Yadkin Center of Surry CC 224 Wednesdays 6-8:30pm EXCEPT March 17th and April 7

In what ways do the spaces we see in everyday life satisfy (and do not satisfy) the geometric laws and axioms of Euclidean geometry that we learn and teach in middle grades and high school classrooms? In this course we will explore spaces such as a tree trunk, ice cream cone, basketball, donut, coral reefs, Yoda from Star Wars, the world of taxi-cab drivers, Pac-Man's game, artistic works, and our universe via hands-on activities, computer explorations, theory, computations, axioms, applications, historical perspectives and pedagogical discussions. In this context we will understand Euclidean geometry at a much deeper level. Pre-requisite: Undergraduate calculus sequence with analytic geometry or permission of the instructor

- Dr. Sarah's Office Hours this week
- ASULearn To access ASULearn (to post messages to Dr. Sarah or classmates, solutions, and grades).
- Syllabus and Grading Policies
- Jump down to tomorrow's homework which is located
above the red lines
**DUE Date****WORK DUE at the beginning of class or lab unless otherwise noted!**Turn in work that follows the guidelines.5 May - Wed - Last Day of Class. Final Project Due.
**Class Activities**Discuss final project. Why is geometry important? History of geometry education. A history of the teaching of elementary geometry by Alva Walker Stamper. Reflection. If it looks like a sphere... Evaluation.

**__________****________________________________________________________________________****__________****________________________________________________________________________**28 Apr - Wed **Homework**: Graded Homework 6: Connections: A Topic Across Geometries**Class Activities**: Discuss homework. Taxi-cab geometry. Yoda from Star Wars. Yoda 1, Yoda 2, Yoda 3, Yoda 4.21 Apr - Wed **Homework**: Prepare to discuss the following readings: NonEuclid: Why Study Hyperbolic Geometry? and Hyperbolic Geometry in the High School Geometry Classroom [p. 1-10 and p. 31-42].

Continue working on the final project. Complete a sample annotation for part 1 of your final project and email it to me for feedback.**Class Activities**: Discuss the homework readings.- SAS in hyperbolic geometry
- Prove that distance is exponential in the hyperbolic annulus model.
- Bhaskara's proof of the Pythagorean theorem in Euclidean geometry and hyperbolic geometry.
- Apps Finish hyperbolic geometry. The Hyperbolic Geometry Exhibit
14 Apr - Wed **Homework**: Begin working on the final project - choose a topic and find preliminary references for the historical timeline portion of the project. Email the topic and the references to me.**Class Activities**: Hyperbolic geometry activities continued. Finish Playfair's axiom in Euclidean geometry.- Next show that the existence portion of Playfair's axiom works in hyperbolic geometry via Sketchpad:

From Sketchpad, use a File/open and find Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp.

Click on the Script View Tool, under the Alphabet Tool. Notice that this sketch comes with pre-made hyperbolic tools, which is what we will use for hyperbolic constructions and measurements.

Under Hyperbolic Tools, choose**Hyperbolic Line**and then create AB.

Choose**Hyperbolic Perpendicular**, click on A, then B, and then on a point off of the line to create a perpendicular [line through D is perpendicular to AB].

The hyperbolic perpendicular is still selected, so now click on two points of your perpendicular and a point off of it [line through F is perpendicular to DE and parallel to AB].

Create points and use the**Hyperbolic Angle**to measure the angles and verify that they are right angles. Click on**Hyperbolic Angle**each time you wish to measure.

Compare your work with the following image of the existence portion of Playfair's axiom to verify that you have the correct diagram and measurements. If not, try again!

Recall that this part of the Euclidean proof only required up to I-27, which did not require Euclid's 5th postulate, so it is not surprising that the construction still works in hyperbolic geometry.- Once your Sketchpad work matches the image above:

Use a**Hyperbolic Segment**to connect I and B.

Use the**Hyperbolic Angle**to measure the alternate interior angles FIB and IBG.

Notice that the angles are not equal.

Compare your work with the image of the hyperbolic measurement of the alternate interior angles. If it does not match, then repeat the construction and the measurements.

Proposition I-29 does not hold in hyperbolic geometry. This should not be surprising, since I-29 required Euclid's 5th postulate, and we showed that Euclid's 5th postulate doesn't always hold in hyperbolic geometry.

- Make a paper model. We can create a 2-holed donut by using a distorted octagon with 45 degree interior angles. Eight of these glue together in a space that looks like Escher's work to form 360 degrees at a vertex and so they tile the space. Now we understand some of the issues that Escher faced. hyperbolic octagon and tiling.

- Prove that the sum of the angles in a triangle is 180 degrees in Euclidean geometry. What goes wrong with the proof in hyperbolic geometry?

- Does the Pythagorean theorem hold true in hyperbolic geometry? Investigate using Sketchpad. We must use the Hyperbolic Distance to measure, but we can use the regular Measure menu calculator to add distances, square them, etc.
31 Mar - Wed **Homework**:

Read Henderson p. 253-261 and summarize what you read and write down any questions.

Read Crocheting the Hyperbolic Plane: An Interview with David Henderson and Daina Taimina CABINET Issue 16 The Sea Winter 2004/05.

Read Henderson p. 59-67.**Class Activities**Revisit the Hexagonal torus. Which surfaces locally look like the sphere? RP^{2}.

Begin hyperbolic geometry via the Escher worksheet.

- What are the shortest distance paths in hyperbolic geometry?

Sketchpad Shortest Distance Paths

Image of Shortest Distance Paths.

- Euclid's 5th postulate in Euclidean geometry. Does Euclid's 5th postulate hold true in hyperbolic space? Download the Sketchpad file (control/click and then download it to the desktop and then open it up from Sketchpad) and then drag C to answer this question.

Sketchpad Euclid's 5th Postulate

Image of Euclid's 5th Postulate

- Prove Playfair's axiom in Euclidean geometry.
24 Mar - Wed **Homework**:

Read the following article annotationThis is an individual assignment. Find a journal article or book from the library on a topic related to geometry education that you are interested in and have full text access to or can order and receive in time (try the library catalog and/or the library databases Eric, Jstor, or Springer) and type up an annotation for it. In addition, be prepared to discuss your work.__Graded Homework 5: Annotation__**Class Activities**: Collect and discuss homework. Review. Discuss any remaining questions from Henderson. Continue which surfaces locally look like the plane. Revisit the twisted pretzel space.

10 Mar - Wed **Homework**:

Review your notes on Henderson p. 245-258, p. 264-266 and try to resolve any questions you had.

Read unfolding polyhedra and Hairy ball theorem and look at the pictures on wikipedia. Briefly summarize what you read.

Begin working on finding/ordering a book or article from the library (see homework for March 24).**Class Activities**:

Collect homework and discuss unfolding polyhedra. Harry ball theorem.

Geodesics on the donut using symmetry, string, and the Maple file (curvature 2.mw). Solving the geodesic equations using numerical solutions (torus-demo.mw). torus, typical geodesic, an approximately closed geodesic that never crosses the inner annulus. Flat Klein bottle tic-tac-toe, by paper, and on Sketchpad. Glass, urn and knitted Klein bottle in R^{3}. Mobius band. Which surfaces locally look like the the plane? pretzel The sphere? Unfolding the pretzel.3 Mar - Wed **Homework**: Graded Homework 4: Reflection**Class Activities**:

Geodesics on the donut using symmetry, string, and the Maple file (curvature 2.mw). Solving the geodesic equations using numerical solutions (torus-demo.mw). torus, typical geodesic, an approximately closed geodesic that never crosses the inner annulus. Flat Klein bottle tic-tac-toe, by paper, and on Sketchpad. Glass, urn and knitted Klein bottle in R^{3}. Mobius band. Which surfaces locally look like the the plane? pretzel The sphere? Unfolding the pretzel. unfolding polyhedra. Harry ball theorem

24 Feb - Wed **Homework**: Read the following sections from Henderson p. 245-258, p. 264-266 and write down (to turn in) some aspects that you found interesting, that you had questions on, or that you disagreed with.- Graded Homework 3: Symmetry Homework
**Class Activities**:

Review area under a curve

Review surface area for a surface of revolution

Finish Archimedes and the volume of a sphere. Cavalieri's Principle

Begin the torus and the Klein bottle

Play some Tic-Tac-Toe.

Explore geodesics and circles on the flat torus via Sketchpad files from Amanda Hawkins and Nathalie Sinclair (Explorations with Sketchpad in Topogeometry, International Journal of Computers for Mathematical Learning, 13(1), March, 2008). Sketchpad files. Torus pool and real-life pool table. Davide Cervone rotating cube projection and Tom Banchoff flat torus movies

17 Feb - Wed **Homework**: Prepare to discuss the following readings:

The Decline and Rise of Geometry in 20th Century North America

The Van Hiele Model

Visuals in the 'hierarchy of learning'**Class Activities**:

Discuss the homework readings

Eratosthenes estimation of the circumference of the earth.

Surface area of a sphere. Orange activity. Archimedes' method

If we slice a perfectly round loaf of bread into equal width slices, where width is defined as usual using a straight edge or ruler, which piece has the most crust (or surface area)? Why?

circumscribed cylinder

Begin Archimedes and the volume of a sphere. Cavalieri's Principle

10 Feb - Wed **Homework**: Read Chapters 1 and 2 in Henderson.

Graded Homework 2: Axiom Systems on the Sphere.**Class Activities**:

Finish the SAS problem in the Sphere Presentations

Sum of the Angles. Examine the Euclidean proof and discuss what goes wrong. Discuss whether the difference between the angle sum and pi is detectable for a 1 mile square area triangle in Kansas. Use triangles to examine the area of regular polygons on the sphere. Discuss Rectangular states. Examine the implications of AAA on the sphere.

Equilateral triangle construction on the plane and on Spherical Easel

Circumference

Orange activity for surface area

3 Feb - Wed **Homework**:

Read Chapter 4 in Henderson on Cylinders and Cones

Graded Homework 1: Cones**Class Activities**: Discuss parametrizations of the cone, plug into the Maple file on geodesic and normal curvatures. Discuss the 450 degree cone. Continue the sphere.

Geodesics on a sphere using the Maple file on geodesic and normal curvatures.

Continue going over the remaining problems in the Sphere Presentations

27 Jan - Wed **Homework**: Obtain the class textbook: Experiencing Geometry: Euclidean and Non-Euclidean with History (3rd Edition) by David Henderson and Daina Taimina. Prentice Hall, August 2004. ISBN: 0131437488 and bring it to class.

Sphere PresentationsBring a 10-12 inch diameter child's ball - these are usually found in bins in stores and cost a couple of dollars. Be sure that this ball is smooth, can bounce, and that you won't mind writing on it during class. **Class Activities**:

Review class from last week including the history, surface and volume area calculations, the cylinder problem, the Maple file on geodesic and normal curvatures, covering arguments for straight paths, the sum of the angles of a triangle, and the Pythagorean theorem.

History of the cylinder including as a surface of revolution versus a surface that locally looks like a plane.

Calculation of geodesic curvature extrinsically using (cos(u), sin(u), v), normal to surface, the unit tangent vector, and dT/ds.

Pac-Man Online in JAVA

Calculations of geodesics intrinsically using local coordinates.

How many geodesics connect two points on a cylinder? What are the equations of the geodesics?

Sphere presentations.

Geodesics on a sphere using the ideas of symmetry, and unrolling onto a cylinder or cone.

How can we tell that the earth is round without technology?

20 Jan - Wed First Day of Class **Class Activities**:

What do you know about a cylinder?

Geometry of the cylinder - instrinsic and extrinsic equations and perspectives: x^2+y^2=1, local coordinates u, v.

Google "History of a cylinder"

Earliest Known Uses of Some of the Words of Mathematics (C)

history including as a surface of revolution and in applications

Surface area of a cylinder 2 pi r h and derivation of the formula

Volume of a cylinder p r^2 h and derivation of the formula

The Cylinder Problem

Cone and Cylinder Volume on Geogebra

Maple file on geodesic and normal curvatures adapted from David Henderson.

g := (x,y) -> [cos(x), sin(x), y]:

a1:=0: a2:=2*Pi: b1:=0: b2:=Pi:

c1 := 1: c2 := 3:

Point := 2:

f1:= (t) -> t:

f2:= (t) -> sin(t):

The yellow curve does not feel straight since the geodesic curvature (the orange vector) is felt as a turning movement.

Straight paths from a covering (unwrapping argument).

Sum of the angles and Pythagorean theorem on the cylinder.

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

Time to work on the sphere presentations.