Orientability and Twists

Objectives:

Discover topological properties of surfaces with twists.
Distinguish between extrinsic and intrinsic properties.

Materials Needed:

Template P
Scissors
Glue and/or tape

Launch:

What happens if you cut a closed path on a band with a twist?

Activity 1:

Using Template P, cut out the two strips containing a centered dotted line. Glue the opposite ends together on both strips so that their arrows coincide. Notice one band has a half twist in it. This surface is called a Möbius band.

Questions:

What is the topological name of the other surface ?
How many boundaries does this surface have ?
How many boundaries does the Mšbius band have ?

Activity 2:

Tile the two bands and count the number of vertices, arcs, and regions each of your tilings have. Determine each surface's Euler characteristic .

Question:

What did you discover ?

Activity 3:

Cut out the figurine provided on the template. Place it so that it coincides with the figure drawn on the band without a twist. Slide the figurine along the dotted line on the band so that the figurine returns to the figure drawn. Record your results. Repeat the procedure for the Möbius band. Remember that a surface has no thickness so the figures drawn on the bands really exist on both sides of the paper simultaneously. (Note: looking for better wording on sides of paper.)

Comments:

The topological characteristic of a surface that you have discovered is called its orientability. An annulus is orientable; a Möbius band is non-orientable.

Activity 4:

Use additional copies of Template P to fill in the chart on page 55. Record any observations that you make. State any generalizations at the bottom of the chart.

Question:

Indicate whether each surface is orientable or non-orientable.

A five half -twist surface ?
An eight half- twist surface ?
A seventeen half-twist surface ?
A fifty-four half-twist surface ?
A spherical surface ?
A torus surface ?
A disc surface ?

Extensions/ Conclusions:

Topological properties of a surface that can be observed from within the same dimensions as the surface (2-D) are called intrinsic, i.e., an insider's point of view. Topological properties that can only be observed from a higher dimensional perspective are called extrinsic, i.e., an outsider's point of view.

Questions:

Indicate whether each of the following properties is an intrinsic or extrinsic chacteristic.
- A full twist versus no twist.
- Odd versus even number of twists.
- A surface with a knot in it versus no knot.
- Number of knots in a surface.
- Bends in a surface versus no bend.
- Stretch in a surface versus no stretch.
- A hole in a surface versus two holes.
- Euler characteristic of a surface.
If the figurine used in the above activities was a 2-D being what could it determine about the shape of the surface it is living in?
Could the figurine determine whether the surface is orientable or non-orientable?
Could the figurine determine whether the surface is a circular disc or a square disc?
Could the figurine determine whether the surface is a rectangular disc or an annulus (tube)?
Could the figurine determine whether the surface is a sphere or a torus?