'Octave' Baseten
It has been suggested that the reason why we evolved
a baseten (decimal) system is probably because we have ten digits
on our hands. Going a step further, I have come to see how nine and
zero reflect the importance of our two thumbs in contradistinction
to our eight fingers. Here you can see how the ten digits of our hands
can be viewed as an 'octave' baseten. The relationship between zero and nine will be detailed further in this article, but recognizing the thumbs as being unique is the first step.
The green & red lines overlayed on the fingers are a reminder of why our eight fingers are different from our two thumbs, namely that the muscles governing the fingers are located in the forearms while the thumbs have their muscles located inside each hand (this being the key feature of opposable thumbs). If we start to think of our baseten in these terms we can see that
it more resembles an octave system with a nine / zero event as the peak of flexibility. From the image we can see that taking 1 from 9 gives us
eight, 2 from 9 gives 7, 3 from 9 gives 6 and 4 from 9 gives 5, through
this we can get a feel of how 5,6,7,8, are like reflections of 1,2,3,4.
Indig Behaviors
R.B.Fuller coined the term indig (meaning integrated digits) as a
shorthand for what is known in mathematics as 'casting out nines'.
It is simply the reduction of multiple digits to a single digit through
addition, e.g. the number 1534 becomes 1+5+3+4 = 13, 1+3 =4.(1534
indig = 4) Notice that when you start adding large figures you can
disregard (or cast out) nines each time nine is reached in the sum,
once nine is disregarded the remaining digits add to the correct sum,
allowing us to quickly reduce numbers to indig value much more quickly.
Try getting the indig value of 453671152 by adding each number together
until a single digit is reached, then try it again, but this time
simply ignore numbers which add to nine and then add the remanding
digits. Casting out nines is sometimes used to check arithmetic quickly,
here are some quick examples of how casting out nines works in addition
and multiplication. Notice that when a number is multiplied by a figure
with an indig value of nine, then the resulting figure will always
add to nine.
Addition Multiplication
Indig Indig
Indig
Indig
34 7 22
4 32
5 234
9
+66 3(12) +77
5(14) x12 x 3
x645
x 6
100 1 99
9 384
6(15) 150930
54
The most significant thing to remember from this is the correlation
between the behavior of zero and nine. When we look at the word nine
in different languages the similarity starts to become a bit clearer
(e.g. 'nine' sounds like 'none' and to similar words in different languages
like 'nein').
Indig nines
Through this method of reducing digits to indig values
reveals some key characteristics of numbers which would not usually
be apparent. An example of this is with the number nine, when we notice
that nine and zero represent the same thing you can see that when
a complete 'cycle' in the number continuum is reached, the number
always reduces to nine. This is because nine and zero are both the
start and the end of our numeric language (baseten) and are therefore
representations of completion or unity.
Some straight forward cyclic numbers include:
One rotation 360° 9
Two rotations 720° 9
These are simple representations of cyclic unity (complete rotations), with that in mind you can
see why zero is represented as a circle and why other significant
cyclic numbers can be shown to fall in this zeronine zone. In my view,
the reason why the number nine is so often referred to as a divine
or holy number is not just because it is the 'highest' of the base
digits, but because, just like the zero, it is a representation of unity
and completion. Here is a small selection of numbers which are usually considered uniquely significant for one reason or the other, but by reducing to their indig value you can get an idea of how they could be a description of the same thing.
Recessional cycle 25920 indig
9
Maya number for the precession 25956 indig
9
Maya companion number 1366560 indig
9
Maya longcount period (days) 1872000 indig
9
Ancient kemi number 1296000 indig
9
Plato's 'perfect number' 5040 indig 9
The Monster M 80801742479... indig 9
The 4 Hindu Yugas (ages)
Satya Yuga 1,728,000 indig 9
Treta Yuga 1,296,000 indig 9
Dvapara Yuga 864,000 indig 9
Kali Yuga 432,000 indig 9
Sumerian King List (Sumerian
mythology)
Aloros  Babylon
36,000 indig
9
Alaparos  Unknown 10,800 indig
9
Amelon  Pautibiblon 46,800 indig
9
Ammenon  Pautibiblon 43,200 indig 9
Amegalaros  Pautibiblon 64,800 indig 9
Daonos  Pautibiblon 36,000 indig
9
Euedorachos  Pautibiblon 64,800 indig 9
Amempsinos  Laragchos 36,000 indig 9
Otiartes  Laragchos 28,800 indig
9
Xisouthros  Unknown 64,800 indig
9
Because we are 'casting out nines' to find these indig values, we can use 0 instead of 9 in the rest of our number reductions. Looking at multiplication tables through the indig method reveals symmetrical repeating patterns and if we take a look at second and third powering progression rates and
apply indig reduction, it's not surprising that repeating patterns emerge.
Second powering (squaring)
N² Indigs 
N² Indigs 
1² = 1
1 
13² = 169 7 
2² = 4
4 
14² = 196 7 
3² = 9
0 
15² = 225 0 
4² = 16
7 
16² = 256 4 
5² = 25
7 
17² = 289 1 
6² = 36
0 
18² = 324 0 
7² = 49
4 
19² = 361 1 
8² = 64
1 
20² = 400 4 
9² = 81
0 
21² = 441 0 
10² = 100 1 
22² = 484 7 
11² = 121 4 
23² = 529 7 
12² = 144 0 
24² = 576 0 


Third powering (cubing)
N³ Indigs 
N³ Indigs 
1³ = 1
1 
13³ = 2197 1 
2³ = 8
8 
14³ = 2744 8 
3³ = 27
0 
15³ = 3375 0 
4³ = 64
1 
16³ = 4096 1 
5³ = 125
8 
17³ = 4913 8 
6³ = 216
0 
18³ = 5832 0 
7³ = 343
1 
19³ = 6859 1 
8³ = 512
8 
20³ = 8000 8 
9³ = 729
0 
21³ = 9261 0 
10³ = 1000 1 
22³ = 10648 1 
11³ = 1331 8 
23³ = 12167 8 
12³ = 1728 0 
24³ = 13824 0 

Second powering (squaring) is a numeric progression
rate which describes surface area growth, while third powering (or
cubing) is a description of something that is growing at a volumetric
rate. For this reason, the pattern which emerges from third powering
(1,8 and 0) has significance for further on in this article.
synchrographics
Through exploring R. B. Fuller's studies of number I became familiar
with a work by Robert Marshall and Iona Miller called 'Syndex I&II'.
This study of number behavior is called 'Numeronomy' in contradistinction
to numerology, whereas numerology finds patterns and correlations
in numbers through subjective means, numeronomy finds patterns and
correlations in the interrelationships between numbers themselves.
This difference can be likened to the difference between astrology
and astronomy, the former looks at the affects of celestial bodies
from a subjective standpoint (from our point of view) while the latter
gains knowledge of the interrelationships between the planets through
study of the motions of the planets themselves.
Marshall and Miller have advanced the numeric studies which Fuller
recognized as being fundamental to the synergetic structuring of the
natural world. Through a discipline called synchrographics, Marshall
created mandalogs or number wheels divided into various axial positions
to visualize some of the underlying geometry inherent in the natural
number 'continuum'. "Geometry and number are separate yet interwoven
disciplines emerging from a single unified source", and through
this method of synchrographics we have an opportunity to make the
invisible structuring of numbers, more apparent.
Cyclic number wheel  nine pointed
Mandalog
Taking what I saw as the most illustrative aspects of the Syndex work,
I decided to make a mandalog divided into nine axes in order to show
the most striking aspects of our baseten  or 'octave' system. Starting
by placing numbers from one through to nine along a circle (or spiral),
the numbers continue spiraling outwardly in natural numeric progression.
Here I've only gone as far as the number 99.
Notice that each number can be reduced, via the indig (casting out
nines) method, to indicate which axes the number resides in. All numbers
which add to one reside in the axes labeled one, all adding to two
in the two axes and so forth. What would usually be called the nine
axes is labeled 0 here, this is to emphasis that the sequence actually
starts at the zero axes and subsequently reaches back around to 9,
then 18 etc.. This also gives us a better idea of how nine and zero
are closely related.
The most important aspect for now, is the distribution of palindromic
and transpalindromic numbers along the mandalog. A palindromic number
is a number which is the same read forward as it is when read backwards
(e.g. 11,22,33,44). Transpalindromics are sets of numbers which are
essentially 'mirrors' of each other (e.g. 5445, 6556). The first
step is to notice that each palindromic number creates an area above
and below it which is occupied by related transpalindromic numbers.
It might be helpful to think of each palindrome as the surface of
a mirror plane.
Notice that each palindrome has been connected with a green &
red spiral, this is to show that not only does the palindrome itself
'reflect' patterns above and below it, but the trajectory of the spiral
derived from the palindromes 'reflect' patterns too. These areas where
the spiral slips between two transpalindromic numbers are colored
yellow. With a bit of exploring you can see that all the numbers shown
are mirrored reflections of each other derived from the green &
red spiral.
Finally, the spiral itself is divided into two sections (green &
red) because after the first rotation (when it gets back to the zero
axes) the area at which it slices through, or where it goes inbetween
two transpalindromes (4554), is exactly half way through the nine/zero
axes and after another full rotation will bring the spiral back to
the zero axes where it 'hits' at number 99.
Using the indig method described earlier and taking the 'octave' baseten
view of the spirals progression, shows some more interesting results,
the stages which the spiral travels through can be seen to go from
zero to 2, to 4, to 3, to 1, at which point it mirrors signs and
goes from 1, to 3, to 4, to 2 and then to zero. From this we can
clearly see that there are four events in one rotation of the spiral
and after it's halfway mark there's another four events for the second
rotation, only this time the values have been mirrored. The spiral
can be seen to start at 0 and go through two rotations before it gets
back to the 0 axes again, because of the relationship between the
the 0 (or 9) axes and the palindromes (starting with 11). Marshall
called this the '911 Basewave', and recognized it to be the fundamental
characteristic of the number continuum.
As a side note to this section on the Syndex work, I found that, through
intense study of these mandalogs (or number wheels), Marshall was
able to identify 12 distinct classes of numbers which remain unrecognized
(or hidden) to standard number theorists' techniques. This is a subject
far beyond the scope of this article, but because of the significance
of this discovery and particularly because there are 12 (which relates
to parts of this article), I've decided to list them here for reference,
although they may sound complicated and strange.
All positive integers fall into these12
archetypal classes:
1) Transpalindromic composites
2) Transpalindromic primes
3) Transpalindromic squares
4) Palindromic composites
5) Palindromic squares
6) Palindromic primes
7) Retrocomposite primes
8) Retrocomposite squares
9) Retrosquare composites
10) Retrosquare primes
11) Retroprime composites
12) Retroprime squares
Natural growth rate  Phi
After examining the patterns in straightforward number progression,
we can go on to take a look at some patterns in nature's system of
counting, through the famous Fibonacci sequence. The Fibonacci (Phi)
sequence is a numeric progression starting with 0,1...the next numbers
are obtained through simply adding the sum of the previous two and
so on. The Phi sequence(also related to the golden ratio, golden mean,
golden section etc.) has been proven to show up in countless forms
throughout nature and culture. So here we take a closer look at the
sequence, through reducing each phi number to indig values and then
translated into the 'octave' system.
Again, repeating patterns emerge and this time the pattern
is very significant, my interpretation of the pattern is not easy
to explain,
so I've put some of the major points of interest into this diagram.
The 1 at the center is surrounded by a plus four and a minus four
'event', this pattern is reminiscent of the pattern that showed up
in the mandalog, where the first spiral had four events (palindromes)
and then the second spiral had another four events but where mirrors
of the first four. At the halfway point of the mandalog, the spirals
'cut' into the zeronine axes, but the spiral didn't touch a specific
number at this point (it slipped between 45 and 54), this relates
to the central '1' seen here in the phi sequence, this central 1 is
like the 'singularity' or point which is implicit in a sphere (as
it's center) yet never reached.
The four positive and four negative events can be seen as the four
faces of the tetrahedron and it's complimentary negative tetrahedron.
The tetrahedron, the minimum volumetric enclosure, always has it's
'invisible' counterpart which can be thought of as the difference
between a convex tetrahedron and a concave tetrahedron, the two are
complimentary. There are countless different ways to conceptualize
the +4, 4 aspect, one which I find can be related to our experience,
is that the four aspects of the tetrahedron can be seen as the well
known four elements of earth, air, water and fire. Applying the four
human senses to these elements you can see that, tactile is earth,
auditory is air, olfactory is water and visual is fire. This can be
seen as the concave tetrahedron, the complementary (negative) tetrahedron
would relate to the four psychological faculties elucidated by C.
G. Jung, which again fit with the four elements; sensation (earth),
thinking (air), feeling (water), Intuition (fire).
As you can see from the above diagram, the number 108 plays a part
here, the sum of the 12 indig values of 9 (where 9 times 12 is 108)
reminds me of the 12 zodiac signs and the 9 planets. This is why the
number 108 is considered the number of the Universe (related to the
word AUM) in ancient spiritual teachings, it's symbolism is scattered
throughout various cultures and has led to much speculation about
it's possible meaning (more).
There are many special qualities inherit in this number, some of which
have been studied and are well known, such as it's geometric relationship
to the equilateral pentagon, pentacles and relating it back to the
golden ratio itself.
If you take the zero as being a representation of a sphere, you can
see that the pattern generated from the phi sequence is like a description
of the motions, or stages of that sphere. If you take a look at how
the numbers seem to mirror themselves at it's zero stage, it's possible
to imagine the sphere (zero) oscillating back and forth, inside and
outside of itself. So here, uncovered in nature's own number progressions,
we have the IO sphere doing it's insideouting, just as described
on Treeincarnation.com.
Going a step further you can deduce from the pattern that it is the
12 zero's (represented above as the twelve 9's) which seem to
have 'deconstructed' or 'separated' themselves into all of the other
numbers of the sequence, you can see that by summing each section
of the sequence, i.e. 'putting it back together' would produce zero
as the sum of the entire sequence. In other words, what you are seeing
as individual numbers in the phi sequence are really sections of what
is a unified array of spheres represented numerically as zero's. Maybe
someday the logic used in traditional mathematics will be put on it's
head with a 'proof' that 0 > N (where N is the set of all natural
numbers).
This new view of the phi sequence also seems to relate to some of
the geometrical patterns identified as being fundamental to nature,
by R.B.Fuller in his works 'Synergetic's I&II'. For example, the
repeating sequence of twelve digits, whether seen as the twelve 'octave'
digits of 0,1,1,2,3,4,1,4,3,2,1,1, or as the 'folded over' 24 digits
(which reduce to twelve 9's), must be related to what Fuller called
the 'vector equilibrium' (VE) or 'jitterbug'. The VE is a polygonal
representation of a system of closepacked spheres, where 12 equally
sized spheres are touching one central sphere, it represents a state
of perfect balance. You must be thinking that 12 digits and 12 spheres
around one central sphere are not the same, and that we are missing
the central one in the phi sequence. But the VE and the jitterbug
are not the same, in order to make a VE do anything ( i.e. 'do the
Jitterbug'), the central sphere needs to be removed. Removing the
central sphere results in a model that looks like this:
This is what allows the VE
to collapse symmetrically into an octahedron, it allows the model
to oscillate in and out, for Fuller, these 'pumping' models (and subsequently
the jitterbug arrays) represented the fundamental dynamism of nature
and became the peak achievement of his explorations in Synergetic's.
It's also interesting to note that Fuller identified the VEJitterbug
as representing "a sphere at equilibrious, ergo zero energized,
ergo unorbited and unspun state" (Synergetic's 982.65
) and due to it's constant pulsations between inside and outside
states, is never found at complete equilibrium in nature. The expanded jitterbug (VE) represents a sphere in convex form while the contracted (octahedron) represents the concave spaces between spheres. To have such
beautiful geometric models reflected in the patterns of nature's numbers
is mind boggling, but there's still a lot more to be uncovered.
Looking at the origins of the arabic numerals (Theory on the Typographical roots of number) reveals some more interesting correlations. When we apply the octave understanding of the baseten to this theory, we see that the numbers 0, 1, 2, 3, 4 reveal some aspects which can be translated into topological relationships i.e. can be seen to be descriptions of shape. Here, the symbol for 0, the circle, represents complete angular unity or a state of unrestricted spherically (note perfect spherically can not exist in any physical form), the symbol for 1, can be seen the have one distinct angle, 2 introduces another angle, 3 brings another angle and finally the numeral 4 has it's four divisions of angle represented in it's distinctive 'cross' feature. The similarities between the cross and a 2dimensional representation of a tetrahedron are too close to be overlooked, the tetrahedron has four faces and four vertices and is the minimum enclosure of space possible. This perspective of the symbolic and topological meaning of number brings up some questions about the nature of number, i.e. is number an abstraction that we use to describe the world or are numbers themselves a description of something much more fundamental, my interpretation is that numbers seem to be describing the nature of energy transformations at a preexistent or apriori level and the funny thing is, numbers often speak for themselves.
Conclusion
Tying together some of the patterns explored previously, brings us
to this final diagram, which is an attempt to incorporate as much
of the key elements described throughout this article as possible
in the simplest way.
So what does all of this mean? There's no short answer for that at
this moment and because this is relatively new territory for me, it's
hard to say where this is leading and if there will be any practical
application's for such observations, maybe exploring number and mathematics
in this more intuitive way is the ultimate point of this exercise.
For me, contemplating this pattern as an underlying structure of the
natural world has led to some interesting concepts, one of which relates
to the idea of unity. Many spiritual traditions have stressed the
importance of unity and related it to the idea of Oneness. For me,
the overlap of the language of numbers and of traditional language
creates some difficulty here. The philosophical question of how separateness
and wholeness (unity) could exist at the same time, has been explored
throughout the ages, but if we take what physics has found to be the
fundamental characteristic of matter we can start to get a deeper
understanding of what we mean by unity.
Physics has found that matter itself is almost entirely empty, or
composed of nothingness (as evidenced in the spaces involved inside
and outside of atoms), with only a tiny percentage of that space being
subatomic 'particles', discreet energy packages or waves. The important
thing to recognize is that these energy events are composed out of
the same nothingness which they reside in, the difference is that
the particle/wave has a sustainable pattern (frequency / wave). Just
like a wave can be thought of as a pattern or a principle which does
not depend upon the substance through which it becomes visible to
the senses (e.g. waves exist in water, milk, oil  but the wave is
independent of these). The building "blocks" of matter have
this same insubstantial quality, namely that the principle of a wave
(or particle) is a self sustaining pattern, and this is it's only
defining property.You could sum it up by saying that matter is made
up of mostly nothing, and then some 'pieces' of nothing that jiggle.
I feel that this gives us some idea of what is meant by the philosophical
statements which deal with separateness and unity. The circle (or
sphere), representing unity and also zero (or nothing) is the same
underlying interconnectedness which physics has found. Thingness
is the first requirement for distinction and therefore separateness,
corresponding to the discreet packages of energy which create subatomic
particles, but the underlying nothingness is the unifying characteristic which
'ties together' all that is seemingly separate.
It may be that this is why the word 'One' is spelt with a big 'O'
and it could be that the subconscious intent behind the philosophical
statement 'All is One' is less of a statement of 1ness (thingness)
and more of a statement of 0ness (nothingness).
Justin Lawless ~ March
25, 2007
