Thursday, December 17, 2009

Hammering on Synergetics

--- In, "John Brawley" wrote:


> > It's an alliance with XYZ that we're forming,
> > I think that's what you'll find. There's no
> > either/or, and both together is going to
> > take us further than either alone. That's
> > Synergetics for ya!
> That'd be the best of all worlds. Good luck to ya'.
> Be sure to hide your roaches.

OK, funny response.

Let's make this easier and just work with a triangle.

Here's an equilateral triangle, call it empty.

A turtle at the lower left corner is going to climb along an edge, up towards the apex, call it climbing the mountain.

As the turtle moves up the mountain, a ray from said turtle to the opposite base vertex is drawn. The slope of this ray increases as the turtle climbs to the apex.

The area under the sloping ray is colored red.

Question: is the linear motion of the turtle paired with an areal change that reflects the phenomenon A x B? Answer: Yes.

By that I mean: the conventional picture of a line going across a square, showing a larger and larger rectangle (becoming a square), matches a sweeping motion causing the angle between two rays to widen, as the gap between their tips grows to complete an equiangular triangle. Either model is adequate for showing A x B, but the triangle does so with fewer edges.

Consider said equilateral triangle to be 100 x 100. If you crosshatch with 3 sets of parallel lines (multiplication by division), you get the right number of little triangles (10000).

As the turtle climbs the mountain, the right number of little triangles turn red (picture pixels). Maybe increase the resolution? Make it all smooth.

So this is a picture of T x B, where T is Turtle position up the mountain and B is Base. Say B = 100. Then if T is 43, we're saying the red triangle of 43 x 100 has just the right area, in triangular units. Could we vary B as well as T? Sure.

This whole thought experiment works again with the tetrahedron, add another turtle, so the important thing to realize is we have a model of A x B x C. They (the three edges) don't have to be the same length. The answers come out the same. The numbers don't change. It's the visualization that changes, leaving the numbers alone.

This is a freedom we choose to exercise without breaking any rules. The use of a square and cube is by convention. Triangles and tetrahedra have a lot going for 'em. "Teach the controversy"(threw that in just to irk ya). Once you're acculturated, then you have all these other whole number volumes to be excited about, as you visit our core sculpture (middle of the castle atrium, right this way folks, watch your step and watch out for treacherous qyoobists, insecure about others visiting here).

The idea of a conversion constant is somewhat by convention. One makes choices. Bucky is saying to the cube guy: what you call 2, I call 1. There's this unity-2 thing going on, where he deliberately does that. Your two radii are my 1 diameter, so my 1 x 1 x 1 = 1 model is a regular tetrahedron, but from your point of view, given these are radii, 2 long, it's like 2 x 2 x 2, except that you, being a Cartesian, fixate on three cube orthogonals and inter-multiply those, so sqrt(2) x sqrt(2) x sqrt(2) where we're seeing 3. His way of doing this gives us a modestly close-to-one constant. I'm not saying one couldn't conceive of it differently. There's a utilitarian aspect to his design.

[ Now you come along and choose a camp, but don't like the pronouns. "You sir, say I should want two, but I want a one, Coke better than Pepsi". So you're not playing a straight Cartesian the way Bucky would. What can we do about it? Nothing. What should we do about it? Just let the cameras roll. ]

Anyway, my point is squares and cubes really aren't as economical, when it comes to modeling A x B and A x B x C. Our civilization could have taken a different turn, and it's not too late to imagine the ETs enjoying this other way of thinking, even if we find it alien.

You can always think of A x B x C as red water in the regular tetrahedron, partially filling it (A, B and C are up to and including the entire edge of what begins as an empty "cup"). As you tilt the thing, the water line changes. The total volume stays the same. If you point it straight down, the water level will be even along all three edges so read off the 3rd root of your original product A x B x C. Like if you had 3 x 2 x 5 for 30 inside a 1000 volume total (10 every edge), and tilt it, you could have 3rd root of 30 at the bottom of your cup, a little more than 3 (a tick mark on 3 edges radiating from a common apex).

Why say "cube root" in the caption? All our angles are 60 degrees (Cheese Tetrahedron scenario).

1 x 3 x 10 would be another way to tilt it, different read-outs (1 x 3 footprint, fourth point at the apex -- same volume still).

Think of Synergetics as a standalone ride at the amusement park. It's from Mars, from the future, and there's no way you could live on it 24/7, especially if you're an old fart or boomer. You ride it, are amazed, and go back to what you were doing. Such is Synergetics, a place to visit. Now, some people, such as myself, have spent more time in that arena. But I'm not denying myself access to all the math I knew before, am still learning, will learn. I'm not steam rolling myself. I like my conventional math as much as the next guy, am a product of my century (the last one).

Synergetics is like hot sauce. Use sparingly. But use it, as it definitely has some of the right stuff. We could use a lot more of it in this day and age. We'll tell kids it's a different sandcastle. We'll show how internally consistent it is, give 'em a whirl wind tour. That's what the gypsies will do, or the Martians or whatever. Call it "being abducted" if you like, but it's not that melodramatic. You go back to whatever you were doing, but find yourself somehow thinking more like a Martian. Call it brainwashing, or call it getting your money's worth.

I always feel smarter in Windows (XYZ) when I work in Linux for awhile, like working out at the gym (IVM). Martian Math might contain our usual XYZ vectors, trig, some Coxeter type polytopes. We could talk about some different meanings of 4D. Philosophy for Children is not a bad idea. We have namespaces, even in math. It's not monolithic. We have different ethnicities. In this case, we're forming an alliance, between IVM and XYZ mathematicians, so that both will come out stronger, have more clout in the classroom, thanks to a more relevant curriculum.


PS: thinking back