# Math 618

(Last Updated May 25, 2022 @ 3pm)

## 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.

## End of the semester (5/16–5/23)

• Monday, May 16 is the last day of class. We will spend the time working on problems from Assignment 12.

• Exam 2 is on Monday, May 23 from 4-6PM and will be in our normal class room, 320 Kiely Hall.

• Exam 2 will not be cumulative; in particular, Exam 2 will cover the entirety of our notes on hyperbolic geometry.

• Exam 2 will be approximately the same length as Exam 1 and will consist of the same format, but with one notable exception: Please bring a graphing calculator to the exam.

• You may bring two standard-sized sheets of notes to the exam. You may write on the front and back of the sheets.

• I will hold my normal (virtual) office hour this week on Thursday from 4-5pm.

• Exam 2 title page and reference sheet

• Solutions to Exam 2

## Week 14 (5/9–5/13)

Discussed hyperbolic trigonometry, lambert quadrilaterals, and right-angled hexagons.

### Videos

This video should be watched after class on Monday, May 9 and before class on Monday, May 16.

Classify conformal hyperbolic rigid motions.

## Week 13 (5/2–5/6)

Proved the formula the area of a triangle in terms of its angles. Since many students were absent on account of the holiday, here is a link to the live video recording from last year's online course

### Videos

These videos should be watched after class on Monday, May 2 and before class on Monday, May 9.

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.

## Week 12 (4/25–4/29)

Proved that hyperbolic geometry is a non-Euclidean absolute geometry. Characterized hyperbolic reflections.

### Videos

These videos should be watched after class on Monday, April 25 and before class on Monday, May 2.

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.

## Week 11 (4/11–4/15)

Characterized all hyperbolic geodesics in the hyperbolic plane.

### Videos

These videos should be watched after class on Monday, April 11 and before class on Monday, April 25.

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.

## Week 10 (4/4–4/9)

Introduced the hyperbolic plane and showed that vertical line segments are geodesic.
• Exam 1 returned Wednesday, April 6

To understand your grade, divide your score by 10 and then use the result as a GPA to figure out the corresponding letter grade.

### Videos

These videos should be watched after class on Monday, April 4 and before class on Monday, April 11.

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.

## Week 8 (3/21–3/25)

Characterize the orthogonality of circles via action of inversions.

### Videos

These videos should be watched after class on Monday, March 21 and before class on Monday, April 4.

Review how we identify the Euclidean plane with R^2 and the definition of arc length from calculus.

## Week 7 (3/14–3/18)

Went over explicit examples computing images of lines and circles under inversions in a circle.

### Videos

These videos should be watched after class on Monday, March 14 and before class on Monday, March 21.

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.

## Week 6 (3/7–3/11)

Proved some basic properties about tangent lines to circles and recalled the definition of similar triangles and a basic proposition.

### Videos

These videos should be watched after class on Monday, March 7 and before class on Monday, March 14.

Introduce inversions in circles.
Prove a theorem characterizing the images of lines and circles under inversions in circles.

## Week 5 (2/28–3/4)

Proved that the composition of two distinct reflections is either a translation or a rotation.

### Videos

These videos should be watched after class on Monday, February 28 and before class on Monday, March 7.

Proved that the product of three reflections is either a reflection or glide reflection.

## Week 4 (2/21–2/25)

• No class Monday, February 21 (college closed in observation of Presidents' day)

### Videos

These videos should be watched after class on Wednesday, February 23 and before class on Monday, February 28.

Proved that every translation is the composition of two reflections about parallel lines. Introduced rotations.

## Week 3 (2/14–2/18)

### Videos

These videos should be watched after class on Monday, February 14 and before class on Wednesday, February 23.

• Video 5 (Notes) [This video is a recording of a live session with students, so there a few questions from students that I dress towards the end.]

We proved that every Euclidean rigid motion is the composition of at most three reflections.
Introduce translations and basic properties.

## Week 2 (2/7–2/11)

We proved that parallel lines exist in every absolute geometry. We then proved the first of Euclid's theorems requiring his 5th postulate; we then discussed Playfair's postulate and proved that the sum of the angles in a triangle is pi.

### Videos

These videos should be watched after class on Monday February 7 and before class on Monday, February 14.

Introduced the notion of rigid motions.
We show that rigid motions preserve lines and are completely determined by their behavior on any one triangle.

## Week 1 (1/31–2/4)

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 proved Euclid's first theorem from the Elements.

### Videos

These videos should be watched after class Wednesday, February 2 and before class on Monday, February 7.

Introduce the notion of congruence and two congruence theorems (side-angle-side and side-side-side).
Introduce the notion of parallelism and establish the Euclidean theorems about parallel lines that hold for all absolute geometries.