More on Surfaces
Adventures on the Mobius Band, the Klein Bottle,
and
the Projective Plane !
"New" Surfaces and The Classification of Surfaces
What about graphs on the Torus?
We have a connected graph on the torus with one vertex, two edges and only
one region. But this information does not match for the euler formula for the
plane or the sphere. SO.. the torus must be topologically different from the
sphere or the plane!
In fact for the torus we see that it is possible for V+R= E or VE+R=
0
We learned about
(and proved) Euler's Formula
for the plane or the sphere. What can we say about a formula for the torus or the Klein bottle?
Surfaces:Mark Sudduth's web page of surfaces.[ A physics master's degree student at UT, Arlington.]
What is a surface?
Bounded, unbounded:
Closed, open:
With or without boundary:
Orientable or Nonorientable:
Consider the moebius band and the Klein Bottle as examples of nonorientable surfaces.
Can be realized (imbedded) in a plane, in 3 space, in 4 space.
Can be visualized (immersed) in ...
Examples:A closed disc, an open disc, a
plane, an annulus cylinder, a mobius band;
Experiments with the mobius
band:
Draw a curve along the center of the band we cover both "sides."
Cut along that curve the band does not fall apart, but gets twice as
long!
a sphere
a torus
a Klein bottle
the projective plane... Why is this a closed and bounded surface?
Review Activity:Graphs on the torus. [redone as homework for this class]
Games and puzzles on the torus and the klein bottle.
From the Fun Fact files,
hosted by the
Harvey Mudd College Math Department , Unbelievable Unlinking
Imagine that the two objects in Figure 1 are solid (with thickness) and made of very flexible and stretchy rubber. Question: is it possible to deform one object into the other in a continuous motion (without tearing or cutting)? Surprise answer: Yes!! Hint: it is important that the object is solid and has thickness; this transformation cannot be done with a onedimensional piece of string. It is also not possible to do this with a piece of rope because even though the rope has thickness, it is not flexible or "stretchy" enough. See below for an explanation and animated gif. Or, don't scroll down if you want to think about it a while! The Math Behind the Fact: Graeme McRae has generously contributed the
animated gif in Figure 2, showing another solution to
this problem! (Thank you, Graeme!) 
Visualizations of surfaces by flattened
 cut apart models.
A cylinder, a mobius band, the torus, the Klein bottle, the projective plane.
Closed Surfaces: Handles and crosscaps attached to the sphere.


A sphere with a handle = a torus 
A Sphere with a cross cap = the projective plane 
The Topological Classification of "closed surfaces."
Every connected closed and bounded surface is topologically equivalent to a sphere with handles and crosscaps attached.