Note: Am updating this page (slowly!). Have some undigested stuff on Pytgagoras tetractys at the bottom of the page (its related). April 2006
The Fibonacci Progression is as follows:
1,1,2,3,5,8,13,21,34,55,89,143.....
In other words, you get each number by adding the previous two. (If you
like, you can think of the first "number" as zero, so 0+1=1...see below).
Now, one significance of the FB is that it is widely found in nature. (The
little discovery that this post reports on was started off by an argument
that the Fibonacci progression was relevant to the Periodic Table of the
Elements). The ratio between each number infinitely approaches the so-called
"Golden Ratio" beloved of the ancients, more recently known to
mathematicians as the irrational number "phi" (approximately 1.618...). Phi
is deeply involved in the traditional canons of human proportion, the
classical position of the navel is at the "phi point" of the human body. (Ie
if navel to top= 1 unit, bottom of feet to navel =phi units. Of course,
actual humans vary and indeed the navel wanders a bit during the
lifetime...). Click on the title to this page for a site with (much) more info on the Fibonacci progression and related topics.
Anyone who remembers G's claims bout the enneagram and its place as a global
rule of nature, and its relation to the canons of human proportion will
expect that phi and the Fibonacci Progression will relate somehow to the
enneagram....
Now, Theosophical Addition. 192 "=" 1+9+2=12 "=" 1+2=3. So, by Theosophical
Addition, 192 reduces to, or "=", 3. We are all familiar with this, nicht
wahr? G argued that this is not a meaningless procedure dependent on the
accident that we use base 10, but a relating of number to the enneagram. So,
10 "=" 1+0=1, is just a way of saying that if we count ten spaces around the
enneagram we get back to one. And if we count 195 spaces around the
enneagram we get back to
6. (1+9+5=15, 1+5=6).
So what about the Fibonacci Progression? What happens when we perform
Theosophical Addition on it? A little thought will show that we must get a
repeating series of numbers. Here it is:
1,1,2,3,5,8,4,3,7,1,8,9,
8,8,7,6,4,1,5,6,2,8,1,9,
1,1...
ie a repeating series of 24 numerals; with an obvious "unwinding"
relationship between the first and second half of the series. And, remember,
each and every numeral in the repeating series is the reduced form of the
equivalent number in the non-repeating Fibonacci Progression, so the 24th
number in the FP reduces to 9 etc. (Those who can't see why this must be can
amuse themselves by testing it in practice!).
Call this the Fibonacci Pattern.
OK...now for the punchline! What happens when you map the Fibonacci Pattern
on the enneagram? Well...maybe you could have a play yourself.
What has amazed me is what happens when you map it in the Compass or Celtic Glyph of
the Enneagram. (Outside 123487651..., 9 in the middle. Ie the "inner line"
numbers are arranged in the order of the powers of 2 in reduced
form..aaaaah, it may seem complicated but have a play yrself...follow the link for
a picture on this site.)
On the Compass Glyph the Fibonacci Pattern forms an ancient symbol. This can
not be an "accident".....
Note: x= phi when: (1+x)=xSQUARED. Or, to put it less symbolically, when you
divide a line such that the smaller section of line has the same ratio to
the larger that the larger has to the whole then that ratio is phi:1, ie the
"Golden Ratio". And there is much else...
But the point is that, so far as I know, this here is the first ever
relation of the Golden Ratio to the enneagram......
"It is clear that the Binary doubling sequence and the
Fibonacci sequence are related by the following construction called
the Pascal
triangle, of Chinese origins:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Each number is obtained by adding together the numbers diagonally
above. If one adds the rows across, the binary sequence is obtained.
If the
numbers are added on a diagonal path, the Fibonacci sequence is
obtained. (vis: 1=1, 1=1, 1+1=2, 1+2=3, 1+3+1=5, 1+4+3=8, 1+5+6+1=13,
etc.)" , If, like me, you are a bit slow on the mathematical uptake and cannot at once see how the Fibonacci sequence is derived then stick with it and it will become clear.
Material on Pascal's Triangle on the web include
: A general site, this link directs you to the history page
A page with a program enabling interactive investigation [Link now lost!]
A page dealing specifically (amidst over stuff) with the relationship to the Triangle of Theosophical Addition or "modular arithemetic" [Link now lost!]
Also a page on the Pascal-related binomial theroem.
I've been looking at the ten dots of Pythagoras lately. It is from time to time, correctly I think, related to the enneagram. It is a simple glyph of ten dots or circles arranged in a triangle: _______#_________ ______#_#________ _____#_#_#_______ ____#_#_#_#______ As an arrangement of coins it is also an old chilren's puzzle "By moving only three coins, make the triangle point down instead of up". The easy way to solve this is to realise that the shape can be seen as a rosette of six coins arranged symmetrically around a center with three extras. I was idly leafing through a book on "pre-socratic" Greek philosphers when I came across this figure with the remark that it was the centre of Pythagorean numerology and expressed all the core musical ratios; that's 4/3, 3/2 and 2/1. These are the core musical ratios from the perspective of the Pythagorean scale with its "circle of fifths" (also known in ancient China), indeed strictly the 4/3 is redundant being and inverse of the 3/2 or "fifth". (In the Pythagorean scale notes are created as "octaves of fifths of fifths"; so eg 3/2 x 3/2 x 1/2 = 9/8 the Pythagorean ratio of re to do)However to create the scale used by Gurdjieff to explain the law of octaves one needs more core ratios than that, and I was unsatisfied with this as the "meaning" of the tetractys. The "Ptolemaic" version of the Just Intonation diatonic scale used by Gurdjieff has the following ratios: 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 If one adds another line of 5 dots to the tetracys one has the ratio 5/4 and can derive the ratio 15/8 (ie 3/2 x 5/4 =15/8). 5/3 could also be derived (ie 5/4 x 4/3 =5/3) (9/8 could already be derived as above). Then I saw that one doesn't need to add five dots get this ratio....the ratio between the sides of the traingle and the total number of dots is 10/4, which is an octave of 5/4 (and hence equivalent for this purpose). However the possibility of adding dots to the tetractys had been raised and I followed this up and I believe I have discovered some interesting stuff about how the tetractys related to the octave and the enneagram, which I'll go into in a future post. I hope others will comment and even play around with the tetractys in the meantime. But before signing off for the moment, I too have heard the Mullah Nasrudin story of how the eagle made of with the meat that was to be Nasrudin's dinner and Nasrudin mocked the eagle saying "I still have the recipe". Playing around with dots is the recipe not the meat, I know that. But the recipe has its point too, if only to serve as a reminding factor about the missing meant! The tetractys is also a "blank" form of Pascal's triangle which has been known since ancient times and is of significance to the octave. I have an observation or two about that, too, and I commend it to the attention of the list. -jeremy PS Some sites on Just Intonation etc, I include them mainly for my own convenience but others may like to have a look. The first link is a good introduction for those (like me) a bit musically challenged. The last is about the application of the octave to colour and proportion and is of extraordianry interest... http://www.kylegann.com/tuning.html http://www.medieval.org/emfaq/zarlino/article1.html http://home22.inet.tele.dk/hightower/ http://www2.hmc.edu/~alves/vicentino.html http://www.gurdjieff.org/werbock1.htm http://www.midicode.com/tunings/index.shtml http://home.vicnet.net.au/~colmusic/opticks3.htmTwo
It seems to me that I now have a useful analysis of the tetractys in terms of the enneagram. I'm not claiming it is the end of the story, on the contrary, just a significant part of the story missed in public so far as I know up till now. The stuff will be interesting to those concerned with the enneagram of process in general, but I believe it will also be found to have a more immediate practical use in formulating exercises. (As I understand it attempting to work directly on the higher intellectual centre is lawful if and only if one does so for the purpose of facilitating the so-called "second shock", the transformation opening the higher feeling centre.) Consider the enclosed tetractys (ie a 7-figure with the inner tetractys marked out, also the inner point enclosed by the tetractys). _____________#_________ ____________#_#________ ___________#_*_#_______ __________#_*_*_#______ _________#_*_#_*_#_____ ________#_*_*_*_*_#____ _______#_#_#_#_#_#_#___ I don't know if Pythagoras saw this, but I'm pretty sure Gurdjieff and his Seekers did and that they would have seen in it the formation of the finer bodies among other things. Anyway, the octave analysis has to be as follows below. I don't claim to have got it all out, just that I'm right so far as I've gone, and that thatis enough to be useful. As usual I use capitals for the fist octave and mixed capitals and lower for the octave beginning at the shock after RE, and then lower case for the octave beginning at the "false shock" at SOL. _______________________DO (field/full figure) _____________#_________RE ____________#_#________MI _______________________SHOCK _________Do ___________#_*_#_______FA_____________Re __________#_*_*_#______SOL____________Mi ______________________________________Shock____do _________#_*_#_*_#_____LA_____________Fa_______re ________#_*_*_*_*_#____TI_____________Sol______mi _______#_#_#_#_#_#_#___DO' At FA the head point, the Re, of a new tetractys appears. At SOL the tetractys is complete, but a second tetractys is being generated within it, if that continues due to a Shock between Mi and Fa then a third tetractys shows, with its re at LA/Fa. At the shock at DO a new headpoint of a new tetractys appears, and I think this may be the nub of the confusion between DO and RE and consideration of this point will I hope help shed light on the nature of the 'conscious shock'at TI-DO. But obviously I am still working through this myself. This should be compared with the Food Diagram. People will the wondering about the enneagram inner lines. What those inner lines say is at least that before moving from RE to MI one should look forward to FA; and before moving from FA to SOL one should look forward to TI. I think that this makes sense here (despite what I wrote in an earlier post). Before moving from: _____#_______ RE to _____#_______ ____#_#______ MI one looks forward to _____#_______ ___#_#______ ___#_#_#_____ FA which gives the shape of things to come, so to speak, an arithmetic progression is happening one counting number at a time. It also seesm acceptable to me, albeit less clear, that before progressing from _____#_______ ___#_#______ ___#_*_#_____ FA to the complete tetractys at SOL __________#_______ ________#_#______ ________#_*_#_____ _______#_*_*_#____ one looks forward to TI. At TI the "baby" tetractys is complete although as yet unenclosed: _____________#_________ ____________#_#________ ___________#_*_#_______ __________#_*_*_#______ _________#_*_#_*_#_____ ________#_*_*_*_*_#____ (I hope my use of the * mark to show the inner tetractys is not causing confusion. SOL is of course the completed tetractys and could have just been written: __________#_______ ________#_#______ ________#_#_#_____ _______#_#_#_#____ , and likewise the other figures could have been written with # instead of *) The full figure of 28 points shown so as to show the growth of the three stacked parts with their headpoints at RE/Re/re marked out. _____________#_________RE ____________#_#________ ___________#_*_#_____________Re __________#_*_*_#______ _________#_*_#_*_#___________________re ________#_*_#_#_*_#____ _______#_*_#_#_#_*_#___ Thanks to Stephan for providing the necessary shock to help me see waht was wrong with the path I was originally taking here. _Jeremy