Tuesday and Thursday 3-21 & 23
Recent review of 'Flatland'
Last Class:
Maps
Coordinates for "earth" - the sphere
Coordinates for the torus!
Activity for maps on Torus.
Locate P and Q on the map! give their coordinates.
|
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 Non-orientable:
Consider the moebius band and the Klein Bottle as examples of non-orientable 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;
Mirror image and orientability - see the animated version (73K) | The Möbius band is one-sided - view the animated version (1.3M) |
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
[Activity:Graphs on the torus]
The projective plane
Spheres with handles:
Spheres with cross caps
Visualizations of surfaces by
flattened - cut apart models.
A cylinder, a mobius band, the torus, the Klein bottle, the projective
plane.
Handles and cross-caps
attached to the sphere.
The Topological Classification of "closed surfaces."
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?
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 one-dimensional 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 cross-caps 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.