1. The left and right squares in the series of pictures below represent a square rubbery sheet with fixed sides. The small squares labeled A and C are little areas of the rubber. Draw on the blank middle square to show how you can stretch the rubbery sheet, while leaving the boundary fixed, to start at the left picture and get to the right picture 2. Refer now to the large square with six letters in it (figure above). Is it possible to draw a curve connecting A to A, another curve connecting B to B and a third curve connecting C to C inside the square so that none of the three curves intersect each other? Ifso, draw the three connccting curves. If not, explain why it can't be done. 3. On a transparent Mbius band, draw two circlcs that cross cxactly oncc. By "transparent" we mcan that when you draw on the Mobius band, it shows through on both "sides," - that is, it stains all the way through. If you took the rectangle below and joined the right and left edges with a twist, you would obtain a Mobius and. Suppose you cut the Mbius Band on the lines drawn. What would you get? Explain in detai 4. Here is a picture of the Borromean rings with two of the circles round and disjoint and the third onc looped connccting them. Notice that if you remove any onc of these rings, the other two come apart. Can you drawa sct of four Borromean rings with the feature that if you remove any one, the other three come apart to be three unknotted, disconnected rings? 5. If the letters R and P are madc from lines, then they are equivalent by distortion. True or False? Explain 6. N-dimensional cubes. A one-dimensional cube is a line scgment, a two-dimensional cube is a square; a three-dimensional cube is a cube, and so on. Complete the following table: Dimension Number of Number of Number of Number of of the Cube VerticesEdgesTwo Dimenlhree-Dimensional Faccs Faccs" 1. The left and right squares in the series of pictures below represent a square rubbery sheet with fixed sides. The small squares labeled A and C are little areas of the rubber. Draw on the blank middle square to show how you can stretch the rubbery sheet, while leaving the boundary fixed, to start at the left picture and get to the right picture 2. Refer now to the large square with six letters in it (figure above). Is it possible to draw a curve connecting A to A, another curve connecting B to B and a third curve connecting C to C inside the square so that none of the three curves intersect each other? Ifso, draw the three connccting curves. If not, explain why it can't be done. 3. On a transparent Mbius band, draw two circlcs that cross cxactly oncc. By "transparent" we mcan that when you draw on the Mobius band, it shows through on both "sides," - that is, it stains all the way through. If you took the rectangle below and joined the right and left edges with a twist, you would obtain a Mobius and. Suppose you cut the Mbius Band on the lines drawn. What would you get? Explain in detai 4. Here is a picture of the Borromean rings with two of the circles round and disjoint and the third onc looped connccting them. Notice that if you remove any onc of these rings, the other two come apart. Can you drawa sct of four Borromean rings with the feature that if you remove any one, the other three come apart to be three unknotted, disconnected rings? 5. If the letters R and P are madc from lines, then they are equivalent by distortion. True or False? Explain 6. N-dimensional cubes. A one-dimensional cube is a line scgment, a two-dimensional cube is a square; a three-dimensional cube is a cube, and so on. Complete the following table: Dimension Number of Number of Number of Number of of the Cube VerticesEdgesTwo Dimenlhree-Dimensional Faccs Faccs
