Pythagorean Figured Numbers:
Oblong, Square, Triangular, etc.
John Opsopaus
Here we go on Oblong numbers...
The problem is that they are much easier to explain with pictures, as the
Pythagoreans did, than with words, to which ASCII is suited. But I'll try to
do pictures.
The easiest to understand are the Square numbers; these are ones that can be
arranged in a square:
* ** *** **** ***** ...
** *** **** *****
*** **** *****
**** *****
*****
1 4 9 16 25 ...
Well, they don't look quite square on my terminal; maybe they do on yours!
Mathematicians still call these numbers squares: 1 squared = 1, 2 squared =
4, 3 squared = 9, etc. There are lots of interesting things about them,
especially from a Pythagorean standpoint, but let's go on to the Oblongs.
An Oblong number can be arranged in a rectangle whose width and height differ
by one unit; for simplicity I'll make them wider than tall. So whereas the
Square numbers have sizes 1 X 1, 2 X 2, 3 X 3, 4 X 4, etc. the Oblong numbers
have sizes 1 X 2, 2 X 3, 3 X 4, etc. So the look like this:
** *** **** ***** ****** ...
*** **** ***** ******
**** ***** ******
***** ******
******
2 6 12 20 30 ...
They also have lots of interesting properties, but the one of relevence here
is that the ratios of their sides give the divisions of the monochord's string
that yield the musical intervals in order of decreasing consonance:
octave per.5th per.4th maj.3rd min.3rd ...
1:2 2:3 3:4 4:5 5:6 ...
I mentioned that each Oblong number is twice a Triangular number. The latter
are, naturally, those that can be arranged into an equilateral triangle:
*
* * *
* * * * * *
* * * * * * * * * *
* * * * * * * * * * * * * * * ...
1 3 6 10 15 ...
Notice that twice these are the Oblongs, 2 X 1 = 2, 2 X 3 = 6, 2 X 6 = 12,
etc. You can also see this geometrically; take the Oblong number 12 and see
how it can be divided into twice the triangular 6:
* * * / *
* * / * *
* / * * *
So there's a taste of Pythagorean number theory. The triangular number Ten,
the Decad, is very important, since it is the Tetraktys:
*
* *
* * *
* * * *
"The Power, Efficacy and Essence of Number is seen in the Decad; It is great,
It realizes all its purposes, and It is the Cause of all Effects. The Power
of the Decad is the Principle and Guide of all Life, Divine, Celestial, or
Human into which It is insinuated; without It everything is undefinite,
obscure, and furtive." (Stobaeus 1.3.8)
Pythagoreans take an oath by saying,
"I swear by the One who hath bestowed the Tetraktys to the coming
generations, source of Eternal Nature, into our Souls."
>From a musical standpoint its ratios (1:2, 2:3, 3:4) generate the octave,
fifth and fourth.
By the way, though Triads are very important in Arithmetical Theology, the
Tetraktys shows us that Dyads and Tetrads are also important, as several other
posters have mentioned.
I hope that's of some use to those not familiar with Pythagorean number
theory, and not to boring for those who are!
Hugieia,
John Opsopaus
P.S. "Hugieia" (well-being, soundness in body and mind) is a Pythagorean
salutation, also used as a talisman when written in the points of a pentagram
so: U/G/I/EI/A (in Greek letters, of course). -- JO