Math 618


(Last updated Wednesday, May 5 at 6pm)

Meetings

The first class meeting will be on Monday, February 1 at 4:40pm.

After the first week of class, we will use Monday's class time as an office hour meeting 4:40p–5:40pm and we will have a live class session on Wednesdays 4:40–5:40pm.

All meetings will occur on Zoom (Zoom Link).

Materials

A gamification of Euclidean geometry. It is a lot of fun and quite challenging. It is a nice introduciton to some of the Euclidean geometry we will discuss in the first part of the semester. It also serves as a nice visualization for proofs via construction, which is how the Greeks thought of much of mathematics.

Week 13 (5/3--5/7)

Prove that if two hyperbolic triangles are equiangular then they are congruent. Prove that any triple of non-negative real numbers whose sum is less than pi can be realized as the angles of a hyperbolic triangle.Discussed hyperbolic trigonometry, lambert quadrilaterals, and right-angled hexagons.

Week 12 (4/26--4/30)

Introduce tools for computing angles between hyperbolic geodesic, and use the to compute the angles in an example hyperbolic triangle. Review basic multivariable calculus and introduce the notion of hyperoblic area.Introduced ideal points, ideal triangles, and computed the area of a hyperbolic triangle as a function of its angles. Showed that the sum of the angles in a hyperbolic triangle is always less than pi.
The hyperbolic plane made by crocheting.

Week 11 (4/19--4/23)

Discuss calculating distance in the hyperbolic plane and prove that every Euclidean circle in the upper half plane is a hyperbolic circle. Prove that every hyperbolic circle is a Euclidean circle.Proved that hyperbolic geometry is a non-Euclidean absolute geometry. Characterized hyperbolic reflections.

Week 10 (4/12--4/16)

Introduce the notion of hyperbolic rigid motion and give some examples. Show that the inversion of a circle centered at the origin is a hyperbolic rigid motion.Characterized all hyperbolic geodesics in the hyperbolic plane

Week 9 (4/5--4/9)

  • On account of spring break and wanting to start hyperbolic geometry during a live session, there is only one pre-recorded video for this week.

  • Video 1 (Notes) --- Watch by Wednesday, April 7

Review how we identify the Euclidean plane with R^2 and the definition of arc length from calculus. Introduce the hyperbolic plane and compute the distance between certain points in the hyperbolic plane.

Week 8 (3/22--3/26)

Prove that inversions negate angles between two lines, two circles, and a line and a circle.Prove a proposition about circles that we will use in class.Characterize the orthogonality of circles via action of inversions.

Week 7 (3/15--3/19)

Introduce inversions in circles.Prove a theorem characterizing the images of lines and circles under inversions in circles. Went over explicit examples computing images of lines and circles under inversions in a circle.

Week 6 (3/8--3/12)

  • Many students are having/had exams around this time, so I thought I would only post one video for this week. It will also allow us to make our transition to a new topic together in the live session.

  • Video 1 (Notes) --- Watch by Wednesday, March 10

Prove that the product of three reflections is either a reflection or glide reflection. Proved some basic properties about tangent lines to circles, and remember the definition of similar triangles and a basic proposition.

Week 5 (3/1--3/5)

Introduce translations and basic properties. Proved that every translation is the composition of two reflections about parallel lines. Introduced reflections. Skip Exercise 1: The definition of translation I gave in the notes is inadequate for guaranteeing translations are rigid motions. The easiest fix is to just assume from the beginning that translations are rigid motions (so in the definition I gave, simply replace the word “transformation” with “rigid motion” and all is good). With this definition, Exercise 1 on Assignment 4 no longer makes sense as a problem, so please skip it. Proved that a Euclidean rigid motion is the composition of two distinct reflections if and only if it is either a translation or rotation.

Week 4 (2/22--2/26)

We show that rigid motions preserve lines and are completely determined by their behavior on any one triangle.Introduce reflections and prove that they are rigid motions. We proved that every Euclidean rigid motion is the composition of at most three reflections.

Week 3 (2/15--2/19)

  • Schedule Change Notice: The college is closed on Monday in observance of President's day. Office hours on Monday are cancelled, but Wednesday's class time will be treated as an office hour. There is only one video for this week, since we would have lost a day during a regular in-person class.

  • Video 1 (Notes) --- Watch by end of the week

Introduced the notion of rigid motions.

Week 2 (2/8--2/12)

Introduce the notion of congruence and two congruence theorems (side-angle-side and side-side-side).Introduct the notion of parallelism and establish the Euclidean theorems about parallel lines that hold for all absolute geometries. We proved that parallel lines exist in every absolute geometry. We then proved the first of Euclid's theorem's requiring his 5th postulate; we then discussed Playfair's postulated and proved that the sum of the angles in a triangle is pi.

Week 1 (2/1--2/5)

As briefly mentioned in class, Euclid's axioms are incomplete. In 1899, Hilbert wrote a book giving a more rigorous and complete axiomatic approach to geometry. Hilbert requires 20 axioms in his approach. Read more on Wikipedia: https://en.wikipedia.org/wiki/Hilbert's_axioms.Went over basic definitions and concepts for Euclidean geometry. Listed and discussed Euclid's axioms and proofed Euclid's first theorem from the Elements.