Reactions to B3D (chapter 2)
Chapter 2 of Beyond the Third Dimension mainly consists of a series of dimensional analogies. For the most part, it deals with computing areas on volums in the second and third dimensions, and providing equations for four-dimensional "hyper-volumes". This is a lot easier to visulaize algebraically than geometrically. However, since on of the goals of the book is to help the reader visualize hyper-objects, both algebraic and geometric examples are used in the demonstrations.
I think it is a worthwhile exercise to divine fourth-dimensional formulae, because (ordinarily) we don't have to understand exactly why a formula works for two or three dimensions. They are pretty much taken for granted. However, in order to construct formulas for four dimensional figures, we must really understand what is going on.
Some good exercises for this chapter would involve raising two and three dimensional formulas to higher dimensions. For example, the volume of a four-dimensional sphere would be interesting to find. I don't know if that's possible. I would think that the book would have mentioned hyper-spheres by now. Other good exercises might involve finding the area covered by black triangles in Sierpinski's gasket, the volume covered by black pyramids in the three and four dimensional versions of Sierpinski's Gasket. The same could be done with the Koch snowflake. It is not hard to visualize either of these in the third-dimension (using pyramids), but the fourth dimensional versions would be hard to describe except algebraically.
Perhaps it would be interesting to come up with different kinds of such figures at Sierpinski's Gasket. Could we do similar experiments with shapes that are not equilateral and equilangular. In this case, there would probably different kinds of shapes formed in the figure. For example, if we decide to construct our shapes by connecting midpoints of adjacent line segments, we could get interesting and varying patterns depending on the shape we start with.
Having fold-out patterns for hyper-objects is an interesting concept. A good exercise would be to build models of fold-outs for hyper-surfaces.
In general, it seems like the main goal of this chapter is to raise familiar three-dimensional formulas and ideas into higher dimensions. Any expansion on this idea would aid the reader in understanding the subject matter.