Thursday, April 22
A quick review of
web page of surfaces.[ A physics master's degree student at UT, Arlington.
Examples:A closed disc, an open disc, a
plane, an annulus- cylinder, a mobius band;
a Klein bottle
the projective plane... Why is this a closed and bounded surface?
A sphere with a cross cap
Activity:Graphs on the torus.
Games and puzzles on the torus and the klein bottle.
Spheres with handles,
Making Klein Bottles activity.
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
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.
The Euler Characteristic of a surface:
This classification determines the Euler characteristic of each surface.
- If the surface is orientable, it is a sphere with n handles, so V-E+R = 1
- 2n +1 = 2-2n. For example the torus has euler characteristic 2-2*1=0.
Notice that a sphere with two cross caps has euler characteristic 0, the
same as the torus. But this was the Euler characteristic of the Klein Bottle.
- If the surface is non-orientable, then is it a sphere with k crosscaps and
n handles, so the euler characteristic is V-E+R = 1 - (k+2n) +1 =2 -2n -k.
So... we should be able to recognize the Klein bottle as a sphere with two cross
This can be done by a single normalization of one pair of edges with the same orientation.
Other interest in surfaces: Examples
Ways to think of surfaces : cross-sections/ projections/moving curves/ using color to see another dimension. ChromaDepthTM 3D
Generalization of surfaces are called "manifolds". cross sections / projections/ moving surfaces-solids.
"minimal surfaces" (FAPP video)
The geometry of shadows...projections
Perspective from a point.... what happens to a circle in a plane?
The points on the circle and the center of perspective determine lines through the center.
The surface of these lines is a cone.
When you pass the cone through Flatland your see curves- the "conic sections".
Check out this applet.
Google "conics." much is found! :)
Show video on Conics FAPP
The characterization of points on a conic by measurements are:
ellipse: AC + BC = constant
Rope Conic Activity.
parabola: EC = EF
hyperbola: |CB-CA| = constant.
Using a rope :form a human ellipse.
After that demonstrate how to use a rope to form a hyperbola and then a parabola.
String drawing conic activity.