This chapter presents numerous facts about area, volume, and the patterns found among these properties. Don't just take Thomas Banchoff's word about these concepts - test them yourself! Fill a cylinder from a cone, slice a cube into congruent pyramids, construct a fold-out pattern for a hyperpyramid. I had trouble comprehending many of the equations in the chapter, and the illustrations helped me conceptualize them. Building models would provide an even clearer understanding.
The demonstration of the Pythagorean theorem on page 27 reminded me of tangrams. An interesting project might be to create various sets of polygons which fit together to form a square and to think about the way the areas of the shapes add up to the area of the square. This problem could also be taken to a higher dimension, by making polyhedra which fill the volume of a cube.
More advanced students could try translating the hieroglyphs on page 31. :)
Yes, yes--building models will indeed reinforce, or enrich, these three- and higher-dimensional ideas. I look forward to seeing what you come up with. I also like the Tangram idea, especially since it is very subtle to see how it might be interpreted in three-space. With a small number of pieces, you can make either a square or an equilateral triangle. How many pieces would you need so that you could assemble them into either a cube or a regular tetrahedron? The answer is surprising (at least to me). On second thought as I look at your tangram template I think it is clear that you can't get an equilateral triangle out of it--all the angles seem to be multiples of 45 degrees so you'll never be able to come up with 60, right? So, is there another decomposition of the square that can be assembled into the equilateral triangle? (Answer: yes. Find one, and maybe find one with the minimum number of pieces?)
Thanks, by the way, for linking our class with the Geometry Forum. Maybe some of the people here would like to form a team to solve some of their problems of the month?