Hello, friends...
After modeling a self-upgrading, co-modifying pair of four axis milling machines using inexpensive Chinese cast iron and establishing to my own satisfaction that I had implemented a self-reproducing machine tool, and after selling one of the pair to Larry Ritchey on October 24, 1997, for the cost of the parts needed to make the pair, which established to the satisfaction of a representative of our local government that I had indeed created a self-reproducing business model, I have come up with a compact notation for this grisly metalworking event, which spanned three years or so:
0 0 2 0The above is a condensed "quantity matrix." Kemeny et al in "Finite Mathematics with Business Applications" explain that when an order vector is placed with a manufacturing concern for a number of finished products *and spares*, a simple matrix inversion and multiplication provide the most efficient way to tot up what components are needed to fill the order. The result of the calculation is the production vector, which is orthogonal to the order vector. That is, one is a column vector, and one a row vector. I don't use orthogonal to mean what it usually means there.
As a quantity matrix the above reads (roughly) "You need two of item one to produce one of item two".
It's the simplest possible non-trivial quantity matrix!
I've written about self-reproducing machine tools before, mostly in rec.crafts.metalworking, but I have never expressed Goncz's Postulate so clearly. I've written "You need two of everything to make one of anything" a few times, and I've written it in similar ways as well.
Note the use of "every" and "any". These terms go back to John Von Neuman's "Theory of Self-Reproducing Automata". "Every" implies a finite number of things; "any" implies an infinite number. What I was saying was you have to start with two of every machine tool and accessory and hand tool in any proposed universal machine shop in order to qualify as a self-reproducing universal shop: a finite embodiment of Drexler's and Von Neumann's "universal constructors".
It's even simpler than screwing a nut onto two bolts with a match head between to build a firecracker. It's mating two *identical* machines with each other; like that crazy Zen Buddhist diagram with the circle divided into two two identical interlocking halves, usually colored black and white.
What fun.
Now I have something to build on.
Somebody (Einstein?) wrote that you should make things as simple as possible, but no simpler. This, I have done. Go ahead, substitute a one for the two, and see if the result is not entirely trivial.
And for those rowdy boyos in rcm who repeatedly remind me that there seems to be something really sexual about what I am saying, consider this matrix:
0 0 0 0 0 0 1 1 0This matrix says (when properly annotated) "You need a male and a female to produce... something new." It's really not the same thing at all. And as far as I can tell it's not the simplest possible non-trivial anything.
Now, who's bringin' the beer? :)
Yours,
Doug Goncz Replikon Research Falls Church, VA 22044-0394
email welcome to DGoncz at aol dot com