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 0

The 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 0

This 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