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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