3. #7 is a pretty good example too.
3. #7: a 4D person can take the rope at any of the crossing points, pull a bit of it into the fourth dimension, pull it around, and put it back so that the crossing is reversed. If I did that to the crossing at the bottom right of the first knot it would then be unknotted. If I did that to the top left and the far right crossing of the middle knot it would be unknotted. If I did that to the top left crossing and the far right crossing of the right knot it would be unknotted.
2. Tell the number of vertices and edges in the 4-D pyramid whose base is below. Explain how you figured it out.
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2. One way to figure it out is to draw it and count vertices and edges:
 vertices: 7 edges: 15 |
 vertices: 7 edges: 18 |
Or you can reason it out like this:
The base has 6 vertices. A pyramid will have one more vertex for a total of 7 vertices. The base has 9 edges, so there are 9 blue/white edges. The base has 6 vertices, so there will be 6 red edges . The pyramid has a total of 9+6=15 edges |
The base has 6 vertices. A pyramid will have one more vertex for a total of 7 vertices. The base has 12 edges, so there are 12 blue/white edges. The base has 6 vertices, so there will be 6 red edges . The pyramid has a total of 12+6=18 edges |
Be able to tell what you get when you remove a pair of points from a 1D something
1. Do #1 on page 338. S means that the solution is in the back of the book, not that a theta is shaped like an S. you cn fnd a theta on page 333.
2. Show/tell how many pieces you could get if you removed a pair of points from this 1D object. Sketch each possible outcome.
Tell how many pieces you could get when you remove a pair of loops from a 2-D something
3. What is the smallest number of pieces you can get when you A. remove 1 loop from each? B. remove 2 loops from each? Sketch the outcomes.
sphere
torus
2-holed torus1. Do #1 on page 338. S means that the solution is in the back of the book, not that a theta is shaped like an S. you cn fnd a theta on page 333. If you remove the two 3-vertices you will get 3 pieces. You can't break a circle into 3 pieces by just removing 2 points, so that meant the two shapes must be different (not equivalent by distortion)
2. Three possible outcomes from removing two points
Tell how many pieces you could get when you remove a pair of loops from a 2-D something
3. A. 1 loop removed:
2 pieces
1 piece
1 pieceB. 2 loops removed:
3 pieces
2 pieces
1 piece
