### Geometry of the Earth and Universe - From the Classroom to Current Research

Sarah J. Greenwald
http://www.mathsci.appstate.edu/~sjg/talks/earthanduniverse.html
With a far-reaching range from students with no math background all the way to PhD researchers in geometry and topology, and with connections to art, philosophy, physics, astronomy, geography, and visualization, this topic has something for everyone!
• The Shape of the Universe published as a 2005 Mathematics Awareness Month theme essay.
• Powerpoint File of Talk at the 2003 Joint Mathematics Meetings in a session on Creative Visualization Labs
• A 4-week Segment in a Liberal Arts Mathematics Class
• Geometry of the Earth and Universe in Different Classes
• ### Activities in Math 1010

• In the Fall of 2006, this segment was the first segment we completed: Daily Activities   Homework Assignments
• Geometry of the Earth and Universe Research Problems Given out while I am still completing my last segment so that the students have at least 2 weeks to complete this. (30 minutes for problem assignment and initial intuition).
• Readings and Activities on Perspective Drawing and Projective Geometry,   Perspective Worksheet (1 hour)
• Homer Writing Assignment that asks whether the Simpsons are 2D or 3D (1 hour)
• IDS Viewpoint Readings, which contains a number of perspectives to try and answer the question - What is Mathematics?. Very useful before beginning the 2D Universes Lab. (30 minutes)
• Introduction to Hyperbolic Geometry (30 minutes)
• Review answers to Homer questions, 2D Universes Lab with directed questions (1 hour). Also see p. 32-35 of the PRIMUS article on r dr r: Engaging Students with Significant Mathematical Content From The Simpsons which further discusses the use of Homer3 in the classroom.
• Review gluing spaces and visualizing a torus and a Klein bottle. Collect geometry reports, students present problems 1-7, and I go over them. (1 hour and 15 minutes)
• 2D Universes continued. Oral quiz on questions 0-7 to finish geometry of the earth. Hyperbolic geometry via demos on sketchpad (see dynamic geometry activities below), Escher. Highlight the differences between Euclidean, hyperbolic and spherical geometries. Polygonal tiling models of the sphere, the plane and hyperbolic space. Crochet model of hyperbolic space. Review gluing spaces from the 2D universes lab and discuss the fact that the torus and Klein bottle are really 2D Euclidean universes while the 2-holed torus is really a 2D hyperbolic universe. (1 hour and 15 minutes).
• Heart of Math reading summary pp. 288-319 (homework)
• Student presentations of Questions 8-11 (15 minutes)
• Shape of the Universe Lab with directed questions in a worksheet (1 hour and 30 minutes).
• PBS Life by the Numbers Shape of the World video Hundreds of years before Columbus set sail, the Greeks used mathematics to determine the size and shape of the planet. Viewers see how mathematics has become a tool to explore the earth and the heavens as the world and the cosmos is charted. (1 hour)
• PBS Life by the Numbers Seeing is Believing Video The special effects in movies are mathematics in action. The first special effects ever created in Renaissance paintings also owe their existence to mathematics. From IMAX films to modern art and virtual reality, mathematics helps define space and helps present visions of our world to us. (1 hour)
• Jeff Weeks worksheet about his influences, support, barriers, diversity issues, and mathematical style, and an exploration of additional mathematics.
• Brief intro to my own research and how it fits into these ideas, and my mathematical style via powerpoint presentation (model for mathematician project). Review the 4th physical dimension and its applications, the hypercube via excerpts from Davide Cervone's Selected Course Notes Why the universe is not thought to be a hypercube, and some of the shapes that might be the shape of the universe: Euclidean 10 Euclidean possibilities, including Escher's "Another World" the Futurama video "I, Roommate" reference to Escher's 1953 "Relativity" when Fry and Bender look for an apartment, and the Futurama video "I Dated a Robot" reference to the edge of the unverse. Futurama Shape of the Universe worksheet Spherical a number of the infinite but known spherical possibilities via Davide Cervone's Spheres Sliced in 2D and 3D and excerpts from Week's paper on Topological Lensing in Spherical Spaces page 1, page 12, and the relationship to the Spherical Applet, and Hyperbolic current mathematical attempts to classify the hyperbolic possibilities, including the Weeks example. The differences between the geometries and current attempts to determine the shape of the universe.
• I also reinforce all of the material with WebCT quiz questions and on the test.

• ### Geometry of the Earth and Universe in Different Classes

• Mat 1010 Class Highlights Page shows exactly how the activities are typically spread out.
• Mat 5980-375 (Geometry for Teachers of Grades 6-8) Additional activities requiring more time (but no additional background) are completed, including some simple Euclidean proofs. Moving up on the page means forward in time.
• Mat 3510 (Junior Honors Seminar in Applications of Geometry) Class Highlights Page Related advanced activities and proofs are included - Calculus and linear algebra are needed for some of the activities, while other advanced topics such as groups, quotient spaces, fundamental domains and the covariant derivative are introduced during other activities. Activities begin on Jan 14 and are interspersed throughout.

Many of these activities do not need any additional background, but they do require more time.
• Euclid's Book 1 Geometry of our Earth questions reorganized and phrased in order to connect with Book 1 of Euclid's Elements and prepare for related proofs.
• Letter from 3D Homer to 2D Marge - Assume that the Simpson's really were 2D creatures living in an x-y plane and that Homer and Bart had made the transformation to 3D creatures. While a 2D Marge can't really understand the 3rd dimension and would feel like there isn't any room for another dimension, she could see weird behavior occurring that suggests that the 3rd dimension exists (for example the "wall" that Homer disappeared into). 2D Marge wouldn't be able to comprehend the concept of depth or an entire 3D Homer, since only 2D pieces would make sense to her. Write a letter from Homer Simpson to his wife Marge which discusses his and Bart's change from 2D into 3D. Be sure to include mathematical explanations in addition to physical descriptions. Compare and contrast Homer and Bart's 2D and 3D appearances.
• The Use of Letter Writing Projects in Teaching Geometry, PRIMUS, Volume X Number 1, March, 2000, p. 1-14 http://www.mathsci.appstate.edu/~sjg/papers/geompaper899.pdf
• Beachball Activity on the sum of the angles of a spherical triangle (50 minutes)
• Sketchpad activities on the hyperbolic disk (From the Sketchpad 4 folder, open up Sketchpad/Samples/Sketches/Investigations/ Poincare Disk.gsp) -- one could demo these to a class, or have them come up with them on their own:
Introduction to Hyperbolic Geometry
1) Given a hyperbolic line and a point off of the line, how many parallels can be formed? (Playfair's axiom) Sketchpad Playfair's,   Playfair's Image,
2) What is the sum of the angles in a hyperbolic triangle? How large can the sum of the angles get? How small can the sum of the angles get?   Sketchpad Sum of Angles,   Sum of Angles Image
3) Is Euclid's 5th postulate ever, always or never true in hyperbolic space?   Sketchpad Euclid's 5th Postulate,   Euclid's 5th Postulate Image
4) Is the Pythagorean theorem ever, always or never true in hyperbolic space?   Sketchpad Pythagorean Theorem ,   Pythagorean Theorem Image