- Model building: one of the best ways to understand 3D and 4D objets is to make physical models of them. We have seen views of a number of 3D and 4D models, and have looked at slices, projections, and fold-outs of them as well. Several interesting sets of models could be produced from these, including:
- A collection of models of the hypercube that highlight the various parts in one way or another. For example, models using several of the views we have had, together with one of the 8 cubes highlighted to show the connections between the different models. Another possibility would be to make careful models of the various views of the hypercube that we will see next week.
- Similar models of the 4-simplex and its parts would be another possibility.
- We have talked about different views of the triangular-prism-based hyperprism. A collection of models of the different views of this object, or of another one, like the 4-simplex, would be very nice.
- Our "sweeping out" process tells us what a 5-cube would be like, and it would be nice to have a physical model of one, perhaps color coded to show the 10 hypercubes that form it.
- A collection of models showing the slices of a cube as it falls through flatland face first, edge first, and corner first. Similar slicing sequences for some other figures (like the tetrahedron) would also be nice.
- Sure to impress would be a sequence of models of the hypercube being sliced (which we will probably see next week). We will see this from several different "viewpoints" in four dimensions, so there are quite a few possible directions for this project.
- Even more impressive would be to make similar models of the 4-simplex and its slices.
- Models of the an unfolded 4-cube, 4-symplex, or other objects, color coded for gluing instructions, for example, would be a great project. An unfolded 5-cube would be very impressive.
- We have spent a lot of time looking at how to understand 4-dimensional prisms and pyramids. A great project would be to analyze a 4D object we haven't looked at in class, make models of it from different 4-dimensional viewpoints, and identify it parts, and so on.
- Slicing: We've seen that slicing is an important technique for understanding a higher-dimensional object. A careful study of the slices of more complicated objects would be an interesting project. You could try to develop some theories about how the slices can change, and what you can tell about an object from its slices. What kinds of changes can occur from one slice to the next? What are the most complicated slices you can get? What are the corresponding ideas for slices of four-dimensional objects?
- Unfolding: We have seen that unfolding is another important way to investigate higher-dimensional objects. An interesting project would be to look at unfolding a variety of familiar 3D objects, and see what can be said about the original from the unfolded version, and vice versa. What would it be like to unfold a human body, for example? A creative approach to this would be to do a photographic study of unfolding. What can your unfolding of 3D objects tell us about unfolding 4D ones?
- 4D Worlds: One of the questions you have asked is what a 4D person in a 4D world would be like. An ambitious project would be to try to develop something for use in a 4D world, along the lines of what we did for our 2D world. A more literary project would be to write a story about a 4D world. One possibility would be to rewrite "The Monster from Nowhere" from the monster's point of view. Another would be to write a continuation of the story that chronicles Burch's adventures once he has been pulled into the 4D world of the monster.
- 2D Worlds: Some of you may be interested in making more detailed 2D creatures and their environments than we did during the first weeks of class. You could work out carefully their internal body structures, and their machines, and other such items, and then explain them carefully in a written report. Or you could work out the details of "Flatland University". This project should include reading "The Planiverse", by A.K. Dewdney, which treats these issues very carefully in the setting of a story. Someone with a more literary bent might want to write a story about people in such a world and how they interact with each other and their surroundings. One possibility would be to take a well-known story (like a familiar fairy tail) and convert it to a 2D story. What is lost and what can be effectively transfered to one dimension lower?
- 4D Art: The fourth dimension has appeared in painting and sculpture in a significant way in twentieth (and now twenty-first) century art. Several examples appear in Beyond the Third Dimension. It would be interesting to compare the use of the fourth dimension by various artists, and when possible, read some of their writings on the subject. Those with a creative bent might want to try making some of their own dimensional artwork.
- The Shape of Space: When we watched the movie "The Shape of Space", we saw how it could be possible to have a finite, but unbounded, 3D universe. The shape of our own universe is not actually know, but within the last decade, satellites have been launched to collect data to try to determine that shape. An interesting project for someone with a more scientific leaning would be to find out the current status of this research, and what can be said about the shape of our universe.