The third chapter of B3D covers the basics of slicing objects and viewing them one piece at a time. It explains how one can put these slices together to form a mental picture of what the original object looked like before it was cut up. Analogies are drawn between the slicing of a three-dimensional object (which we can envision quite easily since we can perceive things both in two and three dimensions) and the slicing of objects in higher dimensions. The clarity of these analogies enable the reader to envision higher dimensional objects a little more easily, for every new angle is helpful.
I reacted most strongly to the first section in which time as a dimension is discussed, in particular as the third dimension as perceived by A Square. This explanation of A Square seeing the sphere as a 'circle for a while' really helped me understand how our ideas about time could be interpreted. Seeing a sphere for a while would be the equivalent--an image which fascinates me. (I really enjoyed the visit from the four-dimensional sphere on Friday, by the way.) It is instinctively clear that time cannot be the fourth dimension, for a mathematical ideal does not necessarily have anything to do with our perception of our environment--which is what our perception of time is, afterall--but I do appreciate such a simple way to envision how different even our perception of 'reality' may be from that very reality.
I have been thinking a lot about whether our universe has homogenous dimensions, some temporal and some spacial, or if they are distinguishable from one another. If one can tell them apart, is it possible that there is only one temporal one? That seems unlikely, but then again, it seems unlikely that they would be distinguishable, too.
3.1 When trying to imagine an object based on how its slices appear, is it helpful to have slices from many different directions or is one sufficient? Are some slicing directions more difficult to envision than others? Does the symmetry of the object influence your answer?
3.2 Is it more useful to slice from one direction than from another? Does slicing from different directions provide different information about the object? Can two different objects appear to be the same when sliced from certain points? If yes, give examples. If no, why not?