# Modern Geometry

## Calendar

This is a tentative schedule that may be changed based on the needs of the students, the needs of the instructor, or any apocalyptic events that may occur during the semester.

### 1:

 Monday Labor Day - no classes Wednesday Introductions groups ~by geometry terms (incomplete)~ Syllabus scavenger hunt and HHH 318 wikispaces: user/user or user330/user330 about our group four truths and a lie what is geometry? what is modern geometry? picture (rated G and read about fair use before posting) favorite equations (using [[math]]) Thursday Read $$\sqrt{2}$$ is irrational (p. 6) LaTeX for "pretty" math LaTeX using widgets proof by contradiction Prove $$\sqrt{2}$$ is irrational Wikibook authors complete proof for $$\sqrt{2}$$ is irrational (no picture required) Friday Read appendix A.3 (p. 424-427) and 10 ways to think like a mathematician Proof tips Starting Assumptions: dot, line, plane, line vs straight line; intersect vs parallel Three point geometry (7.6) examples & non-examples how many lines?

### 2:

 Monday finish proof Three point geometry (7.6) finish "how many lines?" every line is on __ points? Wikibook authors complete proofs for Three point geometry Wednesday Four point geometry (7.6) WLOG examples & non-examples how many lines? Wikibook authors complete proof(s) for Four point geometry Thursday Group quiz Friday Fano's geometry (7.6) example & non-examples how many points? Wikibook authors complete proof(s) for Fano's geometry

### 3:

 Monday Playfair's axiom which lines are parallel in the finite geometries? in which geometries is the postulate true? Start Euclid & GSP Wednesday Read Proposition 2 Euclidean constructions Thursday Group quiz Friday Euclidean constructions Wikibook authors write step-by-step instructions and verification for one construction

### 4:

Monday

Try Bride's chair and/or Pythagoras plugged in

#### Pythagorean theorem

• assumptions: triangle is 180 degrees, parallel lines and transversal
• GeoGebra introduction
• proof of Pythagorean theorem
• group project
Wednesday

#### Pythagorean theorem

• group project work day
Thursday

(optional) bring 8 handouts of group project for peer review

#### Pythagorean theorem

Friday Group quiz

### 5:

Monday

Wednesday

#### Exam 1

• one page of notes (1-sided, handwritten)
Thursday

#### Pythagorean activity

Friday

Symmetry

• introduce notation
• compositions of reflections and rotations
• pentagons & composition of its reflections

Wikibook authors finish pentagon and composition of reflections

### 6:

 Monday Transformations isometry can be recovered from any three points (ggb example) Wikibook authors finish procedure Extra: Read Chapter 8.1 introduction (notice that the book is using more formal notation for transformations that is optional for our proofs) Wednesday Transformations recap: 3 points needed, distances preserved conjectures translation is __ reflections Wikibook authors finish procedure Extra: Read Chapter 8.1.1 and Lemma 8.13 (different from class because vector is given but image is not) Thursday Group quiz Friday Transformations recap: lines were parallel, perpendicular bisectors create fixed point(s) conjectures rotation is __ reflections Wikibook authors finish procedure Extra: Read Chapter 8.1.2 and Theorem 8.14

### 7:

 Monday Transformations conjectures glide reflection is __ reflections Wikibook authors finish procedure Extra: Read Chapters 8.1.3 and 8.1.4 Wednesday Transformations [Hlas] isometries are at most __ reflections compositions of isometries (Activity 8.9, only result, not composition ) Wikibook authors finish proof Extra: Read Chapters 8.1.6 and Theorem 8.18 Thursday Group quiz Friday Transformations discuss quiz discuss compositions triangle congruencies/similarities

### 8:

 Monday Key lock puzzle Triangle centers perpendicular bisectors conjectures? circumcenter Wikibook authors finish proof Extra: Read Theorem 3.26 and Exercise 3.31 Wednesday Triangle centers recap circumcenter: create two perpendicular bisectors, show remaining bisector is perpendicular angle bisectors conjectures? incenter Wikibook authors finish proof Extra: Read Theorem 3.24 and Exercise 3.29 Thursday Group quiz application problem theorems: Any isometry of the plane is a composition of at most 3 reflections triangle congruency/similarity conditions circumcenter incenter Friday Triangle centers recap incenter: create two angle bisectors, show remaining bisects angle triangle altitudes conjectures? orthocenter Wikibook authors finish proof Extra: Read Theorem 3.28

### 9:

 Monday Triangle centers Ceva's theorem start triangle medians Wikibook authors finish proofs Wednesday Triangle centers triangle medians conjectures? centroid: concurrent, equal areas Wikibook authors finish proofs Extra: Read Theorem 3.23 Thursday Group quiz application problem theorems: lemma: anti-medial triangle forms 4 congruent triangles altitudes are concurrent (orthocenter) Ceva's theorem medians are concurrent (centroid) medians form 6 triangles with equal areas Friday Class canceled

### 10:

 Monday Class canceled Wednesday create note sheet for exam Review format 2 random proofs from transformations 2 random proofs from triangle centers definitions or other results for remaining points discuss theorems fix images Thursday Triangle centers Definition: Euler's line Definition: Nine-point circle Extra: Read Theorem 3.29 and Theorem 3.30 (proofs not required for assessments) Friday Exam 2 one page of notes (1-sided, handwritten)

### 11:

 Monday Read fractal introduction (in Canvas) New groups Discuss Fractal project Circles perpendicular bisector of chord Thale's theorem Extra: Read Chapter 3.1, exercise 3.2, Theorem 3.1 Wednesday Circles inscribed angle theorem opposite angles of cyclic quadrilateral … Extra: Theorem 3.3, Definition 3.5, Exercise 3.7 Thursday opposite angles of quadrilateral are supplementary then quadrilateral is cyclic Extra: Friday perpendicular bisectors of quadrilateral are concurrent if perpendicular bisectors of quadrilateral are concurrent, then quadrilateral is cyclic Extra: Theorem 3.7

### 12:

 Monday (optional) Email instructor for feedback on fractal project Fractal dimension curve example with Koch curve/snowflake shape example with Sirpenski's triangle Circles construct tangent to circle through given point (GeoGebra exploration) Extra: Wednesday Fractal Project work day Thursday Thanksgiving break - no classes Friday Thanksgiving break - no classes

### 13:

 Monday Due: Fractal project Spherical geometry introduction: line (great circle), parallel lines? area of a lune Extra: Read Chapter 7.1 Wednesday Spherical geometry area of a triangle Extra: Thursday Group quiz application problem (review volume formulas) one proof from "circles" Friday Spherical geometry sum of angles in triangle congruent triangles Saccheri quadrilateral Extra:

### 14:

 Monday Spherical geometry Saccheri quadrilateral circle inversion GeoGebra tools Extra: Wednesday Read chapter 6.3.1 and 6.3.3 Hyperbolic geometry representation inversion tool hyperbolic lines in GeoGebra Extra: Thursday Formula review area? surface area? volume? interior angles? Extra: Friday Group quiz (bonus)

### 15:

 Monday Review Wednesday Final exam (part 1) One page notes, one sided Format: 1 proof spherical geometry 1 proof circles 1 proof triangle centers 1 proof Pythagorean theorem (based on Wikibook) Assign take home (due to Canvas by end of final exam time)

### Final exam (part 2)

• Return part 1
• Take home part due by end of exam time