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

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

  • GeoGebra advanced: circle with radius, sliders
  • group project work day
Thursday

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

Pythagorean theorem

Friday Group quiz

5:

Monday

Group project due

Review

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

Circles

  • opposite angles of quadrilateral are supplementary then quadrilateral is cyclic

Extra:

Friday

Circles

  • 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

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

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)