Self-replicating factories circa 1790: Granularity

Let's say you were a visionary in 1790 and lathes and Mills existed. We can
adjust the year for when that was true.

Let's say you liked math and knew that a lathe was a cylindrical coordinate
system and the milling machine was a rectangular coordinate system.

Let's say you liked machines enough to want to reproduce an entire factory,
a very small factory, consisting of one mill and one lathe

So you sat and thought about it for 45 years which is what I did

No really.... I did

I came up with this

If every egg has a nest
And every nest is made of sticks
Then
The birds won't die
And we will be eating omelettes

What that means is that each machine has a work envelope let's say a lathe
with an 8x20 work envelope. And that if you make the sticks the right size,
which means if you apply a granularity of say 10 or say 100 to the work
envelope of all the machines including that lathe, you find that the lathe
can make 100 parts each of which has an aspect ratio dependent on the
granule selected but every one of which fits in the overall work envelope.
Likewise a Mill would be capable of making a thousand parts each of which
fits in the overall working envelope and each of which is more or less
distinct from all the others.

So for your project
You have 1,100 parts you can make basically

Well the rest is pretty simple

Aside from the really big frame parts which are made a different way

You just inventory all the parts in both of your machines and if the answer
is less than 1,100 parts you're pretty much good to go.

Isn't that nifty?

Well

It gets really nifty

They're actually 13 orthogonal coordinate systems

So theoretically there are 13 distinct types of machine tools it's all
theory anyhow

So if each of them has a granularity of 10 on each of the three axes

With 13 actual tools in an actual factory you would be able to count to
13,000 and that would be your inventory of parts to make those machines
from or rather remake them after you started with some model machines

That's nifty

It's even more nifty

Did you notice that on the lathe it didn't make a thousand different parts?
That's because it is so unusual to apply granularity to the alpha
coordinate in a cylindrical coordinate system that we just leave that one
out all together. Any kind of cylinder with a wedge removed from it, or any
kind of wedge with a cylindrical outer surface would be candidate part for
going back to including that coordinate.

Likewise each of the many orthogonal systems has axes on which a
granularity of 10 multiplies neatly and other axes on which a granularity
of 10 would be pretty much inappropriate.

So

Since it is possible for each egg to have a nest
And we can choose the sticks we use
A self reproducing factory
Which is non-trivial
And nearly universal
Is indeed possible,
And I think we should build one soon.

Cheers,

Douglas