1. Do # 1, 2, 3 on page 341
#26: These are not equivalent because the one on the left has 6 holes (don't forget to count the middle) and the one on the right has only 5
#27
#28
Be able to tell what you get when you remove a pair of points from a 1D something
Tell whether 3-D objects are equivalent
1. Do # 1, 2, 3 on page 341
#26: These are not equivalent because the one on the left has 6 holes (don't forget to count the middle) and the one on the right has only 5
#27
#28
Draw movies showing how 2D or 3D objects are equivalent
2. Draw movie pictures to show how these are equivalent by distortion:
3. If you start on the bottom of the diagram and follow it to the right, you get to point1, which is glued to point 1 on the top left (green path). continue on to the right from there, and you get to point 2, which is glued to point 2 on the bottom left (blue path). Continuing on to the right, you get to where you started. When you get back to where you started, you have traced over all of the edge pieces, so all of the edge pieces are connected, and there is only one edge.
4. The blue parts below are glued together, and the green parts are glued together, so when you cut down the center, everything will still be in one piece.
You can also thing of cutting first, and then gluing blue to blue (and green to green). Here is a picture showing how the blues get taped together:
(note: the letter that looks like a g is a flipped upside down a)
5. The Moebius strip is non-orientable because if you were a 2-D person holding a flag in your right hand, and you marched around the moebius strip and back to the beginning, when you got there, the flag would now be in your left hand (handed-ness changes)
6. The first has 3 edges: start at the green *, and follow the path to point 1 at the NE, back in at 1 at the N, and back to where you started (one edge). Next, start at the blue *, follow that edge out to 3 (E), and back in from point 3 (W), out at point 5 (SW) which is glued to point 5 (NW); in and out again at point 2 (N), in at point 2 (NE), and back to where you started. Finally, start at the red *, follow that path along, going in the direction of the arrows, and jumping from numbered points to the points they are glued to until you get back where you started. At this point, you have colored all edges, and used 3 colors, so there are 3 separate edges.
The second has only one edge. Follow the arrows around to see how you get to all of the edge points, just by following along long enough:
