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§1 INTRODUCTION TO TOPOLOGY
§1 Introduction to Topology
This section is an introduction to the topology of surfaces.
References:
• The Shape of Space by Je↵Weeks.
• Torus Games at http://www.geometrygames.org/TorusGames/index.html or fromGoogle Play.
Supplies:
• Play-doh
• Transparency fish
• Topology grab bag (pretzel, bagels, banana, cup with two handles, sugar bowl, flipflop, lemon, scissors, blue thingy, yellow and red thingies, jug, shoe with strap,everyone’s shoes)
• Plastic tubes
• Torus games app
• Chess boards and pieces or cut-outs of pieces
• Flatland Movie
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Geometry vs. Topolgy §1 INTRODUCTION TO TOPOLOGY
Geometry vs. Topolgy
• The properties of an object that change when you bend or stretch it are the geometryof the object.
– For example, angles, distances, and curvature are parts of geometry but nottopology.
• The properties of an object that stay the same when you bend or stretch it are calledthe topology of the object.
– Two objects are considered the same topologically if you can deform one intothe other without tearing, cutting, fusing, or other violent actions.
”Mug and Torus morph”. Licensed under Public Domain via Wikimedia Commons
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Geometry vs. Topolgy §1 INTRODUCTION TO TOPOLOGY
Example. Which surfaces are topologically the same?
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Dimension §1 INTRODUCTION TO TOPOLOGY
Dimension
What is an example of something that is
• 1-dimensional?
• 2-dimensional?
• 3-dimensional?
What is meant by
• 1-dimensional?
• 2-dimensional?
• n-dimensional?
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Dimension §1 INTRODUCTION TO TOPOLOGY
Example. What dimension are these?
• Your desktop
• A straight line
• The circumference of a circle
• The inside of a circle
• The surface of the earth
• The air inside this room
• The surface of a doughnut
• The inside of a doughnut
• The surface of your skin
• Time
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Boundary and Extent §1 INTRODUCTION TO TOPOLOGY
Boundary and Extent
• A boundary of a space is ...
• A space is considered finite if ...
• A space is considered infinite if ...
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Boundary and Extent §1 INTRODUCTION TO TOPOLOGY
Example. Draw each of the following 1-dimensional spaces in its correct place in thetable:line segment with two endpoints line ray circle
With boundary No boundary
Finite
Infinite
Now try to think of 2-dimensional spaces to go in each place in the table. What about3-dimensional spaces?
What is the dimension of our universe? Does it have boundary? Is it finite or infinite?
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Surfaces and manifolds §1 INTRODUCTION TO TOPOLOGY
Surfaces and manifolds
• A surface is ...
• A surface with boundary is ...
• A 3-manifold is ...
• A 3-manifold with boundary is ....
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Gluing Diagrams §1 INTRODUCTION TO TOPOLOGY
Gluing Diagrams
Example. What topological surface do you get when you glue (or tape) the edges ofthe triangle together as shown?
Example. What do you get when you glue the edges of the square together like this?
Don’t glue the interior parts of the square together, just the edges!
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Gluing Diagrams §1 INTRODUCTION TO TOPOLOGY
Example. What surface is this?
Example. And this?
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Gluing Diagrams §1 INTRODUCTION TO TOPOLOGY
What happens as this 2-dimensional creature travels through its tiny universe?
What does it see when it looks forwards? Backwards? Left? Right?
See Torus Games animation
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Tic-Tac-Toe on a Torus §1 INTRODUCTION TO TOPOLOGY
Tic-Tac-Toe on a Torus
Where should X go to win?
What if it’s O’s turn?
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Tic-Tac-Toe on a Torus §1 INTRODUCTION TO TOPOLOGY
Example. Which of the following positions are equivalent in torus tic-tac-toe?
From The Shape of Space by Je↵Weeks.
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Tic-Tac-Toe on a Torus §1 INTRODUCTION TO TOPOLOGY
Download Torus Games at http://www.geometrygames.org/TorusGames/or from GooglePlay and play some games with a partner.
1. How many essentially di↵erent first moves are there in Tic Tac Toe on the torus?How many essentially di↵erent second moves are there?
2. Is there a winning strategy for the first player in Tic Tac Toe on the torus? Thatis, is it possible for the first player to win no matter what the second player does?Does it change your answer if the first player is required to start on the upper leftcorner?
3. A Cat’s Game in Tic Tac Toe is a game where neither side wins, even though theboard is filled up with X’s and O’s. Is it possible to have a Cat’s Game in Tic TacToe on the torus?
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Tic-Tac-Toe on a Torus §1 INTRODUCTION TO TOPOLOGY
How can you make a 3-dimensional space that is analogous to the 2-dimensional torus?
• Read the first two chapters of The Shape of Space (pp. 3-24) and work the problems
• Flatland movie (as time permits)
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Gluing Diagram and Torus Game Problems §1 INTRODUCTION TO TOPOLOGY
Gluing Diagram and Torus Game Problems
1. A ladybug on a torus walks in a straight line until she returns to her starting point.Draw her path and find the length of her path if:
(a) she walks 1 unit northward for every 1 unit eastward, as shown at the left.
(b) she walks 2 units northward for every 1 unit eastward as shown in the center.
(c) she walks 3 units northward for every 2 units eastward as shown at right.
2. If the ladybug walks n units northward for every e units eastward, where n and e
are integers, how long will her path be? What if n and e are not integers?
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Gluing Diagram and Torus Game Problems §1 INTRODUCTION TO TOPOLOGY
3. Consider the position in chess on a torus shown below. Which black pieces doesthe white knight threaten? Which black pieces threaten it?
4. Which black pieces are threatened by both the white knight and the white queen?
5. Play a few games of torus chess with a classmate. The usual starting position justwon’t do for torus chess (try it and you’ll see why). Instead, either use the startingposition below or make up a starting position of your own. All the pieces movenormally except the pawns: a pawn moves one space forward, backward, to the
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Gluing Diagram and Torus Game Problems §1 INTRODUCTION TO TOPOLOGY
left or to the right, and captures by moving one space on any diagonal.
6. When a bishop goes out the upper right hand corner of a torus chessboard, wheredoes he return?
7. In torus chess, can a knight and a bishop simultaneously threaten each other?
8. Do the following word search on a torus, or play a few on the Torus Games app.
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Gluing Diagram and Torus Game Problems §1 INTRODUCTION TO TOPOLOGY
9. A society of Spacelanders lives in a 3-torus universe that is made from a rectangularblock of space that is 600 km by 800 km by 2400 km. They want to build two spacestations that are as far away from each other as possible. Where should they placethem and how far apart will they be?
10. Make up your own word search on a torus.
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