"Slicing and Contours"
So far, humans can't see hyper-objects. We can only see lower-dimensional shadows of them--infinitely thin slices. One way to learn to understand these slices is to first learn their simpler analogs. For instance, instead of trying to see four-dimensional objects through its three-dimensional slices, it would help to learn to visualize three-dimensional objects through their two dimensional slices. By analogy and oodles of brain power, this skill (simple enough to learn) should be applied to higher dimensions. A Square would see a sphere if he could see all of the sphere's circular slices at once. Likewise, humans would "see" a hyper-object, if he could visualize all of its three-dimensional slices in one fell swoop. Chapter Three of Beyond the Third Dimension is mostly an endeavor into the two-dimensional images of three-dimensional objects, with spottings of lower- and higher-dimensional analogs.
The picture on the opening page is of the "slices" of a Katmandune hillside. Out of curiosity, are those actually "horizontal slices"? Somewhere else in the introduction, the amoeba analogy is once again utilized. A camera takes periodic pictures of the "two-dimensional amoeba" in its measly "two-dimensional world". By superimposing all of the pictures, one ends up with a three-dimensional version of the amoeba's flat life. Essentially, this makes time a third dimension for the amoeba (but this does not mean that the third dimension is time). If (for example) length and width were the "standard" two dimensions, is "height" the standard third dimension. Why does it have more validity than time as the third dimension? Does it have more validity than time as the third dimension? If it is a matter of homogeneity, height probably would have more validity. A Square also experiences time as a third dimension when the sphere passes through his world. To him, he has just seen the "accelerated lifetime of a high priest". Interestingly enough, one of the writers in The Fourth Dimension Simply Explained describes human life as possibly an unfolding, then refolding out and into the fourth dimension...frightening thought...
The concept of hypercube slices is very difficult to grasp. How exactly do these relate to the models in class. If all we can see as humans are three-dimensional slices of a hyper-object, what do the models on class actually represent. Are they slices from a different direction, or more abstract representations of hyper-objects? Are the slices in the book just individual pieces of the three-dimensional models?
The interlocking circles in the torus slice, while a little hard to see, are very interesting. There's some extra space on the page--it would be nice to see it a little bigger. By the way, why is a horizontal torus a "bagel", and a vertical torus a "doughnut"? How foodist!
Anyway, some good exercises for this chapter might involve looking at two dimensional shapes and determining what figure they are slices of (and from what direction!). The reader might also benefit from making his own slices of three-dimensional objects. Drawing contour maps of the human face would be neat.
Until next week...