# Shape and Number

//Shape and Number

## Shape and Number

If you think that advanced mathematics is far too complicated and that it makes aspects
of science and physics incomprehendeable to the majority of people, then you’re a lot like me.
Most of what I was thought about math in the educational system has long since deserted me,
in my view, this is not because math is irrelevant, it is because the language used to describe
the principles of math is so alien to most. Luckily my interest in math (specifically number)
has been excited through applying a more visual spatial approach to geometry
and it’s counterpart, number.

I’m no expert of mathemetics, ‘but I know what I like’ as they say. This article will attempt to
describe some of the elementary tools (drawing heavily from the work of others) and apply them
to some basic number progressions to reveal some hidden and often meaningful patterns.
A nice way to think of these tools, is in the form of X-ray glasses, which can reveal
‘orderly’ patterns in places where only ‘chaos’ was formely seen.

Justin Lawless ~ March 25, 2007
.Source : http://treeincarnation.com/articles/Number.htm

Natural Numbers

When we look at our baseten system (0,1,2,3,4,5,6,7,8,9) we often take for granted the underlying functions of each number and how they are interrelated. From my own investigations of the behaviors of these universally accepted digits, I have learnt some fascinating and often meaningful patterns exising between these simple numbers.

This is an attempt to tie together some of the overall patterms and relationships in natural numbers, this is easier said than done, as a lot of what will be explained here will require the reader to revaluate some of their preconcieved ideas about what number is. A lot of what will follow defies a purely logical explaination, for this reason a lot of what will be explained needs to be considered largely on the visuals that go along with this, in order to help ‘connect the dots’ and to lend a more aesthetic or intuitive interpretation of the explaination. From what I can see, the reason for this difficulty is with our assumptions of what number and math really are. We tend to visualise three ‘things’ when we think of the number 3, we hardly ever take notice of the language that numbers represent. For this reason we tend to overlook some more basic aspects which might exist between numbers.

My own view is that number is not an objective reality in and of itself, but a conceptual tool which we apply to reality and get results based upon those concepts, for this reason it seems odd to ask the question, how does nature ‘add’ or ‘multiply’? It seems to be self evident simply because our own conceptual models appear to reflect our sensed environment well, but this is still not evidence of the validity of such conceptual models. In other words, a conceptual model is like a game with many different rules for many different situations, the fact that the game works well (the rules don’t contradict) can be very helpful, but it doesn’t mean that the conceptula model has any objective basis in reality. This reminds me of that word which is often used to describe the roots of accepted schools of thought, the word ‘foundational’, which, just like the immutable bricks upon which a building is built, implies a solid or ‘perfect’ state from which further conclusions can be built upon. The problem is that the foundations of such systems of thought are rarely put to scrutiny. In a previous article on this subject (posted here), an equation that describes demonstrateable aspects of space was used to point out one such flaw in our inherited, ‘foundational’ algebra.

Mathematics is defined as ‘the science of structure and pattern in general’, taking this view on, along with my own views on possible misassumptions of such mathematical models, I decided to focus my attention on examining the most fundamental aspects of mathematics, the language of number itself. Amazingly, there will be no talk of binomial’s, Riemann zeta functions, quadratic fields or even prime number distributions, this is nothing like conventional number theory, it is an unorthodox examination of the language of number. The surprising thing for me was, that the language itself seems to reflect the patterns of the natural world in a more elegant way than that which conventional, mechanical measurement and theoretical physics could predict.

‘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.

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 zero-nine 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 long-count 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. 54-45, 65-56). 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 in-between two transpalindromes (45-54), 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 ‘9-11 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) Retro-composite primes
8) Retro-composite squares
9) Retro-square composites
10) Retro-square primes
11) Retro-prime composites
12) Retro-prime 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.

Phi (N)
Indigs Octave
Phi (N)
Indigs Octave
fib(0)=0
0   0
fib(19)=4181
5 -4
fib(1)=1
1 +1
fib(20)=6765
6 -3
fib(2)=1
1 +1
fib(21)=10946
2 +2
fib(3)=2
2 +2
fib(22)=17711
8 -1
fib(4)=3
3 +3
fib(23)=28657
1 +1
fib(5)=5
5 -4
fib(24)=46368
0  0
fib(6)=8
8 -1
fib(25)=75025
1 +1
fib(7)=13
4 +4
fib(26)=121393
1 +1
fib(8)=21
3 +3
fib(27)=196418
2 +2
fib(9)=34
7 -2
fib(28)=317811
3 +3
fib(10)=55
1 +1
fib(29)=514229
5 -4
fib(11)=89
8 -1
fib(30)=832040
8 -1
fib(12)=144
0   0
fib(31)=1346269
4 +4
fib(13)=233
8 -1
fib(32)=2178309
3 +3
fib(14)=377
8 -1
fib(33)=3524578
7 -2
fib(15)=610
7 -2
fib(34)=5702887
1 +1
fib(16)=987
6 -3
fib(35)=9227465
8 -1
fib(17)=1597
4 +4
fib(36)=14930352
0   0
fib(18)=2584
1 +1
fib(37)=24157817
8 -1

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 zero-nine 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 I-O sphere doing it’s inside-outing, 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 close-packed 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 VE-Jitterbug 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 pre-existent or a-priori 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. Thing-ness 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 1-ness (thing-ness) and more of a statement of 0-ness (no-thing-ness).