This chapter invites the reader to search for patterns in the usual two- and three-dimensional volumes that can be generalized to produce ways of finding volumes of objects in four and higher dimensional spaces. It begins with methods of dividing objects into smaller objects whose volumes can be measured and then added, and moves on to some visualization techniques using foldouts. These foldouts are intriguing models, and I look forward to their more in-depth coverage in Chapter 5.
My favorite part of this chapter was the treatment of the binomial theorem. It is wonderful to see such nicely drawn geometric interpretations of the squared and cubed steps of this procedure because I think they link the algebraic with the geometric most clearly, something which is not and cannot always be accomplished. I believe that these pictures go on to help one imagine what a geometric construction of a hypercube out of hypercubes and prisms might look like. Seeing that there would be an analogous construction helps one realize later on in the chapter that the 'four-dimensional artisan' may actually face a problem much like the three-dimensional one solved by the ancient Egyptians.
It is very important to appreciate the accomplishment of the Egyptians when they successfully calculated remaining volumes using the factorization of a difference of two cubes. We are taught these things as already-known facts, but for those making the discoveries, the triumph was immense. I enjoyed the inclusion of the Rhind papyrus as a visual reminder that their methods were quite different from ours and that the solution of a problem of this nature was indeed very serious work.
2.1 Think about finding volumes of various three-dimensional objects by dividing them into objects whose volumes you are able to calculate. Are there methods other than by such a division through which you could arrive at an answer? How could you estimate your own volume?
2.2 Build some fold-outs of different objects. How does the volume of the objects correspond to the area of the faces?
2.3 Imagine you are a laborer (assuming you are not merely a slave whose input is negligible, rather some sort of moderately well-respected foreman) in ancient Egypt whose job is to oversee the excavation of the rock needed to complete a giant pyramid. Write a story about your interactions with the mathematician who tells you how much material you will need. What sort of disagreements would you have had? Would his figures seem mystical to you or rational? Do you think you would have understood his reasoning? How would your powers of reasoning have influenced the exchange?