Monday, September 29, 2014

(w, x, y, z) in Four Dimensions

(Click on the comic if you can't see the full image.)
(C)Copyright 2014, C. Burke.

I may make a poster of this even if my students get neither the math nor the TV reference. They need to learn about both.

As far as the math goes, there are two points on the co-ordinate plane to start. Let's call the first one Point A, with coordinates (-5, 2). The other point we'll call Point B, was at (2, 4) -- so it wasn't a rotational image of Point A. Point B has up and moved itself, which points don't usually do -- but then, they don't sing TV theme ditties, either. The image of Point B is at a translation of up 3 and 5 to the right, or T5, 3, which puts it at (7, 7).

In the second image, B' moves from Quadrant I to Quadrant IV. It's not a reflection over the x-axis (although it could be represented as a glide reflection), so we'll call it another translation. This translation appears to be 3 to the left and 12 down, which we represent with negative numbers, T-3, -12. This puts B'' at (4, -5).

Things get interesting with the third illustration as the point moves two more times, this time moving in three dimensions. We don't have any scale for the z-axis, so let's label the original plane as z = 0 and then increment by 1 as we move up. Using that frame of reference, B'' is located at (4, -5, 0). It then gets translated to B'''(-6, -6, 1), which is a transformation of T-10, -1, 1, and finally to B''''(2, 5, 2), a transformation of T8, 11, 1.

Keeping in mind that this panel shows a Composition of Transformations, we denote it in reverse order, the translation of the translation, using the form

T8, 11, 1 o T-10, -1, 1

The final frame shows a tesseract. I don't get to use those very often, but they are fun to draw (and to say). A tesseract is a way of representing hypothetical four-dimensional spatial coordinates. It might be better represented with a three-dimensional model, in the same way that three dimensions can be modeled in a two-dimensional image, but since I don't have an actual three-dimensional comic, this is the best I can do.

We use the last three letters of the alphabet for the usual three dimensions, (x, y, z), so we're out of letters. How do we represent the fourth spatial dimension? We can use w for it, but it seems awkward using (x, y, z, w) (and maybe people who work with higher mathematics than I do actually do this -- I don't know), but it would seem easier and more logical to use (w, x, y, z), putting it first like move up an office tower increases the first digit of the room number (and different corridors might represent increments of the second digit). Because we can't tell where the point is coming from nor do we have any scale for the new position, we'll skip writing a transformation statement for it. (Or I could leave it as an "obvious" "exercise for the reader".)

One final note and I'll end my seemingly endless mind-dump of this stream of thought: I keep referring to four spatial dimensions. This is to distinguish from the usual four dimensions we live in. Originally, the comic read "(w, x, y, z) in 4D space-time", but something about that bothered me. First, I don't know that tesseracts would be used to represent a time dimension mixed with three spatial ones, although I don't see a reason why they couldn't be. Moreover, the entire point of the comic was to use "(w, x, y, z)", which wouldn't make sense if we were using time, which would more logically be represented as t, which would give us (x, y, z, t), space plus time gives us space-time.

And that doesn't sound like a radio station. Certainly not one Dr. Johnny Fever would work at.




2 comments:

Bill Ricker said...

Tesseract FTW !

My favorite short-story about Tesseracts is of course "'—And He Built a Crooked House—'", Robert A. Heinlein, (Astounding,1941). (PDF at Union.edu)
(GoodReads).

Dali's Corpus_Hypercubus (1954) might be problematic in a no-established-religion setting, but provides a visualization of the standard unfolding within the common European cultural tradition.

Very good point about distinguishing Euclidean 4-space from the T+3 we seem to inhabit.
For students who've seen rotation and projection matrices in geometry with trig, the subtle difference in the Metric tensor from wxyz to xyzt may be helpful. (It also explains the light-cones; particularly elegant in units where c=1.)

Dennis Stout said...

Minkowski geometry addresses space time.