1. Appalachian State University
Geometry of the Earth and Universe Labs: From the Classroom to Current Research
Dr. Sarah J Greenwald
Appalachian State University
Boone, North Carolina

2. Geometry of the Earth and Universe Labs
Develop visualization skills while taking advantage of various technologies and manipulatives
Perspective drawing
Escher and hyperbolic geometry
Spherical geometry
2-D, 3-D and 4-D universes
Current theories on the shape of our universe

3. Engaging the Students Web movies by Davide Cervone
Interactive web games by Jeff Weeks
The Simpsons cartoon
The Geometer’s Sketchpad
Microsoft Excel
Slinkies
Globes
Zome educational construction toys

4. Lab 1: Research Problems
Each student or group has a different problem
Students give initial intuition and then conduct research
Problems connect with Book 1 of Euclid's Elements to prepare for related proofs
A less technical version of this lab is aimed at students in a course for non-majors
Students turn in a report and present their research to the rest of the class and then I go over answers (many only require a globe and string or other manipulatives)

5. Lab 1: Problem 10
Problem 10 In Book 1 of Euclid's Elements, proposition 47 says that in right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle. Assume we have a right-angled spherical triangular plot of land on the surface of a spherical globe between approximately Umanak, Greenland, Goiania, Brazil, and Harare, Zimbabwe, that measures 300 and 400 on its short sides. How long is the long side from Greenland to Zimbabwe?

6. Lab 1: Answer to Problem 10 Initial intuition and student research
My 1st response: String Argument
My 2nd response: Walter Fendt’s Spherical Triangle Applet

7. Student Reactions to Lab 1
Difficulties
Terminology difficulties
Research difficulties
Grading concerns
Benefits
Challenges their notion of what mathematics is
Process is similar to mathematical research
Ownership and excitement about the material
Exposure to a variety of viewpoints and definitions

8. Lab 2: Perspective Drawing and Homer3
Perspective portion adapted from Marc Frantz’s Mathematics and Art
Vanishing points
Perspective Theorem
Perspective Drawing in Excel
One-point perspective and Vanishing Distance
Students perform activities and then answer
worksheet questions

9. Lab 2: Perspective Drawing and Homer3
In the supposedly 2-D Simpsons’ world, what aspects could be used to argue that they are living in a 2-D world? A 3-D world?
Homer Changing Dimensions from Treehouse of Horror VI 3F04 (10/30/95)

10. Student Reactions to Lab 2
Difficulties
Must get close to the computer screen to view the box from the correct vanishing distance
Taking it Seriously
Benefits
Challenges their notion of what mathematics is
Creative and fun
Explore paintings and cartoons in a new mathematical way
What is dimension?

11. Lab 3: 2-D Universes Web readings about 2-D creatures
Davide Cervone’s web movies
Directed questions about a 2-D Marge
Gluing edges to obtain 2-D universes
Jeff Weeks’ torus and Klein bottle tic-tac-toe
Adapted from excerpts taken from:
David Henderson Experiencing Geometry in the Euclidean,
Spherical, and Hyperbolic Spaces
Michio Kaku Hyperspace & A Theory of Everything
Davide Cervone's materials and movies
Jeff Weeks Exploring the Shape of Space

12. Lab 4: Hyperbolic Geometry
Sketchpad explorations of the Poincare Disk model
Escher activities
Henderson and Taimina models: polygonal tiling models of the sphere, the plane, and hyperbolic space, and Crochet model of hyperbolic space

13. Lab 5: Shape of the Universe
Gluing spaces to obtain possible shapes
Real-life attempts to discover the geometry
What is the 4th physical dimension?
Managing data using higher dimensions
Mapping the brain
What might one layer of Homer's skin look like if
he were to change from 3-D to 4-D?
Adapted from excerpts taken from:
Davide Cervone's materials and movies
Cathy Gorini Geometry at Work
David Henderson Experiencing Geometry in the Euclidean, Spherical,
and Hyperbolic Spaces
Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks Is Space Finite?
Diane Martindale Road Map for the Mind
Jeff Weeks Exploring the Shape of Space

14. Appalachian State University
Geometry of the Earth and Universe Labs: From the Classroom to Current Research
Dr. Sarah J Greenwald
Appalachian State University
Boone, North Carolina