FoRK-Fork: Talent (dozens & degrees)

[Dave Long]

> ... as in a talent of silver.

which is the same amount as
60 minas of silver.  Now, a
while back on FoRK, we had
traced 80-column lines back
a century or so, but 60s go
all the way back to Babylon.
(see Plimpton 322)

A "J U S T - S O" S T O R Y
---------------------------

Just as we try to determine
if a particular computation
can be done in a polynomial
number of steps, geometers
in ancient times would try
to construct entities with
compass and straightedge.

When fiddling about with a
compass, it is difficult to
avoid making hexagons: with  [0]
the radius set, rotate the
compass around a point on
the circumference, scribing
an arc through the center;
this will strike off two of
the hexagon's sides.

Now that we have a hexagon,
a triangle is pretty easy:
just connect every second    [1]
corner.  (Connect the rest
and get a Magen David)

On the other hand, by using
perpendiculars, we can make
a square -- and now we have
something more interesting:
by taking the product (if we
may anticipate Descartes) of
the square and triangle (by
superposing vertices and then
marking points until we can't
find new ones), we wind up
with a 12-fold split of the
complete circle.

D O Z E N S
-----------
That's a dozen, and we run
into it in many places --
some mostly obsolete, like
astrological signs, some
still with us, like the 12
hour display for Micros~1
Windows' "set time" panel.

minas and minutes
-----------------
What if we need more points
dividing our complete circle?
We can build a pentagon with [2]
compass and straightedge; it
can be "multiplied" by the
dozen points we already have
to generate a 60-fold split
of the whole.  We run into
this number other places --
some mostly obsolete, like
60 minas in a talent, some
still with us, like the 60
minutes in an hour.

degaussed
---------
Let's take it to the next
level.  Multiply our 60-gon
by a hexagon -- Oops! we're
stuck with those 60 points.
Obviously (with the benefit
of number theory) we need a
new prime polygon.  Well, it
took Gauss to show that the
constructible ones are those
with 2^2^N+1 sides; perhaps
the Babylonians thought that
pentagons were the maximal
constructible polygons (but
then, who would blame them
for ignorance of the 17-gon
or the 257-gon?).

D E G R E E S
-------------
No doubt they felt like they
were going around in circles
after failing to build 7- or
11-gons, and simply decided
to fall back on arithmetic:
3*4*5*6=360 degrees.         [3] 

-Dave

[0] it's fortunate we get the
hexagon for free, as we can't
trisect arbitrary angles.

[1] what if we connect every
third corner?  That's a rather
degenerate polygon, two-sided:
a bit?

[2] this is not so easy as the
constructions above, and uses
the golden section.

[3] it's probably just as well
there was no Babylonian Gauss,
as we'd need 1020 degrees?