shapes of surfaces

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Shapes of Surfaces Yana Mohanty

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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. - PowerPoint PPT Presentation

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Page 1: Shapes of Surfaces

Shapes of Surfaces

Yana Mohanty

Page 2: Shapes of Surfaces

Originator of cut and paste teaching method

Bill ThurstonFields Medalist, 1982

Page 3: Shapes of Surfaces

What is a surface?

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

Page 4: Shapes of Surfaces

Our goal: classify all surfaces!

Botanist: classifies plants Topologist: classifies surfaces

Page 5: Shapes of Surfaces

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.

Page 6: Shapes of Surfaces

Explanation of joke

Michael Freedman, Fields Medal (1986) for his work in 4-dimensional topology

?=

Page 7: Shapes of Surfaces

Which surfaces look the same to a topologist?

Note: no handles

To a topologist, these objects are:

torus

Punctured torus

sphere

Punctured torus

Punctured torus

sphere

Page 8: Shapes of Surfaces

The punctured torusas viewed by various topologists

Page 9: Shapes of Surfaces

http://www.technomagi.com/josh/images/torus8.jpg

Transforming into

Page 10: Shapes of Surfaces

We can make all these shape ourselves!

... topologically speaking

What is this?

Page 11: Shapes of Surfaces

How do we make a two-holed torus?

Hint: It’s two regular tori glued together.

Find the gluing diagram

Page 12: Shapes of Surfaces

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

Page 13: Shapes of Surfaces

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.

Page 14: Shapes of Surfaces

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 an2n-gon.

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

n holes

Page 15: Shapes of Surfaces

What about an n-holed torus with a puncture????

Recall regular torus with hole Now fetch his orange brother

Now glue them together

Voila! A punctured two-holed torus

What can you say about the blue/orange boundary?

Page 16: Shapes of Surfaces

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!

Page 17: Shapes of Surfaces

Is the Moebius strip orientable?

Page 18: Shapes of Surfaces

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

Page 19: Shapes of Surfaces

Are these surfaces orientable??

Page 20: Shapes of Surfaces

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:

Page 21: Shapes of Surfaces

Instructions for making common surfaces