Geometry of the Earth and Universe Labs From

Geometry of the Earth and Universe Labs: From the Classroom to Current Research Dr. Sarah J Greenwald Appalachian State University Boone, North Carolina greenwaldsj@appstate. edu http: //www. cs. appstate. edu/~sjg/joint/

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 http: //www. cs. appstate. edu/~sjg/joint/

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 http: //www. cs. appstate. edu/~sjg/joint/

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) http: //www. cs. appstate. edu/~sjg/joint/

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? http: //www. cs. appstate. edu/~sjg/joint/

Lab 1: Answer to Problem 10 • Initial intuition and student research • My 1 st response: String Argument • My 2 nd response: Walter Fendt’s Spherical Triangle Applet http: //home. a-city. de/walter. fendt/me/sphertriangle. htm http: //www. cs. appstate. edu/~sjg/joint/

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 http: //www. cs. appstate. edu/~sjg/joint/

Lab 2: Perspective Drawing and Homer 3 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 http: //www. cs. appstate. edu/~sjg/joint/

Lab 2: Perspective Drawing and Homer 3 • 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 3 F 04 (10/30/95) http: //www. cs. appstate. edu/~sjg/joint/

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? http: //www. cs. appstate. edu/~sjg/joint/

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

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 http: //www. cs. appstate. edu/~sjg/joint/

Lab 5: Shape of the Universe • • • Gluing spaces to obtain possible shapes Real-life attempts to discover the geometry What is the 4 th 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

Geometry of the Earth and Universe Labs: From the Classroom to Current Research Dr. Sarah J Greenwald Appalachian State University Boone, North Carolina greenwaldsj@appstate. edu http: //www. cs. appstate. edu/~sjg/joint/