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Shapes of Surfaces

Shapes of Surfaces. Yana Mohanty. Originator of cut and paste teaching method. Bill Thurston Fields Medalist, 1982. What is a surface?. Roughly: anything that feels like a plane when you focus on a tiny area of it. Our goal: classify all surfaces!. Botanist: classifies plants.

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Shapes of Surfaces

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  1. Shapes of Surfaces Yana Mohanty

  2. Originator of cut and paste teaching method Bill Thurston Fields Medalist, 1982

  3. What is a surface? Roughly: anything that feels like a plane when you focus on a tiny area of it.

  4. Our goal: classify all surfaces! Botanist: classifies plants Topologist: classifies surfaces

  5. What is topology? • A branch of geometry • Ignores differences in shapes caused by stretching and twisting without tearing or gluing. • Math joke: • Q: What is a topologist? • A: Someone who cannot distinguish between a doughnut and a coffee cup.

  6. Explanation of joke ?= Michael Freedman, Fields Medal (1986) for his work in 4-dimensional topology

  7. Which surfaces look the same to a topologist? To a topologist, these objects are: torus sphere Punctured torus Punctured torus sphere Note: no handles Punctured torus

  8. The punctured torusas viewed by various topologists

  9. Transforming into http://www.technomagi.com/josh/images/torus8.jpg

  10. We can make all these shape ourselves! ... topologically speaking What is this?

  11. How do we make a two-holed torus? Find the gluing diagram Hint: It’s two regular tori glued together.

  12. Pre-operative procedure:making a hole in the torus via its diagram

  13. Making a two-holed torus out of 2 one-holed tori 1. Start with 2 one-holed tori: • 2. Make holes in the diagrams. • 3. Stretch it all out. • 3. Join holes.

  14. Note the pattern • We can make a one-holed torus out of a rectangle. • We can make a two-holed torus out of an octagon. • Therefore, we can make an n-holed torus out of an 2n-gon. n holes We say this is a surface of genus n. Ex: glue sides to get 6-holed torus

  15. What about an n-holed torus with a puncture???? Recall regular torus with hole Now fetch his orange brother Now glue them together What can you say about the blue/orange boundary? Voila! A punctured two-holed torus

  16. Orientability Roughly this means that you can define an arrow pointing “OUT” or “IN” throughout the entire surface. Q: Are all tori orientable? A: Yes!

  17. Is the Moebius strip orientable? NO!!!!!!!!!!

  18. What can we glue to the boundary of the Moebius strip? • Another Moebius strip to get a • Klein bottle • A disk to get a • Projective plane Sliced up version

  19. Are these surfaces orientable?? NO!!!!!!!!!!

  20. Classification of surfaces theorem Any non-infinite surface MUST be made up of a bunch of “bags” (both varieties may be used) and possibly a bunch of holes. For example:

  21. Instructions for making common surfaces

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