Shapes of Surfaces Yana Mohanty. Originator of cut and paste teaching method Bill Thurston Fields...
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Transcript of Shapes of Surfaces Yana Mohanty. Originator of cut and paste teaching method Bill Thurston Fields...
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