1. Geometry out of the Paper Dr. Jessica Purcell University of Texas at Austin An Introduction to Manifolds

2. Based on map of St. Isidore of Seville 600-636 A.D. “The earth is named from its roundness (orbis) which is like a wheel. For the Ocean flows round it on all sides and encircles its boundaries.” 10 th Century Map

3. Dimension “measurement in length, width, and thickness” --Dictionary.com

4. 0-dimensions A point. 1-dimension A line.

5. 2-dimensions A plane. 2 coordinates: (x,y)

6. 3-dimensions Space. 3 coordinates: (x,y,z)

7. 4-dimensions ? Space and time. 4 coordinates: (x, y, z, w) 5-dimensions ? ? 5 coordinates: (x, y, z, w, t)

8. n-dimensional manifold Any point has a neighborhood that looks like a region in n-dimensional space. A circle is a 1-dimensional manifold. x 2 + y 2 = 1

9. A sphere is a 2-dimensional manifold. Sphere x 2 + y 2 + z 2 = 1

10. 3-Sphere 3-dimensional manifold x 2 + y 2 + z 2 + w 2 = 1

11. Manifold with boundary Every point has a neighborhood that either: looks like a region in n-dimensional space, or looks like a region in n-dimensional half space. A disk is a 2-manifold with boundary a circle.

12. Activity 1 Building manifolds.

13. Cylinder

14. Torus

15. Moebius Band - M. C. Escher

16. Klein Bottle http://www.gakushuin.ac.jp/~881791/kuroki/Klein.GIF

17. What’s the difference? Cylinder Two boundaries Two sides Moebius strip One boundary One side

18. What’s the difference? (II) Torus 0 boundaries 2 sides Klein bottle 0 boundaries 1 side

19. What’s the difference? (III) Sphere 0 boundaries 2 sides Torus 0 boundaries 2 sides

20. Break

21. Activity 2 Euler’s Formula

22. Example

23. v=3; e=3; f=2.

24. Answer to #7: 2 Is the answer really always 2, or did it just happen that none of us drew the right picture to get something besides 2? Mathematical proof: Show that no matter how many vertices, edges and faces we have, if we follow the rules, then v – e + f = 2

25. Cauchy’s Proof of Euler’s Formula If there is any face with more than 3 sides, draw a diagonal. Repeat. Eventually, everything is divided into triangles.

26. Cauchy’s Proof Continued Repeat the following two steps: 1.Remove triangles with one edge on the exterior. 2.Remove triangles with two edges on exterior.

27. Cauchy’s Proof Concluded You are left with triangles with 3 exterior edges only. This looks like my example.

28. Euler Characteristic for Moebius Strip v=? e=? f=? v – e + f = ?

29. Concluding Remarks To a bug on an n-dimensional manifold, the world looks n-dimensional. It is hard to tell manifolds apart when you’re standing inside them. Different manifolds have many different interesting properties. Keep exploring!