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Shapes of Surfaces Yana Mohanty

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

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

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Our goal: classify all surfaces! Botanist: classifies plantsTopologist: classifies surfaces

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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.

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

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

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The punctured torus as viewed by various topologists

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Transforminginto

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

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

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

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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. Join holes. 3. Stretch it all out.

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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. Ex: glue sides to get 6-holed torus We say this is a surface of genus n. n holes

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

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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!

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Is the Moebius strip orientable?

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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

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Are these surfaces orientable??

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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:

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Instructions for making common surfaces

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