Calculation, Combinators, Cable Coiling

Dave Long dl at
Wed Apr 9 10:52:43 PDT 2003

Calculation, Combinators, Cable Coiling
(plus chirality, coffee cups and quanta)

For the first few thousand years, counting
was a relatively primitive business, which
consisted largely of shuffling tokens and
following simple formal rules to complete 
a computation. [0]

The abacus was an advance on the counting
board, in that it fixed the digit schema,
but although it served as a good database
to store intermediate results, the digits
had independent representations, and the
logic to run a computation still needed 
an intermediate layer of "business rules"
in the heads of those sliding the beads.

The Pascaline advanced on the abacus, by
figuring out how to couple the beads.  If
one has two wheels, both marked (0-9), it
is possible to add them together to count
up to 10+10=20.  In proper combination [1]
it is possible to multiply them to count
up to 10*10=100, and the Pascaline[2] had
both several dials and carry propagation.

In a way, coupling units and tens with a
carry bit is similar to twisting a loop to
form a Moebius strip: when untangled, there
are two independent wheels, like the normal
loop of two sides; after entangling, there
is one long virtual wheel, like the Moebius
strip with only one single side.

Maybe it would be possible to use a log
scale [3] with a Moebius loop so that it
could also multiply, but normally these 
are just additive loops: they turn what
used to be two edges of length 2pi each
into a single edge of length 4pi.

Loops of length 4pi are not without their
uses; they allow us to:
  - coil a cable so it uncoils nicely
  - rotate a full coffee cup completely
    (without letting go or spilling)
  - leave a fermion in the same state
    in which we found it
and so, sometime soon maybe we'll advance
on the Pascaline, by shuffling entangled
states through simple formal operators to
complete a computation.


:: :: ::

[0] Do not believe that the Romans used
to calculate with Roman numerals.  Sure,
they gave lists of figures in that format
to their computers, and expected answers
back in the same format, but the servers
would have calculated in decimal, with a
conversion for input and output.  If you
find this odd, consider how we give lists
of figures in decimal to our computers,
and expect answers back in that format,
but the calculations are done in binary,
with a conversion for input and output.

[1] BijSO, not TBijSO?

[2] invented by Blaise Pascal in 1642.
In our day, information processing is so
advanced that a government collects taxes
centrally, and uses part of the proceeds
to pay dividends (and return capital) to
people who have advanced it money.  In
Pascal's fathers' day, the relationship
was inverted: someone would advance the
government money, and (as he was better
informed of his neighbors' situations
than a bureaucrat would be) would then
get the privilege of collecting taxes
due (plus a bit for a dividend stream)
in his local area.  Of course, Pascal's
father not only had to collect taxes to
recoup his investment, but he had to do
so in 20's and 12's as well as 10's, so
those digits had their own wheels.

[3] courtesy Napier, 1614

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