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Last Updated 29 May 2007
you're familiar with our recent explorations here on Treeincarnation.com,
you probably reckon that there is a lot more to 'number' than
meets the eye. We have applied scientific method to number
and geometry to reveal underlying characteristics inherent
in number and thereby make some observations about the nature
The Pythagoreans believed that number was the ultimate reality.
Pythagoras himself was quoted as saying "Number is the
ruler of forms and ideas and the cause of gods and demons."
Today physicists, astronomers, cosmologists and scientists
are in general aware of this important and precise language.
But the overspecialized and segmented nature of scientific
inquiry means the tools employed by science are no longer
questioned. It is an unfortunate situation where 'the left
hand doesn't know what the right hand is doing'. We have become
complacent about using these tools, becoming in effect a 'slave'
to those same tools. As has been well observed: "the
important thing is, never to stop questioning".
Our understanding of the most important tool of all: number
itself, has barely changed since our adoption of the Arabic
numerals in the 10th century. But we will be describing here
the discovery of the underlying nature of number. One which,
to my knowledge, has neither been realized or explored until
I'd like to make clear that the explorations of number we've
been describing on Treeincarnation.com are not the same as
numerology. We are employing scientific reason and new conceptual
tools to delve beyond a subjective numerological focus and
uncovering the primal structure and pattern of number itself.
In the article Shape
and Number, we described some underlying characteristics of
our base ten number system, and went on to use this method to view
the Fibonacci sequence in a new way. The relationship between zero
and the number nine turned out to be a key aspect of that exploration.
Our study made it clear that the number 9 is key. But... a funny
thing about keys, they aren't very useful unless you know where
the keyhole is.
And what is a hole anyway? Is it a thing? Or is it the absence of
thing-ness? Does the answer relate somehow to the meaning of the
word whole? And are we going to play around with words here or are
we going to get to the point? Ok, enough of the word play, lets
get on with it.
But not before we explain the meaning of one more
word, which is vital to understanding the difference between what
we have been doing with number and what we should be doing. The
word 'Synergy,' which means: the behavior of whole systems, unpredicted
and unpredictable by the behavior of the individual parts taken
individually. For example, there is no information within an atom
that can predict the formation of complex proteins like DNA; and
there is no information within the DNA itself that can predict the
formation of complex organisms like us. We could study a single
bee for an eternity and still have no clue that when a colony of
bees get together, they cooperate to create a hive; that they create
honeycombs; or that they will swarm to protect their queen. (Some
areas of science and philosophy have adopted using the term 'emergence'
to describe the same principle, although in my opinion, this term
is not as clearly defined.)
This principle is intuitively and commonly understood.
But when it comes to scientific inquiry, we generally revert back
to the unnatural system of intensely studying the parts and compounding
that with information about other parts, in hope of getting some
idea about how the larger 'whole' might be working. Thus we are
constantly surprised when whole systems turn out to have attributes
unforeseen in prior calculations, because we do not realize that
the synergetic principle is always operative.
Buckminster Fuller saw this principle as being fundamental
to nature, and went on to describe a synergetic coordinate system
which was not based on compounding cubes upon cubes as in the Cartesian,
'cubic' XYZ-Axis system. Fuller's synergetic coordinate system was
described in great detail in his geometrical tour de force, 'Synergetic's:
It is a coordinate system based on Nature. It is proven time and
again to be the most faithful representation of how nature coordinates
in an economical and elegant way. It is 'natures own coordinate
system.' Once we begin to recognize the patterns that flow from
this, we realize that space has a very real shape undefined by cubic
metrics. But despite this, cubic metrics are still widely used in
the majority of scientific inquiry.
I have been studying this synergetic coordinate system,
along with the geometry's that flow from it, for almost a year now.
I am constantly amazed at the self-referential and 'holographic'
nature to this approach. But I made a breakthrough in November 2006,
when I stumbled across what I would now call a 'glitch' in our mathematical
I took an equation that dealt with the spherical growth rate of
the shape called 'vector equilibrium' -the centerpiece of synergetic
geometry. Through logical evaluation of each function within that
equation, I was convinced that something was amiss with our understanding
of number. That the nature of space seemed to be operating in a
way which was diametrically opposed to conventional concepts of
mathematics. The inherent properties of space seemed to be saying
equals 2, implying that 'two-ness' is always operative. I was
convinced that this was a fundamental question, blocking our understanding
of how nature and number functioned.
Looking back now, I see that all of what we perceive to be provable
within physics and mathematics is only made possible through the
acceptance, or belief, in one concept or another, and by and large
these concepts cannot be demonstrated as being really present in
Nature. One way of thinking about this is to ask yourself what is
the height of a tree. You could say that the tree is 7 meters tall,
and you could go on to 'prove' this through a measurement. But the
overlooked question is: where's the proof of the measurement itself.
We can use agreed measurements such as a meter or a mile, but this
does not prove there are meters or miles 'out there' to begin with.
It may seem like a pedantic observation, but it's true nonetheless.
Measurements such as these are not proven as real. Although we can
apply the idea of a meter or a mile in order to help us coordinate
our surroundings, this un-provable aspect of our conceptual models
still remains. From those early realizations, I have stopped taking
our math tools at face value and went on to explore patterns of
number behavior which show number to be more than merely descriptive.
I have written about some of these experiments on
Forum, but for brevity's sake, it's sufficient to say that 0
and 9 are effectively the same; and that the function of 0 and 9
is vital to our understanding of number. Number is a description
of how energy operates. The functions which 0 and 9 represent can
be described as angularly unfettered and exhibiting an elusively
perfect sphericality --nonexistent in any physical form-- but vital
to nature's complex progressions. The base10 number system exhibits
the interplay between theses two components, which together provide
us with a progressive means of counting. The numbers 0 and 9 represent
an opened 'junction' which allows information to pass through freely.
Without these pulsations -or states- of freedom, Nature would permanently
crystallize. It would 'freeze up.'
With that background, it's time to explain how this new discovery
came about. It occurred to me that synergetic geometry seemed to
correlate with another interesting subject: the Mayan Calendar.
The numbers that were discovered to be integral to calculating the
time periods involved in this ancient calendric divination system
coincided with some core numbers used in Fuller's geometry. Those
numbers being 13 and 20. The 13 relates to the total number of vertices
(events) in the Vector Equilibrium (VE), which is perhaps the most
important structure in synergetic geometry. The VE is a polyhedral representation of 12 equal radius spheres touching
one central sphere of the same radius. The number 20 was regarded
by Fuller as the total volume of the VE. There are clearly connections
to be made between this ancient system of calculation and the numerical
values occurring in synergetic geometry.
Fuller went to great lengths to stress the importance of the VE.
It is a geometrical representation of the ultimate balancing act
of push and pull (yin and yang): every edge or vector is of equal
'size,' so that radial and circumferential forces are in perfect
The number 9 was also fundamental to the Mayan calendar
-not to mention many other ancient methods of calculation, particularly
in regard to precessional numbers. The Maya built pyramids with
nine levels, reflecting the importance of the same nine 'levels' which are a crucial aspect of their calendar. This is no ordinary calendar, it
gives accurate calculations of all the major development points
in the evolution of life. Scholars point out that the Maya had measured
the age of the universe to be remarkably close to the 6 billion
year figure which modern science now estimates. The Maya were measuring
something far more important than celestial orbits. These nine levels
or 'underworlds of creation' point to the progression of something
beyond the purely physical.
A vital aspect of the geometrical discoveries laid out in Fuller's
Synergetics is his application of whole number values to the ratios
between the volumes of some of the primary platonic geometries.
This is akin to having discovered the musical ratios expressed in
the harmonic series --but in the volume ratios of geometry.
In the (fcc) closest packing of sphere arrangement, we use unit
radius spheres to give us a view of the underlying 'structure' of
space. We model with geometrical representations the inherent shape
of space. This is a truly natural coordinate system, because it
shows us how nature always uses space in the most economical manner,
and how self-structuring systems actually occur.
Displaying the geometry's: Tetrahedron Octahedron Cube-octahedron (VE)
Connecting the centers of all spheres creates the 'Isotropic Vector
Matrix,' where all connecting lines (vectors) are the same length.
Thus every 'point' or crossing has twelve vectors emanating from
it. This is the (4D) synergetic equivalent of the (3D)
XYZ Cartesian coordinate system. The geometrical shapes represented
here are given relative volumes to each other and because each shape
in the IVM has the same length vectors (isotropic - everywhere the
same), the volumes of these shapes are not dependent upon 'size',
they are all relative to each other. These volumetric ratios are
non-metric values which are standardized relationships expressed
numerically. These are relative values not dependent on a metric.
Through Fuller's achievements, we now have a means of accurately
accounting geometric and numerical ratios within this synergetic
framework. In Fuller's words, "Synergetics is a priori nuclear", meaning we are using geometry and number that can
represent states of energy transformations and which are present
before the formation of anything physically observable.
After much thought I came to see that there was something amiss
with some key aspects of Fuller's application of number in Synergetics,
specifically, the way in which he arrived at the numerical values
for the volumes of these geometry's I realize now that Fuller had
not applied synergetic principles to numbers. In other words he
had been using numbers much like the way XYZ coordinates used cubes.
This is classically how we 'count': one number compounded upon another
in a linear fashion, 1 + 1 + 1 +1 etc. Which to me seemed like a
non-synergetic application of number. (I don't mean to suggest that
Fuller was remiss in his explorations. I have a keen awareness that
his foresight and vision was way ahead of his time, and that the
full impact of his contributions to technology and design have yet
to be realized, or even accepted! But to think of his work as being
somehow 'complete' would be to go against all that he would have
hoped for. His spirit of exploration has inspired many to continue
to reevaluate and develop his work.)
Fuller arrived at the Volume of the VE through applying the value
of 1 to the minimum volumetric possible, the tetrahedron. And 20
'tetra volumes' was equal to the volume of the VE because it takes
twenty 1's to 'fill' the volume of a VE. Think of each polyhedra
as a container, like a jug of water. In which case the tetrahedron
is the minimum container and the VE is the maximum. Fuller called
the tetrahedron unity and went on to give the other polyhedra relative
volumes based on the tetrahedron with volume of 1. But I began to
think that the minimum 'unit' being called 1 was not really a synergetic
way of looking at number. Technically it shouldn't matter what value
you apply to these ratios, yet I still felt that there was something
It was at this moment that it struck me: to get truly
synergetic values for these volumetric ratios, instead of building from a tetrahedral
value of 1, we need to start with the volume of the VE and then subdivide
that 'whole'! After all, the 'whole' is the essence of Synergetics..
The question was, what might be the volume of the VE? The answer,
as referred to at the start of this article, is 9. This is the 'key'.
And the key-hole (whole) is the volume of the VE. As you will see,
the number 9 fits the VE like a glove.
Starting from the Whole
If the volume of the VE is taken to be 9, our entire view of number
could synchronize into a new synergetic understanding. The value
of 9 resonates so well with the VE, not least because the VE is
a representation of the whole, but because it is not directly observable
in any physical form. Nine is 'none' and the VE is referred to as
the ultimate synergetic representation of the nonphysical, i.e.
nothing. The VE is that elusive state which could be described as
zero point. Fuller states: "The vector equilibrium is a condition
in which nature never allows herself to tarry. The vector equilibrium
itself is never found exactly symmetrical in nature's crystallography.
Ever pulsive and impulsive, nature never pauses her cycling at equilibrium:
she refuses to get caught irrecoverably at the zero phase of energy."
Now the synchronicity begins, all because of the unique function of the
number 9 (which Fuller himself observed in his number explorations).
Once the number 9 is used as the volume of the VE, something
truly synergetic occurs, namely that all the numerical values for all
of the related volumes become reflections of the number 9.
For example, the octahedron has a 'linear' (or normal) volume of 4
and the VE a volume of 20. (20 / 5 = 4). In a fully synergetic understanding
where the VE is 9, the octahedron has a relative volume 1.8 (9 /
5 = 1.8). Notice that the additive value of the numbers 1 and 8
is 9. This is synergy in number. Starting with the number 9,
the whole is now reflected within each of the parts, almost like
a holographic view of number. Although we are still using the same
ratios which Fuller developed and although applying these synergetic
numbers might seem a trivial adjustment, this is a vital of understanding
what a truly synergetic view of number entails. Here's a chart of
both linear and synergetic volumetric ratios of some important polyhedra:
**The Dodechedron and Icosahedron: Unlike most
of the other whole number values, Fuller calculated the volumes
of these geometry's to be somewhat 'irrational'. In this system
we've estimated their volumes to be 6.9075 and 8.330535 respectively,
although it's hard to say with any certainty if our synergetic
figures are completely accurate, in this new system the additive
value of the figures is still nine..
I'd like to reiterate that the volume numbers are
ratios. These geometric ratios do not depend upon any specific measurement
system. The relationships are how nature operates. We are now dealing
with real world ratios. By comparison, conceptual models which have
no basis in reality, such as flat planes and linear notions of number,
are effectively sophisticated mind games. To quote Fuller, "The
mathematicians feel that they can do anything they want with their
abstraction because they don't relate it to reality. And, of course,
they really can do anything they want with their abstractions, even
though, like masturbation, it is irrelevant to the propagation of
The numbers 9 and 0 are reflections of the same thing. In this synergetic
view of number, we are starting from 0, and just as in other scientific
and mathematical disciplines, 0 is the origin. The difference, however,
is that we are subdividing from this whole,- 0 - and thereby always
keeping our numerical calculations within a synergetic framework.
Although the 9 and 0 represent the same states of energy, we use
9 for our calculations, because in our classic view of number we
cannot divide 0 - (you might break your calculator). But we achieve
effectively the same thing by dividing by 9.
Let's explore this new synergetic understanding to number. Before
we had been using number as a kind of label where 1 is used as the
starting point (much like a 'building block'). Now we see that the
whole (9) is where we must start.
First let's convert numbers from the 'linear' 1-20
into a synergetic system where we go from the 9 of the maximum (the
VE) to the 1 of the minimum unit(tetrahedron). Note that in order
to convert any standard number into this synergetic system you simple
'multiply' linear numbers by 0.45. For example 20 x 0.45 = 9. Conversely,
the inverse of multiplying by 0.45, is to divide by 2.22222....
--which is another interesting number we'll get back to later. For
example, take the value for the octahedron volume (4) and divide
by 2.22222... to get the synergetic number 1.8.
Notice the synergetic number progression, Look
at the synergetic column from top (9) to the bottom (0.45),
and notice that the numbers in the single decimal place
positions go from .1, .2, .3 up to .9. Note also that they
alternate with numbers which have two decimal places.
Note also that when you add tetrahedron volumes
(0.45), it takes 20 additions before the decimal point is
disregarded. The same effect occurs when you compound the
volume of a cube (1.35).
Try this using a calculator; add 1.35 to 1.35 and then add
1.35 to the result and continue adding until the value synchronizes
(i.e. when the decimal is removed and. a whole number is reached).
You find that it takes 20 cube volumes of 1.35 until the value
synchronizes at 27 and another 20 before it reaches the whole
number 54 etc.
By the way, 20 is also integral to synergetic geometry and
the ancient Maya used a Base20 number system, To me it's clear
they had advanced knowledge about the nature of number that
we are only now beginning to catch up with.
Another observation that maybe significant, is that the synergetic
volume of the minimum volume, the tetrahedron, is less than
1, (as is the half octahedron). It takes 3 tetra-volumes of
0.45 to make one cube, which is the first volume to be greater
than 1. Perhaps this might have something to do with the XYZ
coordinate system and cubic metrics, because in the cubic
system dimensionality is 3-fold, (i.e. length breath and height
constitutes dimensionality in the Cartesian view). This may
relate to the three tetra-volumes required to fill one cube.
The cube being the first volume greater than 1.
There is an almost fluid motion to how this system
operates, and studying this synergetic number progression reveals
many interesting patterns. See how the numbers to the right of the
decimal place alternate between two decimal places and one. As if
separating and combining continuously. This numeric progression
is a reflection of the separating and combining aspect within number
and within energy. By decrementing from 9 to 0.45 in steps of 0.45,
we get a mirrored pattern centered on the synergetic 4.5 (equivalent
of linear 10). Also, the value of the units of these synergetic
numbers and the value of the first decimal places, counterpoint
each other in alternating patterns. And always the additive value
of each synergetic number is 9.(See also Appendix II Quantum Bit )
Just for the sake of curiosity, lets examine these numbers through the 'Octave' or Indig method which we described in the previous article. We take all ten digits from 0 to 9 and convert them into a system where we have only +/-1, +/-2, +/-3, +/-4, and then 0, this means that 8 becomes -1, 7 becomes -2, 6 becomes -3 and 5 becomes -4. When we apply this system to the synergetic numbers shown above, we get a clearer picture of how these numbers mirror themselves over their decimal place and how their additive value of 9 operates.
Take 1.8 for example, when we look at this as an Indig it reads 1.-1, that is a positive 1 on one side and a negative 1 on the other, try it on the number 3.6 and you get 3.-3, the same effect, this positive and negative axis at the decimal place is true for each synergetic number.
12 Around One
Taking this view of number and applying it to another
area of synergetic geometry reveals more interesting aspects. As
we've said, the VE has 13 events (or vertices) in total. When we
calculate the number of events at the 'shell' of the VE we use the
equation 10(F)²+2=N, where F is the frequency and N is the
number of events on the surface (Frequency is the number of spheres
at the center of the VE). At frequency 1 the number of events at
the surface is 12. These are the 12 spheres which surround one central
sphere in the closest packing of spheres arrangement. Clearly, viewing
the events at the surface of the VE requires a different approach
than what was used above to describe the VE's volumetric ratios
-simply because these are two different aspects of the system. When
we subdivide the volume of the VE we see that it takes 6 half octahedrons
and 8 tetrahedrons to achieve a volumetric total of 9. We can arrange
these volumes so as to reflect the alternating nature of the synergetic
number sequence and the total 13 events of the VE. Notice that the
center of the arrangement is a star tetrahedron or 2-Tet.
We can see that the central event of the VE has an inherently polarized
quality. This is like the hub at the center of the wheel which gives
the VE it's axis of spin or orientation. By the way, the VE has
no extension or size whatsoever. Representing these forms geometrically
just enables us to visualize the 'invisible'. Below is one way in
which we could construct the VE to represent these 12 surrounding
6 Tetrahedrons and 6 Octahedrons of the VE
~ Plus the polar two ~
Getting back to the equation for calculating the
events on the surface of the VE, we can now see that the 12 events
are divided into two groups of six. In other words, there are 6
half octahedrons and 6 tetrahedrons surrounding the central 2-Tet.
So now we can be more precise in identifying what these 12 events
are. Bringing this into our understanding of the volumes involved
and we can say that they can be categorized as 6 associative and
6 disassociative aspects. This is important when we want to convert
the shell value of 12 into our synergetic view of number, because
instead of simply using 12 standard numerical values, we need to
be aware that there are two unique functions present and that the
12 events are not all the same.
First we divide the 12 into two groups of 6, one to be converted
into tetrahedron volumes, the other to be converted into half octahedron
volumes. In our synergetic view of number the tetrahedron has a
value of 0.45 and the half octahedron has a value of 0.9. We can
see that the numbers themselves are reflecting the two unique types
of energy involved, one is combined - 0.9 (associative) the other
is separated - 0.45 (disassociative). Multiplying these two values
by 6 gives us 2.7 for the tetrahedron group and 5.4 for the octahedron
group. These two group values taken together will give us the total
synergetic value correlating to the 12 events at the surface of
the VE, so 2.7 + 5.4 = 8.1.
This is significant because it shows that the synergetic number
for the expanded shell of the VE is a mirror of the synergetic number
correlating to the contracted (concave) VE: the octahedron(1.8).
It is only when we remove the central sphere in the VE arrangement
that the system comes to life. This is what we call the Jitterbug
transformation. It contracts from an expanded 'hollow' VE into it's
compliment, the octahedron. So the values which we are getting here
is showing us that the shell of this VE is 8.1 --a mirror of the
octahedron which has a synergetic value of 1.8.
If we include the value for the 2-Tet at the center of the VE, we
would come back to our synergetic value for the volume of the VE.
The 2-Tet is 0.45 + 0.45 = 0.9 and added to the VE shell value of
8.1 gives us the total volume of the VE, 9.
This is allowing us to see that calculating the volumes within these
geometry's and calculating the events at their surface may have
a relationship to one another which was not visible to us when we
used a linear system. Now let's see what happens when there are
more spheres close packed into the VE arrangement - what we call
higher frequency VE's. Taking this method of dividing the shell
into two groups (so we can calculate our synergetic values), we
can apply it to VE's of higher frequencies to get the following
Again, N is the number of spheres on the surface of the VE and Frequency
can be thought of as the number of underlying sphere layers which
surround ( and includes) the central sphere. Calculated with the
You can think of each frequency as the size of the
VE, but we can also correlate the frequency here to time. Because
the geometry's we are dealing with must have more to them than static
shape, we need to think of each shape as being a snapshot photograph
of just one of many different stages of development. It's helpful
to think of all of what we are describing here in these terms. Just
as we see the egg and the chicken to be two very different structural
patterns, yet we know they are both the same organism, we should
keep in mind that what we're describing here works similarly i.e.
many different stages of development describing the same overall
It would help to imagine each frequency shown above as a single
frame from a movie reel. The single image doesn't give you a lot
of information, but by taking consecutive images and viewing them
in sequence the picture starts to animate and more information is
displayed. This is what we are doing with these numbers. Shown in
the table above is the surface of the VE as it grows. The pattern
of the synergetic numbers is displaying another oscillating feature.
As we saw, the 12 events on the VE at frequency 1 was translated
into 8.1 - which we saw as the 'mirrored' version of the octahedron.
Next is frequency 2 which has a synergetic value of 28.35. The partial
additive value of the numbers before and after the decimal point
is 10.8 --which further reduces to 1.8. Do this to all of the synergetic
values above and the pattern continues indefinitely from 8.1 to
1.8 to 8.1 to 1.8 etc. This is the VE going through it's jitterbug
phases, flipping from inside to outside or from convex to concave.
Calculating the shell values in this way is showing us more of the
holographic nature of synergetic geometry and this time the underlying
functions of synergetic number are joining in on the act.
This value of 8.1 for the VE shell is actually quite close to the
value of the Icosahedron's volume, this fits well with our understanding
of how these geometry's structure themselves in nature. The Icosahedron
is a spherical enclosure which, unlike the fully expanded VE, has
no room at it's center for another equal radius sphere. Without
an equally-sized central sphere the surrounding twelve spheres are
not being pushed out into the the maximally expanded VE. Thus the
spheres that represented the square faces on the outer shell of
the VE can contract inwardly, whence the VE becomes a fully triangulated
Icosahedron. In fact, it is this function of the 'hollowed' VE (Jitterbug)
which allows synergetic geometry to make the leap into Fivefold
If we were to add the synergetic values for two adjacent frequencies
we get a symmetrical number, with nine on one side of the decimal
and nine on the other side 9.9. Taking the additive value of any
two consecutive numbers from the sequence above always gives us
a number which can be further reduced to the number 9.9. For example
frequency two and three are synergetic 28.35 + 62.1 = 90.45 (9 on
both sides of the decimal). In every case it's the same as summing
8.1 + 1.8 to get 9.9. This is is like a composite of the two states
of this surface and with two 9's on both sides of the division represented
by the decimal. It's like looking at two VE interiors with their
two volumes of 9 side by side.
This combination of the surface values relating to these convex
and concave states is giving us a total value which is symmetrical.
Furthermore if we divide 8.1 by 1.8 we get 4.5, the midpoint of
our synergetic number table. Conversely, if we take the volume of
the octahedron and the volume of the fully expanded VE (with it's
13 spheres), we get a combined total volume of 1.8 + 9 = 10.8 and
it's almost like we've come full circle and we're back to the octahedron
So when we calculate the events on these spherical surfaces we are
converting them into their corresponding values for volume. But
maybe these values are now becoming more generalized i.e. not having
to be descriptions of one specific parameter. This method is allowing
us to see that the ratios we use for volume may also be related
to other aspects of geometry, this number system may allow many
different aspects of number to relate to each other.
Looking back at the synergetic number series from
9 to 0.45 (20 to 1) there is more to be explored. Adding each number
to itself and continuing on until a whole number is reached reveals
another pattern. The table below shows the number of places it takes
consecutively summed synergetic numbers to synchronize i.e. to reach
a whole number. Again, notice the mirrored effect which is produced
from the centerpoint at 4.5,
20 places to synch
10 places to synch
20 places to synch
5 places to synch
4 places to synch
10 places to synch
20 places to synch
5 places to synch
20 places to synch
1 places to synch
20 places to synch
5 places to synch
20 places to synch
10 places to synch
4 places to synch
5 places to synch
20 places to synch
10 places to synch
20 places to synch
This is showing us more evidence that the vector equilibrium
(the whole) is intimately tied to the number 20. But we can reason
that this is not just because it takes 20 tetravolumes to fill the
volume of the VE, but that 20 is the maximum amount of numerical
permutations possible within this synergetic sequence, before a
whole number is reached.
If we continue with this reasoning we can say that
the whole, numerically represented as the number 9, can only be
subdivided into a maximum of 20 different parts. In that case, I
think we can say that the ratio between the whole and all of the
possible parts within that whole is a ratio of 20 to 9 (20 : 9)
, where 20 is all of the possible increments within the whole.
As it turns out, that ratio is very significant, we
can take it as 20 divided by 9, which is equal to 2.2222222222222222...
. What we're actually saying is that the relationship between all
possible things (all the parts) and of the whole, is infinite duality,
or infinite complementarity; and furthermore that relationship is
a ratio and is numeric. Duality is a ratio.
Another aspect to this is that dividing any normal number by all
of those 2's gives us a number which is then a synergetic number
and can be shown to always have an additive value of 9. From me,
this number seems to work like an infinitely precise cutting tool,
except instead of splitting the numbers apart, it seems to work
in reverse, i.e. bringing them back together and showing the numbers
to be related to the whole, 9. As Fuller would say, "Unity
is plural and at minimum two." Again, this relationship may
have something in common with the 'glitch' we saw when exploring 0²=2.
Synergetic Pi ?
This synergetic understanding of number may give us
a new perspective on the meaning of the Pi ratio. Pi is the number
of times a circle's diameter will fit around its circumference.
When the diameter of the circle is 1, it's circumference is the
transcendental number 3.14159265... (Pi can be roughly estimated
by dividing 355 by 113). Taking the synergetic view of number, we see that the
circumference of the circle is the unified 'whole' aspect, while
the diameter correlates to the increments or the parts which 'fill'
that whole. With this in mind, we might be able to get a new perspective
on the Pi ratio.
As we are now going to attempt to get a synergetic understanding
of Pi, we're going to have to keep in mind that the method by which
Pi is classically calculated may have some outdated philosophical
This figure is arrived at through taking the diameter
of the circle as the yardstick. This is the incremental and inherently
separated part(s) with which we attempt to measure the whole. And
not surprisingly, we cannot gain insights into the whole system
(the circle) when the starting point is a part taken separately
(in this case the linear measurement of the diameter).
It isn't possible to reevaluate the ratio of Pi and
study it within a synergetic framework if you strongly hold to the
same ratio which started out with the statement, "when the
diameter of the circle is 1....". Like with so many axioms,
the principle of synergy is not accounted for with this method.
It might help to keep in mind that just like our synergetic values
for volume, Pi is a ratio. But most importantly, it is a ratio between
an incremental (separated) measurement and of the most natural forms
of existence (circles or spheres). What we are trying to do is,
find a way of getting a similar ratio, but one which does not start with
an incremental measurement of the parts. We need to be aware that the principle of synergy is operative.
So, instead of starting from a 'logical' straight
line, we need to start at the opposite end of the spectrum and try
to see what would constitute some numerical value for the whole
system, in this case the circumference of the circle is the system.
But is it a whole system?
The first thing to notice is that this circle is drawn
on a conceptually flat plane. As '2D planes' are nonexistent, we
need to take a look at what would constitute a system with some
demonstrable or dimensional reality. It's clear to me that we need
to be dealing with a sphere instead of a circle. Instead of the
perimeter of a circle we should examine the surface of a sphere.
When we second power any given number, we are effectively
dealing with surface area. Don't be fooled into thinking of this
as the area of a flat square (as in 'squared'). Second powering
any value is giving us a surface area which is 'unrestricted' and
therefore spherical. So when we second power Pi we are displaying
this numerical ratio as if it was 'mapped' to a size-less sphere surface. It becomes an 'omni-directional' instead of a linear
number. So Pi 'squared' is effectively, the spherical surface area
of the ratio relating the circumference to the diameter of the circle.
This is a remarkable stroke of luck, because this ratio is almost identical to the ratio we got from combining the convex and concave ratios of the VE surface (or more accurately, the Icosahedron surface - as there was no central sphere in the calculation). Our experiment with synergetic number and the surface phases of these spherical shells gave us 1.8 for the concave and 8.1 for the convex, the combined total of both convex and concave was 9.9.
Pi² then, is simply the Pi ratio after being
given surface area. This is an important point because this surface
area is itself spherical, just by the fact that it has been raised
to the second power. And this ratio is nearly a perfect match for
our previously discovered synergetic ratio relating to the combined
convex and concave surfaces.
This is a crucial moment in our explorations, because
now we're about to do something almost blasphemous to the transcendental
and infinite number Pi, we're going to round it !. But not without
good reason. We're going to round the value for Pi² to fit
with our previously discovered synergetic value of 9.9. So let's
approximate Pi² = 9.9, being the ratio of the circle to it's
diameter translated to spherical surface area. From this 'clean'
number we now have a ratio which fits with our values for synergetic
number; a ratio relating to the whole system. Now the question is,
what's the diameter of this sphere? If we are going to take a synergetic
view of Pi, we're going to have to be aware that we now have two
Pi values in play, one regular and then this synergetic variation
which we're trying to find a value for.
To find the diameter of this sphere with a surface area of 9.9 we
can use a Pi calculator. But be aware that these conversions are
using the regular Pi ratio. So the diameter we find will be from
the 'metric' point of view:
Entering a diameter of between 1.78 and 1.77 gives
us a sphere with a surface area which is very close to our Pi²
ratio of 9.9. It looks as though we're in luck again, because 1.78
is again very close to a value which we've seen before: 1.8, the
synergetic value which we correlated to the octahedron and to the
convex sphere surface. This number came up when we calculated the
surface progressions from convex to concave. 9.9, 8.1 and 1.8 were
the numbers which showed up when we calculated the properties of
this surface. Two of those three values seem to tie into these Pi-derived
Again, keep in mind that the value of 1.78 was calculated
using the regular value of Pi and therefore would not fully correlate
with the synergetic ratios we described earlier. Once again the
next step is to take this regular-Pi value for the spheres diameter
and round it as we did to 9.8696044, bringing it in line with our synergetic
number system. So 1.78 becomes 1.8. This value was previously the
value for the concave sphere, now we're seeing it as the diameter of our pre-metric-Pi sphere. Maybe the two parameters are really describing
the same aspect?
Now we have all the values we need in order to get a synergetic interpretation of the Pi ratio.
Sphere values from synergetic ratios;
Diameter = 1.8
Total surface - 'area' = 9.9
The next step is to simply divide 9.9 by 1.8 to get
a value for the circumference of the related circle. For us, this
is the ratio between the combined surface and it's concave 'side'.
Simply dividing a spheres surface area by it's diameter gives us
the circumference of it's corresponding circular section.
9.9 / 1.8 = 5.5
Again, we can cross-check these numbers using the
regular Pi calculator and we'll see that the values still come out
relatively close to what we're getting here. But again, keep in
mind that the calculator is using the regular Pi value, so our values
will never match up perfectly.
Now we have the synergetic values for the sphere and the circle worked out, these numbers look almost 'rational', but we haven't worked out a value for synergetic Pi yet.
Surface area = 9.9
Circumference = 5.5
Diameter = 1.8
Diameter = 1.8
These parameters are now intimately tied to the values
we used for our synergetic ratios. You should think of these as
ratios applying to pre-metric or 'a-priori' geometry. Applying parameters
such as diameter and surface area are technically the wrong terms
to use, because all we have here are ratios, not measurements. These
are actually pseudo parameters to help us relate our ratios to Pi.
In fact, if we looked over the ratios which we already saw to relate
to the concave and convex surfaces, we could have produced these
same figures without ever having used Pi. Having said that, using
Pi and these related parameters has been indispensable as a kind
of guide and has been a major help in grasping how these ratios
relate to each other.
Now to go that last step and reevaluate Pi as synergetic ratio.
Pi = C / D (where C is circumference and D is the diameter of the circle).
Synergetic Pi = 5.5 / 1.8 = 3.0555555555555555555555555555555555555555555....
This is a nice number, and very easy to recite, it just depends upon how tired you get from saying 'five'.
Not surprisingly, we see that this ratio has many
unique qualities. The first thing you may notice is that
after the 3.0 there is an infinite amount of 5's and if you can
remember the rule we use for rounding numbers to smaller decimal
places, you might see the funny side of it:
When rounding off digits to fewer decimal places,
the rule is, decide how may digits you need to round to, then locate
the number corresponding to that place. This is the number you are
going to keep after you have increased or decreased it's value.
To figure out if that number goes 'up' or 'down' look at the number
to it's immediate right, if this number is greater that 5 you round
up, if the number is less than 5 you round down but if the number
is exactly 5 you also round UP!
This is excellent news, because with an endless amount
of 5's, this means that synergetic Pi is always positively rounded
(up), no matter how many decimals you choose to count. This seems
to be like an infinite 'curvature' present in this ratio, it always
has this 'lip' at whatever point you choose to stop calculating.
This synergetic variant of Pi then, must always come with this final
6 i.e. 3.0555556.
The 6 is significant in some very subtle ways, we already know the importance of the number 9, and here we see that the 9 is 'split' in two, i.e. 3 and 6. The number 3 is at the very beginning and it's 'partner' is way off at the furthest end.
There has never been a meaningful pattern revealed in the digits of the transcendental Pi, the pattern is 100% chaotic. This synergetic variant obviously differs in that the pattern is immediately obvious, unlimited 5's is about as orderly as you can get, but further investigation of the sequence reveals some less obvious patterns.
From isolating the sequence of 5's and progressively increasing
the amount of 5's we are studying, reveals a pattern. If you imagine
writing this synergetic Pi as 3.056, 3.0556, then as 3.05556 etc.
where you increase the amount of 5's in the sequence. Then, using
a similar process to that which we used in the Phi sequence of the
previous article, we sum each 5 in the sequence to get a single
digit. So here we disregard the other elements of the number and
we increase the number of 5's in synergetic Pi from no 5's up until
twenty 5's. Then we calculate the additive value to see the underlying
Multiples of 5
5 x 0 = 0
5 x 1 = 5
5 x 2 = 10
5 x 3 = 15
5 x 4 = 20
5 x 5 = 25
5 x 6 = 30
5 x 7 = 35
5 x 8 = 40
5 x 9 = 45
5 x 10 = 50
5 x 11 = 55
5 x 12 = 60
5 x 13 = 65
5 x 14 = 70
5 x 15 = 75
5 x 16 = 80
5 x 17 = 85
5 x 18 = 90
5 x 19 = 95
5 x 20 = 100
There are a few ways of looking at the pattern
which has formed. Here we've represented values which add
to 9 as 0. Now start from any 0 and go along the sequence
until you reach another 0. You find that you have to move
9 spaces to get back to where you started. Bringing back the
other four elements of the sequence i.e. not only the 3 at
the start and the 6 at the end of the ratio, but also the
decimal point and the first 0, this gives us a total of 13
Another way of looking at it is, notice the mirrored numbers
which surround the 0. For example the mirror of 4 is 5; the
mirror of 8 is 1. See how each of these compliments have an
additive value of 9.
You can see this mirrored effect more clearly in the Indig
or 'Octave' section, we're displaying the numbers here as
only eight 'frequencies' (positive four and negative four)
and again we're showing the 9 and 0 as the same, this was
the method used in the previous article; Shape
Notice how each number folds around to it's
corresponding compliment, and how the 'Octave' sequence continuously
alternates from positive to negative numbers around the mirror
point (represented as 0). There seems to be a lot going in
the underlying structure of this straightforward sequence.
Whether this is a simple artifact of summing identical digits
or not may be beside the point.
In general, this synergetic Pi ratio may work well with our understanding of synergetic numbers, but clearly it would not be very accurate at measurements within a linear-metric framework. Although the figure is in the same ballpark as regular Pi, trying to use one to prove or disprove the other is like 'comparing apples to oranges'. It seems to me that the two ratios are viewing the same thing but from two different perspectives and of course, they were arrived at through two very different methods.(See also Appendix I Synergetic e? )
Looking back over the process we used to get this ratio, recall
how we started from a sphere surface which had a value of Pi²
.Then once we had identified 9.9 as a viable value for the spheres
surface, we were able to find a correlating value for a diameter
1.8, the concave sphere. We then found that the ratio between the
total surface (concave and convex) and just the concave to be 5.5,
and we related this to the circumference parameter. Finally the
ratio between the diameter (the concave) and the circumference gave
us a synergetic Pi. All possible without a linear metric measurement.
All based on pure ratio. This method didn't assume straight lines
as a starting point, and the resulting ratio may have more in common
with the ratio between a circle's circumference and a section of
a much larger circle or curve, which would appear to us as being
In fact, not only is the ratio between the circle and it's diameter
giving us important number, Pi. But the ratio between the sphere
surface to it's diameter also gives us an important number, 5.5.
This is what would be called the circumference of our circle, there
is something special about this number, it's a perfectly symmetrical,
reflected or palindromic number (it's the same read forwards as
it is read backwards). The two five's that are coming up here seem
to be very significant as fivefold symmetry is the hallmark of living
organisms, and this deserves further investigation at another time.
Icosahedron's opposite poles
display Fivefold symmetry - Twice !!
Getting back to the original application of synergetic
number, where the volume of the VE (the whole) is given the appropriate
value of 9, we can now see good reason why ancient cultures used
calculations that we're intimately tied to the number 9. Perhaps
they were aware of the importance of starting from the whole and
knew that there was no synergy possible from using a system which
compounds 1's upon 1's as our modern understanding has classically
The original connections between synergetic geometry
and the Mayan calendar are continuing at this moment and there are
a fair amount of correlation's appearing, I'm not as well versed
in the mechanics of the Mayan calendar to give anything substantive
at this moment. However, I will say at this early stage, it is looking
to be a description of what Fuller called a 'Jitterbug' transformation.
The calendar seems to be describing the volumetric growth from an
octahedron (concavity) into the vector equilibrium (convexity),
the ancient Maya most definitely had some advanced knowledge about
the nature of time, and it's amazing to have that now tying into
our modern understanding of synergetic geometry. This is an area
which may give us the opportunity to understand the nature of time
from a new perspective. At this early stage, it seems as though
time is not linear, but rather an 'omni-directional' and an exponential
growth in volume.
It's clear to me that now we need to reevaluate and
update our classical understanding of number. We need to use conceptual
tools more suitable to the reality we find ourselves in. Without
getting into too much philosophy, I would like to explain that these
explorations have, for me, revealed some significant insights. One
such observation is that all number (and by definition all 'things')
beyond the illusion of division, are fundamentally related to nine,
nothing, the cypher - or sphere, the epitome of unity and fundamental
to all 'things'. Look at some of the numbers we've come across here
and notice some of the subtle symbolism they suggest. Like the number
1.8. To me, it suggests that in order to have a singular entity,
represented here by the number 1, there must always be this 'everything
else' aspect, represented here by the 8, which corresponds to our
symbol for infinity turned to it's side. So on one side of the decimal
we see singularity, the other side suggests infinity, and the two
combined bring us back to 9 - nothingness, or the 'whole'.
We can broaden our understanding of what is meant by no-thing-ness,
by saying that all of the meanings underlying our communications
and interactions are non-weighable and non-detectable to physical
instruments. There is no quality of thing-ness to any of the meanings
which we experience. It has been said that all meaning is angle,
like the way we use ratios to see relationships between many different
aspects of life. And that these relationships - like angles - are
not things in themselves.
To put it another way, we are the 'sensing mechanism' which can
understand the meanings which life presents to us and that this
nothingness is the interconnected metaphysical reality which we
experience. This unified whole is always reflected in each of the
parts, and funnily enough, if all numbers are divisions of nothing,
you may be right to observe that 'all is none'.
After examining the process which was used to get the synergetic variation of Pi, I wondered if a similar process could be used to find a common ratio in the volumes of these geometries. The synergetic Pi ratio was using the values we had for the surfaces of these geometries, it made sense that using the volumes would give us a similar ratio.
When we calculated synergetic Pi we started from the spherical surface value of 9.9, We got this value from our previous calculations involving the surface of the VE. The synergetic values for the surface gave us two numbers, the convex (8.1) and the concave (1.8), which taken together gave us 9.9. This combined value for the surface of the sphere was our 'whole' and it was from here we started our calculations.
This time, instead of using the total surface value of 9.9, we will only use the original value of 9, which is the volume of the VE. This allows us to use the same method which we used for calculating synergetic Pi, but instead of taking the surface to be the whole, we will take the VE's volume to be the whole.
This is the only thing that needs to change in the calculation, because the value we used in synergetic Pi for the diameter, is the same as the value for the volume of the fully contracted state, the octahedron with it's volume of 1.8.
Now we have all we need to get a ratio similar to synergetic Pi, but one which relates to volume.
This process is like finding ratios within ratios, which is just as valid as any other form of mathematical experiment. The first thing we do is find the ratio between the maximum volume of 9 and it's contracted counterpart, 1.8.
9 / 1.8 = 5
This makes sense, because when we're talking about the volumes which these values relate to, you can see that the ratio between the VE (9) and the octahedron (1.8) is 5. Simply stated, it takes 5 octahedrons to fill the volume of the VE.
In relation to how we calculated synergetic Pi, notice that this ratio is the volumetric equivalent to the 5.5 which we attributed to the circles circumference. And similar to the way we went that final step when we divided the 5.5 by 1.8 to get our synergetic Pi, we can simply divide this volume based ratio of 5 by 1.8 to get our new ratio.
5 / 1.8 = 2.777777777777...
This is the volumetric equivalent to synergetic Pi, and to me it looks like a synergetic variation of the second most important transcendental number in mathematics. Euler's number e is the 'base of natural logarithm's', it is a transcendental and therefore irrational number, just like Pi. I have only a basic understanding of why this number is so important, but it seems that this constant is the rate at which exponential growth is calculated. It is used as a description of the natural increase or growth in frequency of a system.
The ratio which we have extrapolated from the volume of the expanded vector equilibrium in relation to it's contracted state, seems to be a description of a similar ratio to the ratio which e describes i.e. exponential growth. The ratio we've just calculated is tied to the differences in volume from the expansion or contraction of the vector equilibrium, it's this ratio which describes the volumetric growth from concavity to convexity. As you might have figured, I am not a mathematician, but it is my guess that this ratio and the number e, are two ways of describing the same thing.
Similar to the synergetic variation of Pi, we see that there is a repeating number after the decimal, in this case it's 7, and just like with synergetic Pi, this number must always be rounded at whatever decimal place you choose to stop. So this synergetic e always comes with a final 8. i.e. 2.7777778.
This volume based ratio was calculated through simply dividing ratios into ratios, therefore it is not dependant upon any physical measurement. And as with our synergetic Pi, this new ratio is revealing some very synchronistic properties.
Taking this ratio and multiplying it by angles relating to circular degrees gives us some fascinating results.
2.77778 x 45 = 125.0001
2.77778 x 180 = 500.0004
2.77778 x 360 = 1000.0008
2.77778 x 720 = 2000.0016
2.77778 x 1080 = 3000.0024
2.77778 x 1440 = 4000.0032
2.77778 x 1800 = 5000.004
2.77778 x 2160 = 6000.0048
2.77778 x 2520 = 7000.0056
2.77778 x 2880 = 8000.0064
2.77778 x 3240 = 9000.0072
2.77778 x 3600 = 10,000.008
Here we're seeing values which relate one full rotation to the number 1000, meaning that two rotations is 2000, three is 3000 etc. And In every case, the figures after the decimal place makes the total value add to nine. There must be some significance to this, maybe it relates to a method of calculating precessional numbers.
There's also a relationship between synergetic Pi and this volumetric variation. Dividing Synergetic Pi into this new ratio gives us the following.
Besides the fact that this number begins with a 'clear cut' 1.1, there is a pattern in the digits after the decimal place. You can see that there are a lot of 0's and 9's showing up here, but how these numbers are arranged is pretty amazing. The pattern begins after the decimal place;
The first digit is a single 1, after which there are eleven 0's
Then two digits (which add to 8) followed by ten 9's
Then three digits (which add to 1) followed by nine 0's
Then four digits (which add to 8) followed by eight 9's
Then five digits (which add to 1) followed by seven 0's
Then six digits (which add to 8) followed by six 9's ...etc.
The pattern is incrementing in the number of digits that separate the 0's and 9's and at the same time it's decrementing the number of 0's and 9's which come between those same separating digits.
Not only that, but the additive values of those interpenetrating digits always alternates between 8 and 1.
The pattern continues in this wave like motion until it becomes hard to distinguish the pattern from random digits.
Notice that the amount of 0's you find at the start of the sequence depends on how many digits you include in the two ratios you begin with. For example, if you include twenty digits after the decimal place in the ratio 3.0555... and in the ratio 2.7777... , you get a sequence that begins with 1.1 followed by twenty 0's.
This is showing us that these two ratios have an important relationship to each other, maybe future experimentations will allow this relationship to become clearer to us.
A brief detour into some of the more complex areas
of Fuller's work gives us a glimpse of some interesting functions
of this synergetic view of number. The Quantum modules are like
the 'subatomic shapes' of synergetic geometry. From these parts
you can construct all-space-filling polyhedra.
It usually takes
a combination of both tetrahedrons and octahedrons to fill all space
-this would be like the 'skeleton' of the synergetic IVM. But Fuller
went one step further than this when he subdivided tetrahedrons
and octahedrons into 'smaller' volumes. These are the quantum modules
(or mods), and from these mods, more primitive space-filling polyhedra
can be constructed.
By subdividing the tetrahedron we get what are called
the A mods. It takes 24 of these lesser 'irregular' tetrahedrons to
fill the volume of one regular tetrahedron. These can be categorized
as 12 left handed and 12 right handed modules. This means that each
A mod is 1/24th the volume of a regular tetrahedron.
In a similar process, we subdivide a regular octahedron
to find our B mod. But in this case we find that the octahedron
is composed of both A and B mods. The A and B mods are equal in
volume (both are 1/24th the volume of a regular tetrahedron) and It
takes 48 A mods and 48 B mods, giving a total volume of 96 mods
for the construction of one regular octahedron. So to get the full volume of one octahedron, it takes
96 mods (48A, 48B) where each of volume is1/24th of a regular tetrahedron
Both mods are irregular tetrahedrons, they are elongated
wedge shaped tetrahedrons. With a combination of
these two mods all other synergetic polyhedra can be constructed
(excepting those with 5-fold symmetries).
Something very interesting happens when we apply our
synergetic values for volume into to these Quantum Modules. Our
regular tetrahedron's volume has a synergetic value of 0.45. Dividing this by 24 will give
us the synergetic value for a single A or B mod.
So, 0.45 / 24 = 0.01875
Something unexpected has happened here, because for
the first time our synergetic numbers are not adding to 9. In this
case the additive value is 3.
But we haven't mentioned another important feature of these mods yet, which is that these mods taken individually cannot fill all space, these mods are like your puzzle pieces with which you can then create new all space filling polyhedra. One of these mods cannot fill all space by itself, but there are a number of ways which these polyhedra can be combined in order to achieve this.
For a comprehensive visual tour of these geometry's
and how they are constructed, I recommend visiting Richard Hawkins' A&B
Quanta Module Hypertoons page.
For the purposes of our number exploration, we will
just examine one of these all-space-filling polyhedra. The minimum
tetrahedron is known as the 'MITE.' This irregular tetrahedron is
the simplest all space-filler and is composed of 2 A mods and 1
B mod. Now this is fitting with our synergetic understanding of
number, because these mods taken individually do not make a whole
(i.e. add to 9), but taken in the form of the MITE, the values for
the total volume comes to 9.
What we're seeing here is the result of simply adding
one mod to another where each separate mod has an additive value
of 3. This means that we're simply adding up 3's with the volumes
of these mods, so if we were to continue on with this process of adding mods, it
would look like this: 3, 6, 9, 3, 6, 9, etc.
The synergetic values of these modules seem to be operating in such a way as to reflect the required assembly of these volumes in order for them to fill all space. The numbers relating to these mods seem to have the property of combining in very specific ways.
Here it looks as though the A mods come in pairs, while the B's come alone, These two 'groups' taken separately would have an additive value of 6 and 3 respectively.
Could this relate to the property of 'spin-1/2' which
physicists attribute to subatomic particles?
graphic based on Menger Sponge by Paul
Bourke. Copyright applies. Use only with permission.