Chapter 2 of Beyond the Third Dimension , Measurement and Scaling, deals with the similarities of various types of figures and their properties across different dimensions. The relationships between length, area, volume, and hypervolumes are examined, as are the relationships between cones and cylinders or pyramids and prisms. Copious illustrations, including fold-outs and geometrical diagrams of familiar mathematical theorems, are provided. I found the diagrams of the binomial and Pythagorean theorems quite fascinating and a wonderful way to conceptualize the ideas being presented in each. The graphics in this chapter, from the multicolored fold-out pyramids to the Sierpinski gaskets to the Koch snowflake, are engaging and clear demonstrations of the points at hand. My one complaint in this area regards the fold-out hyperpyramid, which I would have liked to see in more than one view so as to better understand what a three-dimensional model would look like.
One curious mathematical device mentioned in this chapter was Pascal's triangle, something which I remember vaguely from my high school days but have since forgotten all about. The pattern, mentioned on page 26, arises from (we are told) the "theory of combinations. Each number in the pattern is the sum of the two numbers above it. " This led me to wonder how Pascal discovered it and whether it was he who made the connection to binomial expansions. It is curious that this pattern should prove useful in other contexts, and I would love to know more about where it pops up again and what the significance of that reassuring mathematical regularity is, particularly in other dimensions. I have a feeling we will see it again in B3D.
Another convention in this chapter which struck me as I read it was the continued use of three-dimensional terminology to refer to objects in dimensions higher than the third one. We may tack on a prefix, such as "hyper-," but effectively we are still using the root "cube" which implies three dimensions. We face the same problem when we refer to a "hyperpyramid" or a "hypersphere". "But," one may argue, "a hyper-cube is just that--a higher cube, or sort of a beyond-a-cube figure." The problem with the term "hypercube" is that it does not specify to what dimension one is referring. It seems that convention would have the user attach a dimensional prefix, i.e.,"five-dimensional hypercube," if one is refers to something larger than a hypercube. The generic term for something in the nth dimension analogous to a cube is an "n-cube". But a fifth dimensional "cube" is not a cube any more than a cube is a three-dimensional square--i.e., it sort of is, but not really! Somehow it seems un-politically correct and 3D-centric to refer to everything in terms of a cube. It seems as though we need a new vocabulary to use in talking about these figures. (Perhaps I have just been at Brown too long :-) ). A related problem: how do we refer to the "volume" of four-dimensional figures? Clearly the word, the concept, we want is not volume. What, then, is it? And in threespace, we use the words "north, south, east, west, up, down" to orient ourselves. I have heard that there are corresponding terms for directionality in the fourth dimension. What are they? If you are giving someone instructions to look in the fourth-dimensional directions, where should you tell them to look? (Of course, whether or not they'll understand you is another matter entirely. But we have to print SOME kind of instructions on our fold-out 4-D pyramid.)
Questions On Chapter 2
1. Use the technique illustrated in The Egyptian Triumph: The Volume of an Incomplete Pyramid to determine the volume of an incomplete cone of radius r. (The answer should include a subtraction of the volumes of two cones.) Can this be generalized for higher dimensions, i.e., the fourth dimension? How?
2. Write out a "proof" of the Pythagorean theorem based upon the picture at the bottom of page 27.
3. (Challenge problem) Develop an infinite, reiterative process that results in a figure with similar properties to the Sierpinski gasket or the Koch snowflake. Illustrate a few iterations in color. :-)