (note special day AND time AND room).
Review for midterm 2, part 2.

• November 20
Midterm 2: Everything so far except hyperbolic geometry. HWs covered: up to HW8. Euclidean and spherical geometry, with emphasis on spherical geometry.

• November 27 - until end of semester
Various topics related to hyperbolic geometry.