Chapter 6: Chapter O' Cool Pictures!/P>
To begin with, I really like the idea of the hundred-meter horse. (Is that like the hundred-acre wood? Sorry, it's been a long day.) Being a vision person, I love illusions and think that 3-D illusions are no exception. In fact, they're especially cool. In the mentions of the trompe l'oeil dome paintings, I am reminded of the building in downtown Providence that appears to have lots of cool windows and to be really office-building-like. Upon closer inspection (you're cued in by the fact that the layer of windows appears to be dramatically peeling skyward), you can tell that it's a clever painting on a brick facade. Cool. And it's true that you often have to regard such a 2-D picture from more than one perspective (vantage point) before your eye is no longer tricked and you realize, hey, I'm moving and the relative position of those objects just isn't changing...
An interesting point is made near the end of the section entitled, "Viewing In Perspective". The reason these illusions fool us, and the reason that we expect things to change in a certain way when we move around them, is solely because of our EXPERIENCE in how things actually DO move. This is why, much as I love them, rotating or other changing-perspective views of higher dimensional objects is necessarily of limited usefulness because of our own limited experience. In other words, watching Prof. Banchoff inflate and then deflate a balloon to simulate a hypersphere passing through our space is useful for thinking about slices, but decidedly unhelpful in our actual comprehension of, or visualization of, the final object. Note that if we, creatures used to perceiving 3-D, were shown 2-D slices of a cube or even some irregular 3-D object, the graphics might aid us in forming a whole and complete picture in our minds of what the object is. We can never have that sense of understanding with the hypercube. Why? Experience. Our brains don't tell us how to put together 4-D objects, even if we have enough experience to develop expectations about which slice we might view next. I find this to be the most frustrating aspect of higher dimensions. I do, however, love pictures anyway. Animations, perspective drawings, whatever--they help us to achieve as much understanding as we can ever hope for. Or at least, so we understand now...
Speaking of visual presentation, I would love to see a Schlegel diagram of a polyhedron being made on a computer. In other words, I'd like to see the whole thing, central projection and all, and be able to interactively rotate the figure so I can see what's going on. I'm _so_ bad at visualizing things as it is. (The pictures on page 117 are nice, though.)
The picture on page 119 is cool, too--the Brisson watercolor. Tell me, are the images on either side of the thin vertical line supposed to be free-fused (i.e., viewed by crossing your eyes and focusing on the "third" image which appears in the middle)? Is it supposed to be a stereogram in our dimension, or only from a fourth-dimensional point of view? I guess it wouldn't do us much good if it was, since once again we're limited by our wonderful perceptual systems. As a note of interest, it CAN be free-fused to view something that looks kind of like a hypercube projection, but it's really hard to hold onto the image since the two views are so different. For more information on stereograms, both convergent (like the kind in the book, where you have to cross your eyes) and divergent (like the ones on posters in the mall, where you sort of diverge your eyes), click here, over here, or right here. (Yay, I finally figured out how to use links!!) Enjoy--it's a great way to display three-dimensional info within the confines of the two-dimensional page. They used to use them to help organic chem students visualize 3-D structure...