Our world is three-dimensional (or four, if you count time as a dimension). We have been looking at worlds that have a different number of spacial dimensions. In the the Flatland video, A. Square lives in a two-dimensional world and visits worlds of zero, one, and three dimensions. There is also some speculation about the possibility of a four-dimensional world.
Two- and four-dimensional worlds are discussed in Sections 6.1 and 6.2 of Symmetry, Shape, and Space. You should read through these sections, but you won't need to know everything that is covered in them. From Section 6.1, you might be interested in thinking about some other possible conceptions of 2D worlds, especially the 2D house from Arde, on page 191. You should read Section 6.2 more carefully and think about four dimensional worlds and the directions ana and kata.
This assignment is due in class on Friday, November 21. You will have some time in class for group discussion of the assignment, but you should turn in your own individual answers to the exercises.
| Class will meet in Gulick 208 on Friday, November 21 for a computer lab. |
Problem 1. Exercise 1 in Section 6.2 of Symmetry, Shape, and Space reviews three-dimensional coordinates for points in 3D space, such as (5,-5,5) or (1,3,-7). Describe the position of the point in 4D space that has four-dimensional coordinates (0,0,0,1). (You can use the directions named ana and kata. You have to decide which of the directions ana and kata goes in the positive direction and which goes in the negative direction.) Do the same for the 4D point with coordinates (5,5,5,5) and for the 4D point with coordinates (5,-5,5,-5).
Problem 2. In many ways, the Flatland video does not present a realistic view of a 2D world. Write a short essay (not more than one page long) discussing some of the specific things that you saw in the video that really do not make sense in the two-dimensional world depicted in Flatland. Explain why they don't make sense.
Problem 3. Carefully read Section 6.2 of Symmetry, Shape, and Space from the bottom of page 196 through Exercises 5 and 6 on page 197. Answer Exercises 5 and 6.
Problem 4. Take a look at Exercise 16 in Section 6.1 of Symmetry, Shape, and Space (page 187). The issue here is that a rigid asymmetric shape in a plane cannot be converted to its mirror image by moving it around in the plane. Why not? What's the answer to Exercise 16? (That is, how does the 3D entity, the Sphere, change a mongrel dog into its mirror image?) Now answer Exercise 7 in Section 6.2: Explain how a left glove (in our 3D world) could be converted to a right glove using four-dimensional hyperspace.
Problem 5. We have looked at the possibility that Flatland is not really flat. That is, perhaps it is not a plane but instead is equivalent to the surface of a sphere or of a torus (doughnut shape). Another possibility for a 2D world is that the world is a (very large) Möbius strip. Suppose that an asymmetric figure such as a letter "R" or a mongrel dog wanders all the way around the strip. What would happen to it when it returns to its starting point? It might help to look at the strip cut from edge to edge and laid out as a rectangle:
Look at a letter "R" starting at the center and moving left to right along the (uncut) Möbius strip. When it reaches the (conceptual) cut that we have made, it actually continues through the cut and appears on the other side of the "cut." Draw several pictures that show the "R" moving along the strip, passing through the "cut," and finally arriving back where it started. What has happened to it (besides being upside down)? What does this have to do with Problem 3? Do you think that you could send a left glove on a long trip through outer space and have it come back as a right glove? (Turn in your drawings along with an essay answering the questions in this problem.)
Problem 6. We encountered the artist M.C. Escher earlier in the course because of his work that uses tessellations and wallpaper patterns. Escher also has some work that illustrates the problem of dimension. A basic issue in art is the problem of representing a 3D world on a 2D surface. Take a look at the following works of M.C. Escher, which comment on this problem:
Pick one of these works and write a short essay discussing what the artwork has to say about the relation between two and three dimensions in art.
