2 NON-ORIENTABLE SURFACES §
2 Non-orientable Surfaces §
This section explores stranger surfaces made from gluing diagrams.
Supplies: Glass Klein bottle • Scarf and hat • Transparency fish • Large pieces of posterboard to cut • Markers • Colored paper grid for making the room a gluing diagram • Plastic tubes • Mobius band templates • Cube templates from Exploring the Shape of Space •
24 Mobius Bands 2 NON-ORIENTABLE SURFACES §
Mobius Bands
1. Cut a blank sheet of paper into four long strips. Make one strip into a cylinder by taping the ends with no twist, and make a second strip into a Mobius band by taping the ends together with a half twist (a twist through 180 degrees).
2. Mark an X somewhere on your cylinder. Starting at the X, draw a line down the center of the strip until you return to the starting point. Do the same for the Mobius band. What happens? 3. Make a gluing diagram for a cylinder by drawing a rectangle with arrows. Do the same for a Mobius band. 4. The gluing diagram you made defines a virtual Mobius band, which is a little di↵erent from a paper Mobius band. A paper Mobius band has a slight thickness and occupies a small volume; there is a small separation between its ”two sides”. The virtual Mobius band has zero thickness; it is truly 2-dimensional. Mark an X on your virtual Mobius band and trace down the centerline. You’ll get back to your starting point after only one trip around!
25 Multiple twists 2 NON-ORIENTABLE SURFACES §
5. A cylinder has two boundary circles. How many boundary circles does a Mobius band have? 6. Take a pair of scissors and cut your paper Mobius band down its centerline. What do you get? 7. Take the result from the previous step and cut down its centerline. What do you get now? 8. Take another strip of paper and crease it to divide it into thirds lengthwise and then tape the ends to make a Mobius band. (a) Predict what will happen when you cut your Mobius band along the creases. (b) Cut along the pieces and see what happens.
Multiple twists
9. Make a loop with two half twists instead of one. (a) How many sides does it have? (b) What happens if you cut it down the middle? (c) Do you get one loop, two loops, or more? (d) How long are these loop(s)?
26 Multiple twists 2 NON-ORIENTABLE SURFACES §
(e) How many half-twists do your resulting loop(s) have? 10. What about a loop with three half twists? 11. Experiment with di↵erent numbers of half twists and try to come up with rules that will let you predict what would happen with n half-twists. You can use the chart below to organize your findings.
# loops Length(s) # half-twists # half-twists # sides after cutting after cutting after cutting Notes . 0 1 2 3 4
27 Mobius Band Dissection 2 NON-ORIENTABLE SURFACES §
Mobius Band Dissection 1 12. Experiment with cutting a Mobius band lengthwise into strips of width for n di↵erent values of n. You can use the chart below to organize your observations.
Cut into strips # loops Length(s) # half-twists 1 of length after cutting after cutting after cutting Notes . n 1 2 1 3 1 4 1 5
28 Mobius Chains 2 NON-ORIENTABLE SURFACES §
Mobius Chains
13. Tape two ordinary loops together as shown. Cut each loop down the middle. What happens?
14. What happens if you tape one ordinary loop and one Mobius band together, and then cut down the middle of both? 15. Does it matter which direction you make the half-twist? 16. What happens if you tape two Mobius bands together, and then cut down the middle of both? 17. Does it matter which direction you make the half-twists? 18. Experiment with longer chains, or di↵erent numbers of twists.
Watch Vi Hart’s video: Wind and Mr. Ug
29 More Non-Orientable Surfaces 2 NON-ORIENTABLE SURFACES §
More Non-Orientable Surfaces
What surface do you get when you glue together the sides of the square as shown?
30 More Non-Orientable Surfaces 2 NON-ORIENTABLE SURFACES §
What happens as this creature travels through its Klein bottle universe?
A path that brings a traveler back to his starting point mirror-reversed is called an • orientation-reversing path. A surface that contains an orientation-reversing path is called non-orientable. • Can you find more than one orientation reversing path in this surface?
31 Tic-Tac-Toe on a Klein Bottle 2 NON-ORIENTABLE SURFACES §
Tic-Tac-Toe on a Klein Bottle
Which of these are winning positions in Klein bottle Tic-Tac-Toe? •
Where should X go to win? •
32 Tic-Tac-Toe on a Klein Bottle 2 NON-ORIENTABLE SURFACES §
What are the best moves for X in these positions? •
Play a few rounds of Klein bottle Tic-Tac-Toe. • – Is there a winning strategy? – Is it possible to get a cat’s game? – How many essentially di↵erent first moves are there?
33 Tic-Tac-Toe on a Klein Bottle 2 NON-ORIENTABLE SURFACES §
What happens when you cut a Klein bottle in half?
It depends on how you cut it. Cutting a Klein bottle in half - animation •
Cutting a Klein bottle in half - IRL •
34 Non-orientable 3-manifolds 2 NON-ORIENTABLE SURFACES §
Non-orientable 3-manifolds
Is there a 3-dimensional analog to the Klein bottle?
35 Surfaces made from a square 2 NON-ORIENTABLE SURFACES §
Surfaces made from a square
We have seen that the following gluing diagrams describe a topological sphere, torus, and Klein bottle.
What other topological surfaces result from gluing the edges of a square?
36 The Projective Plane 2 NON-ORIENTABLE SURFACES §
The Projective Plane
37 Identify Surfaces Made from Gluing Squares 2 NON-ORIENTABLE SURFACES §
Identify Surfaces Made from Gluing Squares
Example. What surfaces do these gluing diagrams represent?
38 Orientability and 2-Sidedness 2 NON-ORIENTABLE SURFACES §
Orientability and 2-Sidedness
Use the four provided templates to build four cubes. The markings on the sides • indicate how opposite sides of the cube should be identified to create four 3- manifolds. These manifolds are called the 3-torus, Klein space, quarter-turn space, and half-turn space. Identify which space is which.
39 Orientability and 2-Sidedness 2 NON-ORIENTABLE SURFACES §
For each of the four spaces, imagine a square that sits in the middle of the cube, • halfway up the cube. There are multiple ways to do this, depending how you position the cube. (How many ways for each cube?) For each way, this middle square gets glued up to form a surface in the 3-manifold. For each example, identify the surface and say whether it is orientable or non-orientable, and whether it is one-sided or two-sided. Is it possible to have an orientable surface that is one-sided? two-sided? Is it • possible to have a non-orientable surface that is one-sided? two-sided? If we only consider orientable universes, is a surface non-orientable if and only if • it is one-sided? Prove it or give a counterexample to show it is not true.
40 Shape of Space Video 2 NON-ORIENTABLE SURFACES §
Shape of Space Video
The Shape of Space Video
41 Gluing diagram and Klein Bottle Game Problems 2 NON-ORIENTABLE SURFACES §
Gluing diagram and Klein Bottle Game Problems
1. A ladybug on a Klein bottle walks in a straight line until she returns to her starting point. She walks 1 unit northward for every 1 unit eastward. Draw her path.
2. What surfaces do these gluing diagrams represent?
3. What surfaces do these gluing diagrams represent?
42 Gluing diagram and Klein Bottle Game Problems 2 NON-ORIENTABLE SURFACES §
43 Gluing diagram and Klein Bottle Game Problems 2 NON-ORIENTABLE SURFACES §
4. Try this word search on the Klein bottle. The arrows show how the sides are glued together.
5. Play some more word searches or other games on the Klein bottle on Torus Games.
44 Gluing diagram and Klein Bottle Game Problems 2 NON-ORIENTABLE SURFACES §
6. Find a classmate and try chess on a Klein bottle. Use the starting position below, and glue the left and right side as you would for a torus, but glue the top and bottom with a flip.
7. If a bishop goes out the upper right-hand corner in Klein bottle chess, where does it return? 8. In Klein bottle chess, starting from the initial position above, can the white bishop capture the black rook in one move? Starting from the initial position, can a black knight capture a white rook in one move? 9. Make up your own crossword puzzle on a Klein bottle. There should be a cross-
45 Gluing diagram and Klein Bottle Game Problems 2 NON-ORIENTABLE SURFACES §
word puzzle editor built into the Torus Games App.
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