Dr. Sarah's Math 3610 Class Highlights
### Dr. Sarah's Math 3610 Class Highlights

The following is NOT HOMEWORK unless you miss part or all of the class.
See the Main Class Web Page for ALL homework and due dates.

** Tues Nov 30** Test 2

**Tues Nov 23** Present Euclidean proofs. If time remains,
take questions on study guide 2.

**Thur Nov 18**
Discuss project 7.
Applications of hyperbolic
geometry
How to
sew a 2-holed torus.

**Tues Nov 16**
Save each Sketchpad file (control/click and then download it to the
documents folder) and then open it up from Sketchpad and follow the directions:
What are the shortest distance paths in hyperbolic geometry?

Sketchpad Shortest
Distance Paths

Image of Shortest
Distance Paths.

Begin
hyperbolic activities.

**Thur Nov 11**
Discuss parallels in the books.
Playfair's Postulate in Euclidean geometry - constructing a parallel by
using perpendiculars. Which Euclidean proposition are we using? Why
are the lines parallel? Discuss what goes wrong with Playfair's on the sphere.
Prove that Euclid's 5th Postulate plus Euclid's other axioms before I-31 prove Playfair's. Euclid's 5th Postulate is vacuously true on the sphere so unlike what is listed on the web and in books, the statements are different.
Begin hyperbolic geometry via the
Escher worksheet.
**Tues Nov 9**
Review the Archimedian solids and their symmetries. Discuss Felix Klein -
revolutionizing geometry by understanding a space by its symmetries and
transformations.
Review Minesweeper and create an inconsistent game. Fill in a partial game to show that consistency does not imply uniqueness. Discuss Godel's 1930 theorem. Discuss various ideas of parallel. Use a folding argument to show that parallel implies the sum of the angles in a triangle is 180 degrees, and then complete a Euclidean proof of I-32.
Discuss what goes wrong with the proof of I-32 on the sphere.
Discuss different definitions of parallel.
Review parallel ideas including same side interior angles being supplementary, alternate interior and corresponding angles being the same, equidistant lines, etc.
Prove that parallel lines imply that they are equidistant.

**Thur Nov 4**
Taxicab activities in Sketchpad.
**Tues Nov 2**
Discuss homework.
Metric for taxcab geometry.
US law is Euclidean.
SAS in taxicab geometry.
Euclid's proof of SAS and what goes wrong in taxicab geometry.

**Thur Oct 28**
Review the proof that the
perpendicular bisectors are concurrent.
Play a few games of
taxicab treasure hunt.
Introduce taxicab geometry via moving in Tivo and relate to
taxicab treasure hunt.
Highlight the possible number of intersections of taxicab circles for
different examples.
**Tues Oct 26** Review measurement. The missing square activity.
Discuss metric perspectives and coordinate geometry.
Water Reservoir Problems.
Review the proof that the perpendicular bisectors are concurrent.
Play a few games of
taxicab treasure hunt.

**Tues Oct 19** Research presentations

**Thur Oct 14**
Discuss the search for history references.
Sphere activity.
Sphere activity and examine consequences, including AAA on the
sphere implying congruence.
Consequences for the formula for the area of a spherical triangle - whether the difference between the angle sum and pi is detectable for a 1 mile square area triangle in Kansas. Use the triangles to examine the area of regular polygons on the sphere. Discuss
Colorado
and Wyoming.

**Tues Oct 12** Revisit
Platonic Solids.
Measurements with and without metric perspectives. How were circumference,
area and volume formulas obtained via axiomatic perspectives and before
coordinate geometry and calculus II?
Circumference.
Orange
Activity and Archimedes polygonal method.
Archimedes
and Cavalieri's Principle.
Sphere activity.

**Thur Oct 7**
[Of the five Platonic solids - the earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron,
and dodecahedron as a model for the universe.] *So
their combinations with themselves and with
each other give rise to endless complexities, which anyone who is to give a
likely account of reality must survey*. [Plato, The Timaeus]
Review the proof that there are only 5 regular
Platonic Solids and discuss why there
are infinitely many on the sphere.
Euclidean angle defect.
Nets
applet 1
and
applet 2.
Begin measurement.
Quotations from Archimedes.
Measurements with and without metric perspectives. How were circumference,
area and volume formulas obtained via axiomatic perspectives and before
coordinate geometry and calculus II?
Circumference.
Orange Activity and Archimedes polygonal method.
Archimedes and Cavalieri's Principle.

**Tues Oct 5** Test 1

**Thur Sep 30**
Take questions on test 1. Review the platonic solids - and how to
remember the faces and vertices (and from there calculate the edges using
Euler's formula).
[Of the five Platonic solids - the earth was associated with the cube, air with the octahedron, water with the icosahedron, fire with the tetrahedron,
and dodecahedron as a model for the universe.] *So
their combinations with themselves and with
each other give rise to endless complexities, which anyone who is to give a
likely account of reality must survey*. [Plato, The Timaeus]
Continue Platonic solids.

**Tues Sep 28**
Burden of Proof.
Each group builds a model
of a polyhedra and presents V, E, and F as well as a way to
help remember the name. Euler's formula.

**Thur Sep 23**
Go over similarity in both books. Prove AAA.
Introduction to geometric similarity and its application to geometric
modeling via. Mathematics Methods and Modeling for
Today's Mathematics Classroom 6.3. Go over p. 214 Project 1, and
the example on p. 212. Work on models for p. 216 number 4 (Loggers).

**Tues Sep 21** Collect and discuss Wile
- how did they ensure the chase would always begin? That he would continue to see him? How did they ensure Wile would catch the RR when the RR runs faster?
Review and continue
with similarity:
SSS, SAS, AA, SSA, AAS, ASA, HL (Hypotenuse and
leg of a right triangle - ie SSA in a right triangle).
Use the Triangle_Similarity.gsp
file (control click and save the file. Then open it from Sketchpad)
to complete the Similar Triangles - SSS, SAS, SSA worksheet.
Similarity of quadrilaterals.
Look at a proof of SAS
and discuss what goes wrong on the
sphere for large triangles.
Applications of similarity: Sibley *The Geometric
Viewpoint* p. 55 number 6.
Sliding a Ribbon Wrapped around a Rectangle
and Sliding a
Ribbon Wrapped around a Box.
Read the proof of
the trig identity and then fill in the details
and reasons using similarity, trig and the Pythagorean theorem. Note that
the Pythagorean theorem is a consequence of similarity as in
the next project.

**Thur Sep 16** Andrew Wiles and Proof. Notes.
Henderson
A second example.
Begin similarity. Introduction to "same shape".
Fig 8.4
Fig 8.21
Fig 8.32
Groups prepare short presentations on
SSS, SAS, AA, SSA, AAS, ASA, HL (Hypotenuse and
leg of a right triangle - ie SSA in a right triangle).
Use the Triangle_Similarity.gsp
file (control click and save the file. Then open it from Sketchpad)
to complete the Similar Triangles - SSS, SAS, SSA worksheet.
**Tues Sep 14**
Go over
images and quotations. Highlight that the
Yale tablet is Sibley *The Geometric Viewpoint* 1.1 3
and The 'hsuan-thu' [Zhou Bi Suan Jing] is similar to Bhaskara's
diagram in Sibley *The Geometric Viewpoint* 1.1 10, and
the connection of Eratosthenes to Wallace and West *Roads to Geometry*
1.1 8.
Fermat's
Last Theorem.
Nova's "The Proof" video.
Notes.

**Thur Sep 9**

Go through a modern version of Euclid's proof of the Pythagorean
theorem.
Euclid's proof of the Pythagorean theorem
Pappus on Sketchpad.
A review of the Greenwaldian
Theorem, as well as
the Scarecrow's Theorem.

**Tues Sep 7**
Discuss the homework readings.
Go to the lab.
Go over Sketchpad's built in version of Proposition 11 as well as a ray
versus a line in Sketchpad.
Review the paper folding argument for proposition 11.

Go over an application - a
proof that the perpendicular
bisectors are concurrent

Go to the lab.
Build a right triangle in Sketchpad and investigate the Pythagorean
Theorem.

Go to Applications/Sketchpad/ Samples/Sketches/Geometry/Pythagoras.gsp

Go through Behold Pythagoras!, Puzzled Pythagoras, and then Shear
Pythagoras. Click on Contents to get to the other Sketches.

Go through a modern version of Euclid's proof of the Pythagorean
theorem.

Come back together and go over
Euclid's proof of the Pythagorean theorem
Discuss Sibley *Geometric Viewpoint*
p. 7 # 10 on Project 2.
If time remains, then an
introduction to extensions of the Pythagorean Theorem including
Pappus on Sketchpad.

**Thur Sep 2**
Review Project 1 solutions.
Review equilateral triangle construction.
Replicate the construction in
Spherical Easel
and compare with the Euclidean proof.
Go to 209b and each student does the Euclidean construction
of proposition 1.
Work on proposition 11.

**Tues Aug 31** Project 1 Presentations and Peer Review.

**Thur Aug 26** Take questions on the syllabus.
Discuss suggestions from last semester.
Students are called on in random order to state and
then prove something about a specific square in
game 2 of minesweeper.
Euclid's Proposition 1 in Sketchpad.
Use a paper folding argument for Proposition 11.
Work on project 1.

**Tues Jan 12**
Fill out information sheet.
Form groups of 2 people
and discuss how can we tell the earth is round without technology?

Mention
the related problem on Project 2 [Wallace and West Roads to Geometry 1.1 8].

Where is North? Also discuss 8/08 article *Cows Tend To Face
North-South*

Begin the Geometry of the Earth Project.
Groups choose their top four problems.

Induction versus deduction.
An introduction to minesweeper games as an axiomatic system.

Axiom 1) Each square is a number or a mine.

Axiom 2) A numbered square represents the number of
neighboring mines in the blocks immediately above, below, left, right, or
diagonally touching.

Examine game 1.
History of Euclid's elements and the societal
context of philosophy and debate
within Greek society.
Intro to Geometric
Constructions.
Begin Euclid's Proposition 1 by hand
and by a proof.