Slicing and Contours

Jeremy Kahn

Banchoff's responses are in italics.

A clarifying thought

Most people probably recognize this, but since the editors of B3D don't like using real math, it didn't make it explicitly into this chapter: slicing is the same as examining all the points in a shape that have a certain fixed value (K) in one dimension or another. We can rotate the axes so that the slice is at whatever angle we like.

You're quite right that the editors were the culprits. They wouldn't even let me put the equations for my surfaces and hypersurfaces in an appendix. In dtext, we can run some nice interactive demos to show how equations are turned into contours for any fixed value (we use "k") of the height (in whatever dimension).

Slicing people

Banchoff discusses, in the opening sentence, a technique for botanical (and other) observation of the internal structure of an organism: one embeds the organism in plastic and then slices the plastic. There is a working biology project called "The Visible Man" which has accomplished the same thing with a human cadaver (I believe the subject was executed by lethal injection, so the toxin made his organs useless for anything else). Each slice's information was digitized and the compiled information was put on a computer, allowing the user of the machine to navigate through the human body with the click of a mouse. A similar project called "The Visible Woman" is underway. Notice that this method of observation does not require careful selection of the "critical" sections. If one wanted to put the "Visible Man" CAT-scan-like in a textbook (or a Scientific American Library edition!) one would have to limit the number of slides, though, and then it would be necessary to select certain critical sections to make one's meaning clear.

Presumably the visible person can be examined from different viewpoints? Perhaps it would be less disturbing to have a "Visible Frog" or "Visible Earthworm", without the attendant formaldehyde odors.

Axial, sagittal, and coronal go to 2-space

We have three axes of slicing in the 3-space we live in, and we give three names to the three ways we can slice our bodies (or at least we can in the abstract) along those axes: sagittal, as we slice by keeping a value on the line between our ears constant and observing the resulting plane, axial, as we slice by keeping a value on the line from crown to toes constant, and coronal, as we slice by keeping a value on the line from the bridge of the nose to the occiput constant. The 2-space creatures in Flatland, Arde, and Sphereland, would definitely have analogous (although only two of them) and they, too, would orient relative to the structure of the bodies involved, not some external coordinate system (coronal is the same whether one is lying down or standing).

What would they call these dimensions? The answer would probably vary by the kind of space involved. The corresponding 2-CAT-scans would yield line-images of the internal structure of a 2-spacer, of course. A Flatlander would probably not understand our description of him as an "axial" view of a 2-creature, since the dimension of axiality (if I may use such a word for the idea of up-down for us) is out of his perception. In fact, he can probably understand perfectly well the idea of sagittal (cutting slices from left to right) and coronal (cutting slices from front to back) but would get lost when one tried to explain the idea of axial (cutting slices from top to bottom). One can imagine him asking "which top? do you mean Northward? or forward? I can make those two coincide, or I can change them so they don't match. what do you mean?"

The fact that we can use the words "sagittal" and "coronal" in a way that A. Square can understand is in fact probably a good diagnostic to tell us that he is a member of the third type: a coronal slice. Similarly, the humanoid creature in Sphereland probably understands "axial" (head to toe, top to bottom) and for that matter, like A. Square, can understand "sagittal," too (one hand [leg, shoulder, hip] to the other), but would have real trouble with the idea of "coronal." "How do you mean 'front'?" one can imagine it asking. "From my head to my toes? That's "axial," you've just described it. Where else would I look? Certainly not from one hand to the other--that's doesn't even yield symmetrical slices," a characteristic of both axial and coronal slices.

Since bilaterally symmetric 3-creatures like ourselves show symmetry in both axial and coronal slices, we would expect 2-creatures based on coronal or axial slices of ourselves to show symmetry as well. Those based on sagittal slicing, though, wouldn't necessarily be that way. The rocketship in "The Planiverse," for example, has an airfoil's profile, which necessitates a definable up vs. down (though it is flipped for landing) and a front vs. back since it has a rocket in one end (and not in the other!)

Your linguistic sensibilities are showing through. I agree that more operationally descriptive terms would be more useful. "Sagitta" means "arrow". What does that have to do with anything? We use things like "x-slice" or "y-slice" but that depends on the initial orientation of the individual being sliced.

Slicing a Hypercube

Slicing along the diagonal of an n-cube seems to yield an (n-1)-simplex that is then truncated by the sides of another (n-1)-simplex, larger than the first, that is slowly moving in and slicing off segments at the corners. An interesting way to think about the second set of opposite faces appearing in the intersection space.

Right, the diagonal slice of an n-cube will start with an (n-1)-simplex and then experience truncation when the next set of n-cubes comes into the picture. What happens for a 5-cube by the way? The slices are still within visual range.

Jeremy Kahn x6753