Shapes of Surfaces

Yana Mohanty

Originator of cut and paste teaching method

Bill ThurstonFields 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 Topologist: classifies 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.

Explanation of joke

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

?=

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

The punctured torusas viewed by various topologists

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

Transforming into

We can make all these shape ourselves!

... topologically speaking

What is this?

How do we make a two-holed torus?

Hint: It’s two regular tori glued together.

Find the gluing diagram

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

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.

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

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?

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!

Is the Moebius strip orientable?

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

Are these surfaces orientable??

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:

Instructions for making common surfaces