## E-Book - Part 1 The Numerical Structure "Simplicity is the ultimate sophistication"

- Leonardo Da Vinci

# Chapter 1: Modular Arithmetic

### Introduction

My story begins on Wednesday September 1st 2010, a day I will never forget.

After a sabbatical following a long and arduous contract with a leading hedge fund during the financial crisis, I was back to my desk and considering the best way forward for my career. I had been sat down for just 15 minutes when I happened across a quote attributed to Nikola Tesla that changed my life:

'If you only knew the power of the 3 6 and 9 you would have the power of the Universe at your disposal.'

I sat there perplexed as I took the quote in. It didn't make any sense. I knew a little about Tesla, enough to know he was thought to be an exceptional genius, one of the greatest to walk the Earth in modern times.

I googled the quote and for 3 weeks solid, in a zone like trance, for 18 hours a day, I became transfixed by this enigma, quickly finding my way to a rather obscure and esoteric, fringe, branch of mathematics named Vortex Based Mathematics (VBM) pioneered by the eccentric Marko Rodin and brought forward by Randy Powell who was a particular rock of support throughout the very early stages of this work. Vortex Base Mathematics is certainly well outside mainstream academia and yet I knew they were on to something very deep and very important.

Very briefly, VBM can be defined as the study of resonance and ratio that attempts to identify the natural flow of energy, modelled on what is called a torus, a geometric object that looks like a doughnut.

### The Torus  I have no scientific background whatsoever, and being totally ignorant about how electricity and magnetism work, I found it hard to take in what they were talking about. However, what I really zeroed in, and what really caught my attention, was their use of Modular Arithmetic, also known as clock arithmetic or as I prefer, the arithmetic of remainders. Of particular interest was Mod 9 which is central to VBM and of course the only Modular base to include all the numbers we have - 0 1 2 3 4 5 6 7 8 9.

It was this that has led me to more of an understanding of what Tesla may have meant by that quotation above, and this research is dedicated to that end.

His quote unequivocally infers that Numbers are real in some causal way and it is in understanding the archetypal properties or 'idiosyncrasies' of each Number 0 to 9 that we can potentially know, and not guess at, how everything works, everywhere in the Universe.

It will hopefully become clear that Mod 9 can and must be used to extract useful information from data available to us that would not otherwise be readily identifiable and that reveals, clearly, the causal explanation for the effects we see and measure.

Included here as evidence for my assertions, I start by showing some of the basic VBM work pioneered by Marko Rodin and progressed by Randy Powell and others, which covers Cellular Mitosis and part of the section on the Fibonacci Sequence.

This will then lead on to my original applications of Mod 9 to other Integer Sequences, to Musical tuning scales, to Prime Numbers, to Music and ultimately to Particle Physics and the numerical and geometric basis of Light.

It has been quite a journey.

Throughout this section I want to show the patient reader how the numbers we have, 0 to 9 behave specifically as a system that pervades all of our natural world. In order to do this I am going to use modular arithmetic which is much simpler than it sounds.

In mathematics, modular arithmetic is a system of arithmetic for integers (the term used for whole numbers), where numbers "wrap around" after reaching a certain value - the modulus (plural moduli).

The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

One use of modular arithmetic that everyone will be familiar with is in the 12-hour clock, where the day is divided into two 12-hour periods.

If the time is 7 o'clock now, then 8 hours later it will be 3 o'clock. Usual addition would suggest that the later time should be 7 + 8 = 15, but this is not the answer because clock time "wraps around" every 12 hours. Because the hour number starts over after it reaches 12, this is arithmetic, modulus 12.

I like to think of modular arithmetic as the arithmetic of remainders. To illustrate modular arithmetic very simply, let us take the number 12 as an example:

In Mod 9 (Mod being short for Modulus), the number 12 is 3 because 9 goes into 12 once, leaving a remainder of 3.

In mathematical terms we write 12 = 3 Mod 9.

35= 8 Mod 9 because 9 goes into 35 three times, leaving a remainder of 8.

When working in other Moduli the answer would be achieved in the same way.

For example, take that number 35 again, but this time in Mod 8.

8 goes into 35 four times and leaves a remainder of 3 so we would write 35 = 3 Mod 8.

There is a neat trick for quickly computing the Mod 9 answer for any number: - By simply adding up the digits of the number we can ascertain the answer. So for 35 we would simply go 3 + 5 = 8 Mod 9 and the answer is correct. If we take the number 1231 for example, the answer would just be 1 + 2 + 3 + 1 = 7 Mod 9. A number greater than 9 as a sum of its digits would be treated in the same way so that 3459 for example would be 3 + 4 + 5 + 9 = 21 = 2 + 1 = 3 Mod 9.

While the mathematician might argue that the patterns I show in the following sections can be found whatever modulus one might use, for the purposes of my research I am only interested in Mod 9 because of its exclusive properties where all of our numbers are included and utilised. A lower modulus, for example Mod 6 can only make use of numbers 0 to 6 which obviously leaves out the numbers 7, 8 and 9.

As we will see, Mod 9 shows how the decimal system of numbers behaves as a very specific system from which we can infer deep insights into the nature of Nature.

## Chapter 2: Number Pairs and Number Groups

### Multiplication Tables - Mod 9 In the graphic above I have used a circle as a control for the 9 natural numbers of our decimal system and then used the multiplication tables for each number. When we get a result, for example, 4 x 4, it gives the natural result 16. Mod 9 is then applied to the result to get the answer. So, in this case, 4 x 4 = 16 = 7 Mod 9 because 9 goes into 16 leaving a remainder of 7.

To create the geometries you see above, we simply go from result to result for each number's multiplication table and join the lines up to form geometric shapes. Above we can clearly see the geometric qualities exhibited by each of the first 8 numbers when we apply Mod 9.

For those who are new to modular arithmetic and were wondering what happened to the 9 and 0? Well, whatever Modular base you work in will necessarily result back to itself or zero as there is no remainder. In Mod 9, the Number 9 is paired with the Number 0.

To clarify absolutely how we get to the above geometries, let's start with the 1 times table: 1 x 1 = 1, 2 x 1 = 2, 3 x 1 = 3 etc. You can see that the one times table creates a geometric shape called a nonagon and you can see that the direction of the flow is clockwise.

Now the 2 times table goes 2, 4, 6, 8, 10 is 1 (Mod 9), 12 is 3 (Mod 9), 14 is 5 (Mod 9) etc. Here we have a different geometry. Again, we see the flow is in a clockwise direction.

The 4 times table goes 4, 8, 12 is 3 (Mod 9), 16 is (7 Mod 9), 20 is 2 (Mod 9) etc. It produces a 9 pointed star and again flowing in a clockwise direction.

The 3 times table goes 3, 6, 9, 12 is 3 (Mod 9) 15 is 6 (Mod 9), 18 is 9 (Mod 9) and so on, creating the triangle and again the flow is clockwise.

The 8 times table goes 8, 16 is 7 (Mod 9) 24 is 6 (Mod 9), 32 is 5 (Mod 9), 40 is 4 (Mod 9) etc. Notice the 8 shares the same geometry as the 1, except it is flowing anticlockwise.

Similarly for the 7, 5 and 6 which produce the same geometry as the 2, 4 and 3 respectively.

We can see clearly that the numbers are paired geometrically and have a Positive / Negative, Clockwise / Anti-clockwise, relationship. These form what are called the Number Pairs. Mod 9 is the only modular base system that displays this kind of symmetry where ALL the numbers we have 0 to 9 are paired off geometrically to form 5 pairs.

### Two Distinct 'Circuits'

You can see from the graphic that there are 2 distinct 'circuits' at work here by noticing that the geometries of the numbers 1 2 4 5 7 and 8 all touch each of the 9 numbers around the circle, whereas the 3 6 and 9 keep to themselves. Seeing numbers pictorially in this way was very new and interesting to me as I am sure it will be to many readers.

### Number Groups - Numerical Analysis

In addition to operating in Pairs, the Numbers also operate in specific groupings being 1 4 7 and 2 5 8, created from one of the 'circuits' outlined above and finally the 3 6 and 9 (0) grouping which makes up the other 'circuit'.

Proof of the Number Groups operating in this way will be delivered in the next chapter by looking at all the important number sequences found in Nature and their respective Mod 9 sequences.

## Chapter 3: Important Number Sequences

### Cellular Mitosis

Essential to life, cellular mitosis is essentially the process of cell division that allows humans to grow and for cells to repair themselves. It gives the following sequence: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, etc. - i.e. the cells simply double and double and double.

When we apply Mod 9 to this basic sequence we see a regular repeating sequence of 6 numbers emerge.

The Mod 9 results are in brackets. When we arrange this 6 number sequence in two rows of 3 numbers with the first three above the second three and then read vertically, we see the Number Pairs that were produced by the multiplication tables for each number in each column. Where one takes an even numbered Mod 9 sequence and splits it in two to display the Number Pairs, I call this two dimensional numerical symmetry.

#### 8 7 5

Further, when we arrange this sequence in two rows of 3 numbers and read vertically we see the Number Groups I mentioned in the previous chapter. Where one takes an odd numbered Mod 9 sequence and splits it into three to display the Number Groups, I call this three dimensional numerical symmetry. We will see more of this going forward in other Integer sequences and Prime Numbers.

#### 1 24 87 5

As we saw from the multiplication table results Mod 9, one of the 'circuits' 1 2 4 8 7 5 is represented by this particular sequence. This circuit breaks down into two of the three distinct Number Groups 1 4 7, and 2 5 8. When these two Groups of Numbers operate together we call them the Doubling Circuit named after the Mitosis sequence which doubles and doubles.

The other circuit in the multiplication table results Mod 9, represented by the 3 and 6 is not represented in this sequence but will appear further along in other important sequences. These two numbers are joined by the interoperable 0 and 9 to form the third Number Group.

To recap I have shown the numbers behaving in Pairs and Groups and displaying 2 and 3 dimensional symmetry in a number sequence essential to Life.

### The Fibonacci Sequence

The Fibonacci sequence appears in Indian mathematics where the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135):

From Wikipedia - 'Outside India, the Fibonacci sequence first appears in the book Liber Abaci (1202) by Fibonacci after whom the sequence is named. Fibonacci considered the growth of an idealised, though biologically unrealistic rabbit population, assuming that: a newly born pair of rabbits, one male, one female, are put in a field; rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits; rabbits never die and a mating pair always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was: how many pairs will there be in one year?

At the end of the nth month, the number of pairs of rabbits is equal to the number of new pairs (that is, the number of pairs in month n - 2) plus the number of pairs alive last month (that is, n - 1). This is the nth Fibonacci number.'

So, to clarify simply:-

• At the end of the first month, they mate, but there is still only 1 pair.

• At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field.

• At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field.

• At the end of the fourth month, the original female has produced yet another new pair, and the female born two months ago also produces her first pair, making 5 pairs.

For those that do not know, the Fibonacci sequence is simply formed by adding the previous two terms of the sequence to get the next term. It begins:-

#### 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 etc.

It is the organising structure found throughout nature and evident in every living thing and all physical phenomena. As an example it is found throughout the human body, even in our DNA. The DNA molecule, the program for all life, measures 34 angstroms long by 21 angstroms wide for each full cycle of its double helix spiral.

When we divide any term in the sequence by the previous term we approximate to what is called the Phi or Golden Mean or Ratio relationship of 1.618.

When we divide any term in the sequence by the next term we approximate to 0.618, also called phi but with a little p.

Further, when we divide any term in the sequence by the term two before it we find that they are in a relationship that approximates to 2.618.

Having more than a passing acquaintance with Fibonacci and the Golden Mean through my work on financial markets, I immediately looked at the sequence in a new light having thrown the Mod 9 lens over it, discovering to my amazement that if you go far enough out you soon discover that there is an infinitely repeating 24 number sequence:

### #### 1 1 2 3 5 8 4 3 7 1 8 9 8 8 7 6 4 1 5 6 2 8 1 9

Arranged as 2 lines of 12 digits with the first 12 numbers above the second, we can see the Number Pairs

1 1 2 3 5 8 4 3 7 1 8 9

8 8 7 6 4 1 5 6 2 8 1 9

Arranged as 3 lines of 8 digits we can see the Number Groups

1 1 2 3 5 8 4 3

7 1 8 9 8 8 7 6

4 1 5 6 2 8 1 9

Arranged as 6 lines of 4 digits we can see the Doubling Circuit that we see in the mitosis sequence in the First Column, the Number Pair 1 and 8 in the second, another Doubling Circuit, and finally the 3 6 9 Number Group.

1 1 2 3

5 8 4 3

7 1 8 9

8 8 7 6

4 1 5 6

2 8 1 9

The Fibonacci sequence is found everywhere in Nature. Below we see it in the hand, the sequence displayed as a spiral and the sunflower.   ### Other Natural Integer Sequences

Below I have applied the same technique to all the other important number sequences, using Mod 9, to find repeating cycles of numbers, or moduli, underpinning them all and all either showing the Number Pairs or Number Groups in two or three dimensional numerical symmetry. Text from Image in Italics

MOD 9 REPEATING SEQUENCES HIGHLIGHTED IN TAN AND YELLOW

Lucas Sequence - Mod 9

2 1 3 4 7 2 0 2 2 4 6 1 7 8 6 5 2 7 0 7 7 5 3 8

24 digit sequence Mod 9 showing Number Pairs and Groups.

2 1 3 4 7 2 0 2 2 4 6 1

7 8 6 5 2 7 0 7 7 5 3 8

2 1 3 4 7 2 0 2

2 4 6 1 7 8 6 5

2 7 0 7 7 5 3 8

Pell Numbers - Mod 9

1 2 5 3 2 7 7 3 4 2 8 9 8 7 4 6 7 2 2 6 5 7 1 9

24 digit sequence Mod 9 showing Number Pairs and Groups

1 2 5 3 2 7 7 3 4 2 8 9

8 7 4 6 7 2 2 6 5 7 1 9

1 2 5 3 2 7 7 3

4 2 8 9 8 7 4 6

7 2 2 6 5 7 1 9

Jacobsthal Numbers - Mod 9

1 1 3 5 2 3 7 4 9 8 8 6 4 7 6 2 5 9

18 digit sequence Mod 9 showing the Number Pairs and Groups.

1 1 3 5 2 3 7 4 9

8 8 6 4 7 6 2 5 9

1 1 3 5 2 3

7 4 9 8 8 6

4 7 6 2 5 9

1 9 9 1 9 1 1 1 2 2 3 4 5 7 9 3 7 3 1 1 4 2 5 6 7 2 4 9 6 4 6 1 1 7 2 8 9 1 8

39 digit sequence Mod 9 showing the Number Groups.

1 9 9 1 9 1 1 1 2 2 3 4 5

7 9 3 7 3 1 1 4 2 5 6 7 2

4 9 6 4 6 1 1 7 2 8 9 1 8

The Lucas Sequence is the same as the Fibonacci series except it begins 2 1 instead of 0 1 that often occurs when working with the Fibonacci series.

The Pell numbers are an infinite sequence of integers that have been known since ancient times, the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12, and 29. Pell numbers grow exponentially, proportionally to powers of the silver ratio 1 + sq root 2.

The Jacobsthal sequence starts with 0 and 1, then each following number is found by adding the number before it to twice the number before that.

The Padovan sequence produces the side lengths of a spiral of equilateral triangles. https://commons.wikimedia.org/w/index.php?curid=24501622 Text from Image in Italics

#### Maximal Number of Regions into which Spheres Divide Space

This sequence returns a repeating 27 digit Mod 9 sequence showing the Number Groups.

0 2 4 8 7 3 7 3 2 6 8 1 5 4 0 4 0 8 3 5 7 2 1 6 1 6 5

0 2 4 8 7 3 7 3 2

6 8 1 5 4 0 4 0 8

3 5 7 2 1 6 1 6 5

Maximal Number of Regions into which Planes Divide Space

This sequence returns a repeating 27 digit Mod 9 sequence showing the Number Groups.

1 2 4 8 6 8 6 1 3 4 5 7 2 0 2 0 4 6 7 8 1 5 3 5 3 7 0

1 2 4 8 6 8 6 1 3

4 5 7 2 0 2 0 4 6

7 8 1 5 3 5 3 7 0

### Geometric Sequences & Figurate Numbers

The mathematical study of figurate numbers is said to have originated with Pythagoras, possibly based on Babylonian or Egyptian precursors. Figuracy of number is something all children should be taught about early on, so that they form a relationship with numbers through the visual nature of geometry as opposed to a meaningless symbol good only for use at the shops.

A number can be described as figurate if a regular geometric shape may be made up from the available quantity of building blocks or counters. For example, 3 blocks or counters may be arranged in a triangle, four in a square, etc. This gives rise to sequences of numbers, triangular, square, pentagonal etc. Some numbers can be represented by two or more regular shapes. These are known as poly-figurate numbers. If two shapes can be created, the number is called Bi-figurate. There are however only two Tri-Figurate Numbers, numbers which can make three regular geometric shapes.

### 91 as Triangle, Centred Hexagon and 5 Layer Square Pyramid. Image Credit: http://homepage.virgin.net/vernon.jenkins/Symb.htm

Below we can see that all figurate numbers display a 9 number sequence that repeats Mod 9.  Notice again that each of these sequences possess the three dimensional numerical symmetry that display the Number Groups. Text from Image in Italics

Triangle Numbers

1 3 6 1 6 3 1 0 0

1 3 6

1 6 3

1 0 0

Square Numbers

1 4 0 7 7 0 4 1 0

1 4 0

7 7 0

4 1 0

Pentagonal Numbers

1 5 3 4 8 6 7 2 0

1 5 3

4 8 6

7 2 0

Hexagonal Numbers

1 6 6 1 0 3 1 3 0

1 6 6

1 0 3

1 3 0

Heptagonal Numbers

1 7 0 7 1 0 4 4 0

1 7 0

7 1 0

4 4 0 #### Text from Image in Italics

Octagonal Numbers

1 8 3 4 2 6 7 5 0

1 8 3

4 2 6

7 5 0

Nonagonal Numbers

1 0 6 1 3 3 1 6 0

1 0 6

1 3 3

1 6 0

Decagonal Numbers

1 1 0 7 4 0 4 7 0

1 1 0

7 4 0

4 7 0

Hendecagonal Numbers

1 2 3 4 5 6 7 8 0

1 2 3

4 5 6

7 8 0

Dodecagonal Numbers

1 3 6 1 6 3 1 0 0

1 3 6

1 6 3

1 0 0

We can see that when we apply Mod 9 analysis to figurate numbers there are 3 distinct groups that are separated by 3.

Triangle Hexagonal Nonagonal Dodecagonal - 3 6 9 12 and so on. The 3 6 9 Grouping.

Square Heptagonal Decagonal - 4 7 10 and so on. The 1 4 7 Grouping.

Pentagonal Octagonal Hendecagonal - 5 8 11 and so on. The 2 5 8 Grouping

### Platonic Solids

The Greek philosopher Plato, born around 430 B.C., wrote about these five solids in a work called Timaeus. Historical accounts vary a bit, but it is usually agreed that the solids were discovered by the early Pythagoreans; perhaps by 450 B.C. Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid as shown below. Of the fifth Platonic solid, the dodecahedron, Plato obscurely remarks, "...the god used [it] for arranging the constellations on the whole heaven". Image credit - https://www.technologyuk.net/mathematics/geometry/platonic-solids.shtml

The Platonic Solids each show a repeating sequence Mod 9 again illustrating the Number Groups 1 4 7, 2 5 8 and 3 6 9 (0). Text from Image in Italics

Platonic Solid Moduli

Tetrahedral Numbers

1 4 1 2 8 2 3 3 3 4 7 4 5 2 5 6 6 6 7 1 7 8 5 8 0 0 0

1 4 1 2 8 2 3 3 3

4 7 4 5 2 5 6 6 6

7 1 7 8 5 8 0 0 0

Octahedral Numbers.

1 6 1 8 4 2 6 2 3 4 0 4 2 7 5 0 5 6 7 3 7 5 1 8 3 8 0

1 6 1 8 4 2 6 2 3

4 0 4 2 7 5 0 5 6

7 3 7 5 1 8 3 8 0

Cube Numbers.

1 8 0

Icosahedral Numbers.

1 3 3 7 3 6 4 3 0

1 3 3

7 3 6

4 3 0

Dodecahedral Numbers.

1 2 3 4 5 6 7 8 0

1 2 3

4 5 6

7 8 0

## Chapter 4: Music

I preface the following by the admission that I have developed little to no musical aptitude as yet and have never studied music theory. However, that does not seem to have stopped me from uncovering what looks to be some very interesting observations having applied Mod 9 to the frequencies generated by the black and white keys of the musical scale.

1955 saw the introduction of the International Standard Tuning of 440 Hz on the A of Middle C Octave. This standardised all instruments around the world. There are 4 tuning systems but the one I looked at was the natural tuning, called Just. Just tuning is deeply connected to the numbers that pervade all of my work. The C's were all 3 & 6, same for C sharp, then D's are all 9's and on until I came to F which revealed the 1 2 4 8 7 5 sequence, in order. This was very exciting yet obviously made sense as the one thing I did know was that frequencies double octave to octave.

You will notice that using this tuning at 440 Hz we see that:

5 sections of the octave are 1 2 4 8 7 5

4 sections are 3 & 6

3 sections are 9

Immediately the Pythagorean 3 4 5 triangle springs to mind.

It also became quickly apparent that I had no need to show as many octaves as I have shown opposite because the Mod 9 sequence repeats every 6 octaves or 6 x 12 = 72 notes. 72 is a very important number in my overall theory, as we shall see. For example, the Mod 9 sequence is the same for the Middle octave, the octave 6 Above and the octave 6 Below. As I did with the multiplication tables for the numbers 1 to 8, I decided to draw each of the octaves, again, using the circle as the control to see what geometry they produced and if any information might be gleaned. Above, we can clearly see that, as with the numbers, the octaves are paired up symmetrically and reflected, separated by 3 octaves.

For example the octave 1 Below is qualitatively or 'archetypically' the same as 2 Above and the Middle octave is the 'same' as 3 Above but they are mirrors, or reflections, of each other.

Just before the end of 2010 I discovered, in my ignorance, and let me assure you I have plenty of that, that the current International Standard tuning of 440 Hz on Middle A had only been around since 1955. Given what I had discovered so far, I was very interested to investigate what came before and what that would look like under the same microscope.

Interestingly, I discovered that the first proponent of the current standard was none other than Nazi Propaganda chief, Josef Goebbels, in 1939, which was incidentally the same year that televisions began to be mass produced and widely available. The introduction of a major Nazi player aroused my interest as those boys knew exactly what they were doing and had all the brainpower in the world at their disposal. Germans won a ridiculous percentage of all Nobel Prizes prior to 1939.

I naturally wondered what the Nazi motivation was to get it changed and of course now I had to know what had been in use before, that they wanted to get away from. Very quickly, I discovered the Pythagorean Musical Scale, the tuning favoured by Mozart & Bach. This tuning is based on 432Hz on the A of the Middle octave instead of the current standard which is at 440 Hz that we analysed above. Look how the numbers turn out when we again apply Mod 9 to the frequencies and see the difference in the sections of the octave occur between the two tunings. We see that:-

2 sections that are 1 2 4 8 7 5, versus 5, using the 440 Hz tuning.

2 sections that are 3 6, versus 4, using the 440 Hz tuning.

8 sections that are 9 versus only 3, using the 440 Hz tuning.

Here too I saw there was a 72 number sequence. Again we see that the octaves are paired and mirrored as they were in the 440 Hz tuning scale. Looking at the respective tuning scales, we can see clearly that there is a lot more 9 and a lot less 1 2 4 8 7 5 and 3 6 in the 432 Hz scale when you compare it to the 440 Hz tuning. My conclusion, using Mod 9, is that this 432 Hz tuning scale is unique in that, it, alone, produces the least amount of electro-magnetic interference versus the largest amount of Source or pure energy. In other words this is the purest direct experience a human being can have of Source and why the works of Bach and Mozart are so imbued with the sublimity that they are.

Now, turning our attention back to the 440 Hz tuning we operate on now. It clearly has too much electro-magnetic interference, with likely, highly negative, disruptive, physical and non-physical side effects.

When we put the octaves together in their respective 72 number sequences we see something interesting.

Here below, we see the geometry created by the 72 number sequence created by the 440 Hz scale. The electro-magnetic interference effects can be seen where we have the perverted 1 6, 8 3, 4 6, 3 5 connections that are removed by the 432 Hz scale.

Below is the geometry created by the 432 Hz scale. Beautiful isn't it? ### Music of the Decimal System

Everyone can agree that the laws of Music are not Man made and they were instead discovered by Man. It is my contention that these musical laws represent the underlying harmony responsible for the formation of the Cosmos and that Numbers, and the Decimal System within which they operate, also, are not Man Made, but were discovered in just the same way as Music, except metaphysically.

When we consider these important number pairs, that I showed earlier, we can see that the Decimal System is nothing but Musical and here is how.

(00)   09   18   27   36   45   54   63   72   81   90   (99)

I simply looked at the differences between each of the Number Pairs, or rather how one may travel from one to the next, to see all the Music of the Universe there in the Number Pairs of the Decimal System.

09 to 18 is 09 x 2 which is the Octave

18 to 27 is 18 x 3/2 is the Perfect Fifth

27 to 36 is 27 x 4/3 is the Perfect Fourth

36 to 45 is 36 x 5/4 is the Major Third

45 to 54 is 45 x 6/5 is the Minor Third

54 to 63 is 54 x 7/6 is the Septimal Minor Third

63 to 72 is 63 x 8/7 is the Septimal Whole Tone

72 to 81 is 72 x 9/8 is the Major Tone

81 to 90 is 81 x 10/9 is the Minor Tone

9 / 8 : 10 / 9 = 81 : 80

Their size differs by exactly one Syntonic comma 81:80

where it is important to note that:-

80/81 = 0.987654321 and 1/81 = 0.123456789

More simply, visually, we see the rather elegant:-

2      1 1/2     1 1/3      1 1/4      1 1/5      1 1/6       1 1/7      1 1/8      1 1/9      1 1/10   etc.

These number pairs, and therefore the decimal system are musical and are therefore inherent and universal.

### Seeking a Vibrational Foundation to Music

Collaborator Julian Shelbourne has kindly allowed me to reproduce some of his excellent work.

'The foundation of music is harmonics: Just touching a vibrating string at a whole number division of its length yields another tone.

For example, touching a vibrating guitar string at precisely 1/3 of its length yields a musical note called "the fifth" (the fifth note in the western scale - do-re-mi-fa-so) which has a vibrational frequency 3 times higher than the original. You can continue touching the vibrating string at whole-number divisions of its length to yield five tones of the western harmonic scale: ½ (octave), ¼ (octave), 1/5 (major "third" - do-re-mi), 1/6 (an octave of a fifth), 1/7 (minor seventh), 1/8 (octave), 1/9 (ninth - do-re).  Doing this from a single string doesn't provide all 7 of the notes necessary to construct the western scale; but doing this on two strings - one tuned to the loudest harmonic of the first string (its "fifth") - does.

Each note generated in this way is an exact, mathematically-aligned reflection of the starting vibration - but it begs the question: which vibrational frequency should we start with?

Anthony has explored the differences between the modern standard where A=440 Hertz, and the naturally resonant standard of A=432 Hertz - which was documented by Zarlino in the 16th century and in other sources.

I wanted to add my own personal experience which bolsters the notion of A being correct at 432 Hertz, but also indicates that A is just part of a universal musical scale for which the foundational vibration is B-flat, at 7.2 Hertz - or some sub-audible octave of this.

This insight came as a result of exploring sub-audible frequencies on a tone generator app on my iPhone using Bluetooth headphones. I discovered that at around 11 Hertz there would be a slow, swooshing sound that would slow to a stop at precisely 10.8 Hertz.  The swooshing sound would speed up and then stop again as I lowered the frequency to 7.2 Hertz, and again at 5.4 Hertz.

In music, the term "beating" is used to describe the oscillation heard when two tones are just slightly out of tune with each other. I believe the swooshing sound I was witnessing was the beating between the frequency I was generating on my tone-generator and the overtones of some kind of "background radiation" of sub-audio sound.

The frequencies themselves are interesting. 108, 72 and 54 are all numbers that appear in sacred geometry, show Anthony's Number Pairs and all return 0 or 9 Mod 9.

If you take 7.2 Hertz as your starting frequency, and generate whole number harmonics from it, you realize that 10.8 is exactly 3/2 (or a musical fifth), and 5.4 Hertz is exactly an octave below that. So, this apparently random standing-wave phenomenon is musically harmonic.

If you construct a harmonic series based on these frequencies, you get the following five frequencies. (Here I have started 32 octaves above the frequency I found: 7.2 Hertz x 32 = 230.4 Hz):

 Multiplier Interval First Harmonic Series Note Harmonic Frequency Mod 9 1 Unison B-flat 230.4 2+3+4=9 3 Fifth F 345.6 3+4+5+6=18=9 5 Third D 288 2+8+8=18=9 7 Seventh G#/Ab 201.6 2+1+6=9 9 Ninth C 518.4 5+1+8+4=18=9

Opposite we can see the second harmonic series - based on the second frequency I discovered, raised by 32 octaves to give us more familiar frequencies. (10.8 Hertz x 32 = 345.6). Note that A comes out to be 432 Hertz!

 Multiplier Interval Second Harmonic Series Note Harmonic Frequency Mod 9 1 Unison F 345.6 3+4+5+6=18=9 3 Fifth C 518.4 5+1+8+4=18=9 5 Third A 432 4+3+2=9 7 Seventh D#/Eb 302.4 3+2+4=9 9 Ninth G 388 .8 3+8+8+8=27=9

Finding A = 432 Hz in this way was not my intent. It just so happens that A = 432 Hertz is the major-third harmonic of 10.8 Hertz standing wave frequency that I had discovered. But this discovery bolsters the notion that A should be 432 Hertz rather than 440 Hertz, while also indicating that it is B-flat which is the fundamental resonance of the universe from which all other vibrations are generated - rather than A. Putting these two harmonic series together yields us a musical scale of 7 notes which I believe is resonant with the core of the universe:-

 Multiplier Interval Harmonic Musical Scale Note Harmonic Frequency Mod 9 1 Unison B-flat 230.4 2+3+4=9 9 2nd/9th C 259.2 2+5+9+2=18=9 5 3rd D 288 2+8+8=18=9 3 x 7 4th D#/Eb 302.4 3+2+4=9 3 5th F 345.6 3+4+5+6=18=9 3 x 9 6th G 388.8 3+8+8+8=27=9 7 7th G#/Ab 403.2 4+3+2=9 3 x 5 Maj-7th A 432 4+3+2=9

Another interesting thing I discovered is that 8 octaves below my "magic frequency" of 7.2 Hertz, B-flat is 0.9 Hertz. Working from the bottom of this table upwards, if you multiply 0.9 Hertz by whole numbers according to the harmonic series, the frequencies you get each have the number on the left of the decimal point rising by one, while the number on the right of the decimal point goes down by one. They're all harmonics of our magic frequency 7.2 hertz and they're all exhibit 0 or 9 Mod 9.

 Multiplier Note Harmonic Frequency Mod 9 Interval 10 D 9.0 9+0=9 3rd 9 C 8.1 8+1=9 9th 8 B-flat 7.2 7+2=9 Octave 7 A-flat 6.3 6+3=9 7th 6 F 5.4 5+4=9 5th 5 D 4.5 4+5=9 3rd 4 B-flat 3.6 3+6=9 Octave 3 F 2.7 2+7=9 5th 2 B-flat 1.8 1+8=9 Octave 1 B-flat 0.9 0+9=9 Unison

In a universe constructed based on 9s, is it really so surprising that the fundamental frequency that drives the sympathetic vibrations of all others might itself be 0.9 Hertz?! I have identified other phenomena which correspond to these "magic frequencies" on my web-site at HarmonicsOfNature.com - including NASA's recordings of B-flat being generated by black-holes, ancient musical instruments tuned to these frequencies and cymatics which first form at these frequencies.' - Julian Shelbourne.

### Cymatics

Cymatics basically reveals the geometry of sound through vibration. This was originally researched and published by Dr. Hans Jenny in 1967 in his book The Study of Wave Phenomena. Ever since I first saw this information I knew it was fantastic and a major piece of the puzzle. I just hope my work brings those involved in this field more attention. I need to quickly cover this area as it is consistent with the insights gleaned from the work on the Pythagorean Musical Scale. The Cymascope is a new type of scientific instrument that makes sound visible. Its development by John Stuart Reid and Erik Larson began in 2002 with a prototype that featured a thin, circular, P.V.C. membrane; later they used latex. Fine particulate matter was used as the revealing media. However, it was soon discovered that far greater detail could be obtained by imprinting sonic vibrations on the surface of ultra-pure water. The surface tension of water has high flexibility and fast response to imposed vibrations, with transients even as short-lived as a few milliseconds.'

When they looked at the geometry of 432 Hz under the Cymascope they found that it pops out as a triangle, every time they imaged it. They initially thought there was something wrong with the CymaScope but after trying for more than an hour they concluded that the number 3 was somehow universally connected to 432 Hz. I realised that this was in fact the same shape as one can see when one draws 3 intersecting circles as we can see below. Reid researched the reason why it takes up this geometry and it turns out to be an interesting case:-

'When A is tuned to 432Hz the frequencies of the other A's shift (within a decimal point) to 27 Hz, 54, 108, 216, 864, 1728 in other octaves. D becomes 576 Hz which becomes 9 Hz, 18, 36, 72, 144, 288, 1152 in other octaves. E becomes 324 Hz which becomes 81 Hz, 162, 648, 1296 in other octaves. All of these frequencies are divisible by 3.' - John Stuart Reid.

Brian Collins, who wrote to John about taking a look at 432Hz under the Cymascope, said John had confirmed 'what seems to be a scientific interpretation of what I believe in my personal opinion, what Rudolph Steiner in his esoteric knowledge of the Mysteries of Golgotha and cosmic principals stated; that a prime centre of tone for living enlightenment in mankind is a C = 128 Hz note with the octave at C = 256 Hz (within 432 Hz Concert pitch) or Michael Sun tone A = 432 Hz which ties the spiritual tetra or trinity of spherical scalar wave patterns and evolution of man ascendant together in sound and geometry of the super hologram of consciousness. The medium of 432 Hz is certain evidence of divine intelligent design of the universe and light.

A 53 note temperament scale based on A = 432 Hz pitch goes less than 1 Hz difference out of harmonic synchronicity over the entire 53 note scale, most other pitches are out of sync after only two or three notes up the octave, an excited engineer confirmed that scales tuned to A=432 Hz are surely the fingerprint evidence of an intelligent divine creator.'

Please do your own research into this exceptional, transformative new Science. I believe it has the ability to create a whole new way of healing disease in the body by using vibration and in fact this has been borne out by recent research in this field and is being used to treat tumours - http://www.cancerresearchuk.org/about-cancer/cancer-in-general/treatment/other/high-intensity-focused-ultrasound-hifu As the body has a high water content and the organisation of water responds by creating beautiful patterns as seen above, might it explain why the classical music of Mozart and Bach, tuned at 432Hz, has such a profound effect upon people who listen to it?

## Chapter 5: The Re-Classification of Prime and Composite Numbers Text from Image in Italics

PRIME NUMBERS.

In order to properly understand the system of Numbers it is crucial that each number be classified correctly. I am asserting that this is not currently the case.

It is thought that Numbers may easily be divided into two groups, those that are called Prime and those that are called Composite.

A Prime Number is recognised as a Number that has factors of itself and 1 only.

A Composite Number is recognised as a Number that has more than these 2 factors.

As we shall see, this is not the whole story or the right way of looking at the picture.

The first thing I did was to simply apply Mod 9 to the first 65000 Prime numbers.

This revealed the surprise that results exclusively returned 1 2 4 5 7 or 8 for all Prime numbers greater than or equal to 5.

However, as with all the previous number sequences, there was no regularity to these Mod 9 results, that is there was no repeating sequence that was immediately discernible. Text from Image in Italics.

Prime Numbers in Red - Composite Numbers is Black

The 72 Number Sequence that I had discovered in Music, 6 octaves of 12, led me to arrange the Numbers 1 to 72 into 6 columns as laid out above.

On the left we have the numbers and on the right we can see the Mod 9 results.

Laying the numbers out like this quickly shows us why Prime numbers will exclusively return these remainder values infinitely, because the Prime numbers only occur in Column 1 and Column 5 which simply repeat 1 7 4 and 5 2 8 respectively, showing that these Number Groups are specifically related to Prime numbers.

The 'problem' with the outlying numbers 2 and 3 will be dealt with shortly. Text from Image in Italics.

By removing columns 2, 3, 4 & 6 and leaving only columns 1 & 5, we are left with only Primes (red) and some 'Other' numbers (black).

These 'Other' numbers are immediately identifiable as either:

Products of Prime Numbers or Exponents of Prime Numbers

5 x 7 = 35 5^2 = 25

5 x 11 = 55 7^2 = 49 etc.

5 x 13 = 65 etc.

These 'Other' numbers are currently classified as Composite Numbers but should rightly be re-classified as Prime Composites and thought of exclusively as part of the Prime Number system, not the Composite Number system as they currently are. Text from Image in Italics.

COMPOSITE NUMBERS.
In order to further cement my case that Mod 9 is useful at inferring information not otherwise obtainable, we must look to the other side of the coin if you like and examine the 'not Prime' numbers or what are known as the Composite numbers.

Composite Numbers have multiple factors, not just itself and 1, as Primes do.

For example, 12 is a Composite Number and has factors of 1, 2, 3, 4, 6 and 12, while 13 is Prime as its only factors are 1 and 13.

As I examined the sequence of Composite Numbers as is currently posited by academia, Mod 9 analysis revealed a sequence that comes out like this:-

4 6 8 0 1 3 5 6 7 0 2 3

4 6 7 8 0 1 3 5 6 7 8 0 2 3

4 6 8 0 1 3 4 5 6

Can you see that these numbers are attempting to organise themselves into a repeating sequence but it's not quite happening?

I immediately realised that you simply have to include the numbers 2 and 3 traditionally thought of as Prime and then take out the Prime Composite Numbers, as described earlier.

So, for Primes 5 and greater, starting with the 5 x 5 = 25 and then 5 x 7 = 35 etc. we take these out and we should hopefully get some order appearing in the Composite Number Sequence.

This would prove that I was on the mark with the Prime Number analysis and my conjecture as to how to treat thos 'Other' numbers in Prime number columns. Text from Image in Italics.

CORRECT COMPOSITE NUMBER SEQUENCE.

In the table on the left you will see that I have added in numbers 2 and 3 and then excluded those numbers that are traditionally thought of as Composite numbers, but which should rightly be classed as a Prime Composite, for all Primes, greater than 5, starting with the 5^5 = 25 and then 5 x 7 = 35 etc.

Where there was no modulus before, there is now.

A 12 digit repeating sequence:-

2 3 4 6 8 0 1 3 5 6 7 0

This can be broken down into 3 columns of 4 to illustrate the Number Groups when reading vertically.

2 3 4 6

8 0 1 3

5 6 7 0

Further, when we break the sequence down into 2 columns of 6 we show the Number Groups again illuminated very well.

2 3

4 6

8 0

1 3

5 6

7 0 Text from Image in Italics.

CORRECT PRIME NUMBER FAMILY SEQUENCE.

Prime Numbers are in fact a very simple group and have a simple congruence exposed by analysis using Mod 9.

When properly classified we see a six number sequence that repeats Mod 9.

1 5 7 2 4 8

Again we see in this sequence the three dimensional numerical symmetry that shows the Number groups.

1 5

7 2

4 8

### Prime Number Re-Classification

Prime Numbers, by virtue of their indivisibility, are the building blocks of all material phenomena, or matter.

I had gleaned from the VBM research that as far as they are concerned the 9 0 represents Source, the central axis, and together with the 3 and 6, representing the polarity or boundaries, this number group represented magnetism. In their estimation the 1 2 4 8 7 5 that we see in the cellular mitosis sequence represents electricity. What they are saying, and I think they are right, is that the combination of the numbers 0 1 2 3 4 5 6 7 8 9 interact in such a way as to produce electro-magnetism.

In my theorised structure of the Universe, the Prime Numbers represent electricity as defined by the interaction of number groups 1 4 7 and 2 5 8, while the 3 6 9 are the containing magnetic infrastructure upon which electricity plays out her dance.

I believe we live in an electric universe, mathematically unfolding with great precision and accuracy through Time, that is totally governed by Prime Numbers and therefore it is Prime Numbers that we must understand thoroughly if we are to master an understanding of the universe and everything in it.

I want to argue vigorously that these previously referred to Prime Composite numbers, identified above, are really no different at all to these traditionally thought of Prime numbers, and actually hold the key to better understanding how Prime Numbers and the whole number system can work as a machine.

The Prime Composites are created exclusively by combinations only of each other. They are all, therefore, inherently Prime, in and of themselves, and nothing else. They are simply iterations of the Prime number sequence itself and all the possible combinations thereof. This allows and accounts for the infinite complexity of Physical Reality.

Reality is created by an infinite fractal expansion of Prime numbers that creates a grid, much like a spider that spins its web. I think that if we are to deny the above, it is surely like saying that what comes of a union between two human beings is not human somehow and that it could be defined as anything else.

### A Priori / Metaphysical Numbers

If we think that all the other kinds of Primes greater than 5, so Seeds (those numbers traditionally thought of as Prime, so divisible only themselves and 1), Composites (Products of Prime Numbers) and Exponents (Primes multiplied by themselves), are Effects, then these, A Priori Numbers, 1 2 3 & 4 can be thought of as the Causes of those Effects.

#### Number 1

The first number in this A Priori group is the Number 1 - which I am calling the Void.

Prime is omitted from the chosen name because immediately there is a question mark raised over even its strict belonging to a Prime number classification at all, A Priori or otherwise.

Even though 1 fits as a member of the 1 4 7 group, obviously; the key here is that the number 1 has an additional factor, exclusive to it as a Prime, other than 1 or itself.

That being -1.

-1^2 = (-1 x -1) = 1

No other Prime number, A Priori or otherwise, can boast such a property and it is this, immediately, that puts it squarely into its own grouping within even the A Priori group of numbers.

#### Numbers 2 & 3

The Composite Numbers 2 and 3, traditionally given the classification of Prime, are really the workhorses, the building blocks of the building blocks of all physical reality and the system of numbers.

All of physical creation comes down to the dance and weave of these two numbers, the 2 and the 3.

They should rightly be called the Causative or A Priori Composite Numbers within the A Priori Group of numbers 1 to 4.

#### Number 4

4 is Composite also, and is important in that it is the first exponent being the first number with a square root - 2^2, and as such, introduces a new dimension or level of complexity by allowing for area. (One might argue that the first squared number could be said to be the number 1 also.)

#### Prime Seeds

Prime Seeds are defined as being all those Prime numbers, 5 and above, traditionally thought of as Prime, i.e. divisible only by themselves and 1.

5 is the first of the Seed Primes that make up all matter in physical reality. It is the first effect created by the additive function of the 2 and 3.

5 is also directly connected to Phi or the Golden Mean, as expressed by the Fibonacci sequence, and which is found everywhere in nature and throughout the Human body.

It is connected thus:- ### Prime Composites & Exponents

There are two types of these Prime Composite numbers, 'Pure Prime Composites' and 'Prime Exponents'.

'Prime Exponents' express the change in dimension of the 'Prime Seeds' whereas the Pure Prime Composites are simply the multiplication or product of Prime Seeds.

### Visual Evidence

As proof of the validity and worth of the above analysis and conjecture, please consider the following.

Below is a Square of 9 Number Spiral - drawn simply by placing the 1 in the middle, the 2 above, 3 to the left of the 2, 4 below the 3 etc. Spiralling around and around. If we expand the spiral of numbers and highlight only the prime numbers traditionally thought of as prime, then we arrive at something called Ulam's Spiral which is show below:- ##### http://en.wikipedia.org/wiki/Ulam_spiral

The numbers, traditionally called Prime, produce, a grainy picture, for sure, but one which definitely shows preponderance by the Prime Numbers to appear on certain clear diagonals. Now, if we add in the Prime Composites and Exponents as I suggest, then we shall see something much more defined and quite incredible…. If Ulam's Spiral is important, then the amazing coherence and congruence displayed here must surely be of significant worth?

The grid or lattice is made up of diamonds that consist of 4 by 4 prime number nodes along the sides and 7 on the diagonals, containing a total of 25 numbers in each. 12 numbers are shared with other adjacent sectors and 13 numbers could be considered to be 'contained', one of which is a single Prime or Prime Composite and that I have coloured pink. Each diamond is constructed as either a pair of Golden Triangles or a Golden, or Square, Rhombus.

Each diamond contains a single Prime, that I have coloured pink, which strangely occurs in each diamond at the location of an eye, with the effect that the diamonds look like little fish swimming in a shoal, in an anti-clockwise direction. Clearly delineated are 4 definite quadrants. These can be visualised as the interacting 4 vortices or 2 pairs of poles that can be found in Walter Russell's cosmogony. North - South, and East - West. Next, please notice, on the South West diagonal, an intriguing and mysterious absence of this coherence, so striking elsewhere. Here we find a curious bunch of numbers highlighted in pink also, yet with a demonstrably even and predictable periodic occurrence. If you join these dots you would conceivably have the effect of a snake slithering up or down a pyramid edge. We know that some pyramids are built to exhibit this precise effect at solstices and equinoxes. The South West diagonal is dominated by the dimension expansion phase for numbers, and is announced by the presence of the Prime Exponent Numbers in the form of 5^2 for 25, 7^2 for 49, 11^2 for 121, 13^2 for 169, 17^2 for 289 etc. on up through the powers, for example 5^3 for 625 and on and on.

### Square Pyramid Number Sequence

If we imagine the square of nine number spiral in three dimensions we can see that the structure of Prime numbers is Square Pyramidal like the Great Pyramid at Giza. Text from Image in Italics.

Square Pyramid Number Sequence.

1 5 5 3 1 1 5 6 6

7 2 2 0 7 7 2 3 3

4 8 8 6 4 4 8 0 0

We find a 27 digit sequence for the Mod 9 results of this sequence.

Here again we see the Number Groups emerge in the columns.

Interestingly the 3 Number Groups are balanced with each appearing exactly 3 times.

Where we find a 27 digit sequence in the Mod 9 sequence my feeling is that the sequence tells us something about Light.

Why this may be the case will become clearer in the last part of the book.

In conclusion, it seems absolutely clear to me that a re-classification of Prime and Composite numbers is in order. A Mod 9 analysis of numbers illuminates the perfect congruence that exists in the system of Prime and Composite Numbers when correctly classified.

## Chapter 6: The Numerical Structure

During the early part of this research period I spent some time in Australia with old friends Tim & Liz Farrell, my studies were still in relatively nascent form, and we had been discussing the Harmonic Grid I use to time global asset markets and my Mod 9 analysis, and how I felt that the two were directly connected in that they both expose an order one otherwise cannot see.

In this section I advance what resembles the structure for a type of string theory to explain the structure of temporal quanta which contains and expresses physical reality.

As I remember, one night, early on in my stay, Tim had a dream that burst out of him in the morning and he drew what resembled the picture below to describe what he saw. It had the 9 0 serving as the central axis that the 1 2 4 5 7 8 spiral around, bounded by the magnetic poles of the 3 and 6.

This cemented something in my mind and instant connections were made and new horizons opened up in my understanding. It was a critical moment that led to the following offering which can be applied to the DNA Helix. The 9 is the Central Axis - The 9 Forms.

The 3 and 6 are the Boundaries - The 3 and 6 Contain and Confine.

The 1 4 7 and 2 5 8 Number Groups, are Controlled by Prime Numbers
The Prime Number Strings spiral, from opposite directions, around the Central Axis, the 9, within the confines of the 3 and 6.

In Terms of an Apple. The 0 9 is represented by the core and the attachment to the tree.

The 3 and 6 are the boundaries that determine the size of the apple.

The 1 4 7 and 2 5 8 make up the flesh of the apple as it is formed / coalesced / integrated by spiraling from opposite directions around the core.

So it is with all 'things' in Physical Manifestation. The 3 6 and 9 are the conduit for the 1 4 7 and 2 5 8 in the same way as the pipework in your house is the conduit that causes water to be available to you at different locations within it.

In this case though, the pipes are obviously not carrying water to all points, they are carrying, or providing the potential for, Electricity, to play out its dance.

I provide this metaphor mainly so that you may appreciate that the plumbing has nothing directly to do, or in common, with, the water, per se, other than its ability to provide a conduit for its delivery.

It is sort of the same with Electricity except, as we shall see in the next part, Magnetic Light and Electricity, it turns out, are the reciprocal of each other, which, of course, pipes and water are not.

### Octaves of Perception

During the course of my research I discovered the work of the late, great, Walter Russell who wrote that:-

'The known octaves which lie in the range of perception are five and one half. These begin with the third, or hydrogen octave, and end with the uranium group which are isotopes of Actinium and Tomium in the last octave.

The invisible octaves of finely divided matter of space are three and one half in number.

These octaves are beyond our range of perception, but they are not beyond our knowing.' - A New Concept of the Universe - Walter Russell.

When we examine these numbers we see that:-

Visible Spectrum = 5.5 Octaves or 5.5 x 12 = 66 notes

Invisible Spectrum = 3.5 Octaves or 3.5 x 12 = 42 notes

5.5 + 3.5 = 9 Octaves x 12 = 108 notes

When we look at the relationship between these two numbers we see:-

5.5 / 3.5 = 11 / 7 = 1.571428 R - Electricity

1 4 7 and 2 5 8 Number Groups that control Prime Numbers.

3.5 / 5.5 = 7 / 11 = 0.636363 R - Magnetism

### 3 6 9 Number Group

Electricity is numerically shown to be the Inverse or Reciprocal of Magnetism.

Russell continues - 'The Electric we can perceive through the Electric senses of our Electric bodies while the Magnetic we cannot directly perceive, yet can know.

Electricity is the only force which God makes use of to create this universe and the only two "tools" God makes use of for creating His universe of matter and motion are two pairs of opposed spiral vortices.

One of these opposite pairs meets at apices at wave amplitudes to create spheres of matter and the other opposed pair meets at cone bases upon wave axes to void both matter and motion'.

### 1 4 7 & 2 5 8

The Prime Number, Electric, Number Groups Are Opposed Spiral Vortices.

'These two pairs of opposed electric spiral vortices are the basic units which construct all matter. Together they form the electric waves of motion which create the various pressure conditions which are needed to produce the many seemingly different elements of visible and invisible matter.

Electricity is divided into two equal-and-opposite forces which thrust away from each other to build this polarized universe. When inability to thrust away from each other takes its sequential turn in the pulse of the universal heartbeat, depolarization voids all opposition. Thus this universe consists of cycles of life followed by death - growth followed by decay, and generation followed by radiation - each expressed simultaneously and repeated sequentially forever without end. That which science calls magnetism, and believes to be a force which has the power of lifting tons of steel, is God's still Light which balances and controls the equality of electric division, but electricity alone performs all of the work of this universe.

The magnetic Light which controls the universal balance performs no work whatsoever.

A bar magnet picks up nails because of the electric current which divided that steel into its activated polarized conditions, and not because of its focal poles of stillness which center its two activities even though the electric current has been withdrawn. The steel retains its electric activity for long periods and acts as though the current still remained.

Magnetic Light control might be likened to the rudder of a ship which controls the direction of the ship's motion without in any way motivating that motion. It might again be likened to the fulcrum which extends its power of expression through motion to a lever, without in any way acting to motivate that expressed motion of the lever.

God's still magnetic Light is the fulcrum of this creating universe. Electricity is the two-way lever which extends from that fulcrum to give the universe its pulsing heartbeat of simulated life-death sequences.

Wherever God's Light appears in matter, there stillness centers motion, but there is no motion at that point. The center of gravity in a spherical sun or earth is one locatable point where God's Light is.

Likewise, the two still centers of north and south spiral vortices are other locatable balancing points of control.

Likewise, the shaft which connects all pairs of opposed poles is an extension of stillness from the zero of wave beginnings to the zero of wave amplitudes, and the return of motion to the zero of its beginnings in the stillness of its homing place on its wave axis.'

This is a universe of Light-at-rest from which two opposed lights-of-motion appear to manifest the IDEA which is eternally sealed in the Light-at-rest.

All matter is electric. Electricity conditions all matter under the measured control of the ONE MAGNETIC LIGHT which forever balances the TWO electrically-divided, conditioned lights of matter and space.

All matter is but pressure-conditioned motion.

Varying pressure conditions yield varying states of motion.

Varying states of motion are what science misinterprets as the elements of matter.

Varying pressures in a wave are tonal. In each octave wave there are four pairs of tones, each of which has the same relative position in its octave color spectrum as it has in its octaves of chemical elements.

Waves are, therefore, electric pressure-conditioned octaves of tones.'

### The Divided Light

'In the light of the Creator's Mind is DESIRE to dramatize his ONE IDEA by dividing its one unconditioned, unchanging unity of balance and rest into pairs of oppositely-conditioned units, which must forever interchange with each other to seek balance and rest.' - A New Concept of the Universe - Walter Russell.

### Walter Russell's Octave Wave Theory of the Chemical Elements ## Chapter 7: The Vortex Theory of Atoms

Collaborator Talal Ghannam PhD has written a truly brave and important book - The Mystery of Numbers. He has very kindly allowed me to extract from the book at will and reproduce here, which sets the scene for a greater understanding of the numerical aspect of particle physics.

His whole book is about the Number Groups 1 4 7, 2 5 8 and 3 6 9 as seen from the point of view of a PhD Physicist and as such our work touches on many of the same discoveries and themes albeit using differing nomenclature.

In his book, I was delighted to discover that for millennia a vortex theory of atoms had been generally accepted and that a substance called Ether or Aether permeated the universe through which Light was able to propagate freely, just as waves propagate through water. Many great scientists including Maxwell, Descartes, Einstein, Tesla and Faraday believed in the Ether.

That was, up until 1887, when the Michelson - Morley experiment seemed to disprove this theory, though even Michelson, one of the named architects of this experiment, refused to give up his belief in the ether. Others have also pointed to problems with the experiment but the Vortex theory of atoms was consigned to the dustbin.

I think it needs to be taken out again and dusted down because for me this theory maps perfectly to the number system exposed throughout my work and that of those upon whose shoulders I stand.

Talal writes that ''Tesla believed that the ether is made of carriers (negative and positive) immersed in an insulating fluid and are always in a whirling motion forming kind of 'micro helices'.

The properties of this ether vary corresponding to movement, mass, and the existence of electromagnetic fields. In fact he believed that all the energy that exists in matter is received from the surrounding ether.  Hence the ether is more like an energy repository capable of imparting its energy to all forms of matter, living and non-living. The reason behind the inertia of masses was believed to be due to the interaction between the ether and mass. However, once a mass starts moving within the ether, the charges of the atoms that form this mass will start creating rotary electromagnetic fields creating holes within the ether and forming what is called 'tube of force' that are free from any ether inside them. This will eliminate the existence of the ether inside the mass, and consequently make the movement of mass smoother and less resistive.

JJ Thomson's thoughts about these moving tubes of force can be summarized in the following points:

1. - that the tubes exist everywhere in space, either on closed circuits or terminating on atoms.
2. - that electric force becomes perceivable only when electric tubes have greater tendency to lie in one direction.
3. - that a beam of light is a group of electric tubes moving at the speed of light.
4. 4 - if moving tubes entering a conductor are dissolved in it, mechanical momentum will be given to the conductor. This momentum is stored in a unit volume of the field being proportional to the vector product of the electric and magnetic vectors.

The ether was also used to explain the forces of nature such as:- electricity and magnetism, which are nothing but disturbances through the ether; and gravity, which is nothing but the pressure of the ether flow. The ether was also used to describe the elementary particles and atoms that the universe is made of. This theory was called - The Vortex Theory of Atoms.

First, let us explain what we mean by a vortex. A vortex is a spinning, funnel shaped, 3 dimensional spiral motion of any kind of matter and fluid, just like a tornado, and where energy is directed in a rotational configuration and concentrated at the center of the vortex as shown overleaf.

The basic idea of the vortex theory of atoms is that the subatomic particles, the electron, the proton, the neutron etc., are nothing but spirals, or more precisely, self-sustaining vortices of energy, having toroid-like shapes, similar to a doughnut, spinning in viscosity-free ether medium.

The direction of the spinning determines whether the etheric energy is flowing in or out of the particle, and whether this particle is positive or negative in terms of charge. In fact, mass, charge, magnetism etc. are all nothing but different states of the ether depending on its concentration, rotation, speed, etc.''

This theory was championed by many of the most brilliant scientific minds of the 19th and 20th century, including Lord Kelvin (1824 - 1907), Hermann Von Helmholz (1921 - 1984) James Clark Maxwell (1831 - 1879), Nikola Tesla (1856 - 1943), and John W Keely (1827 - 1898)

Actually, Keely discovered that the proton is made of 3 sub-particles, long before modern physicists had proposed the quark model. He even went further by claiming that each of these sub-particles (the quarks) is made of another set of 3 particles, and so on, all being vortices in the ether. He claimed to have controlled 27 levels of these triple sub-particles.

Nikola Tesla, who is one of, if not the, most brilliant minds in the field of electromagnetism, also believed strongly in Vortex Theory. He envisioned the atom being differentiated from the ether by a spinning motion, just like a whirl of water in a calm lake.

Some scientists postulate that the flow of the ether through atomic particles and atoms is necessary for them to keep their spin and energy. They even argue that the flow of this ether through the stars and planets is necessary for them to keep their internal molten cores alive. Moreover, spirals and vortices permeate everything in nature, from the smallest to the largest. So why not also in the atomic world? ''- The Mystery of Numbers - Talal Ghannam

#### Numbers & Particle Physics

Just as the Numbers and Octaves pair, so too do the elementary particles!

A particle and its pair have exactly the same mass and fall downwards in the Earth's gravitational field. They are effectively mirrors of each other.

'The most important condition when recognising a particle and anti-particle is that when they combine, they should annihilate each other with a burst of complete pure energy, in other words a burst of self-mirrored photons' - Talal Ghannam - The Mystery of Numbers. The Photon is considered its own Anti Particle which I liken to the 0 9 Number Pair relationship so we have a clear rationale for comparing the Numbers and Particles simply if we only knew which numbers to apply to which particle.

Talal proposes the following answer which I think is right. So now we can attribute each number to an atomic particle or anti-particle. This will be important later on.

Talal then goes on to test these numbers against important reactions in the subatomic world, namely the B- Decay, the B+ Decay, and the electron capture 'K capture' reaction.

Due to the laws of conservation of energy, these reactions must tally up and we can see that they do if we use Mod 9.

#### B- Decay

The Neutron spontaneously decays into the three subatomic particles, proton, electron and anti-neutrino.

Neutron decays to Proton + Electron + Anti - Neutrino

1 decays to   2  + 3  + 5 = 10  or  1 Mod 9 - Check

#### B+ Decay

The energy induced decay of the Proton.

Energy + Proton decays to Neutron + Electron + Neutrino

9  +  2 decays to  1  +  6  +  4

11  = 11 - Check

Here we must also check energy with a value of zero:-

0  +  2 decays to  1  + 6  +  4

2 = 11 or 2 Mod 9 - Check

#### 'K capture'

Where, under energy induced conditions, a Proton will absorb an Electron and turn into a Neutron.

Energy + Proton + Electron turns into a Neutron + Neutrino

9  +  2  +  3  turns into 1 + 4

14   =   5 Mod 9  =  5 - Check

Here too we must check when Energy is zero:-

0 +  2  +  3  turns into 1 + 4

5   =  5  Check

'When a Proton meets an Electron they bond to each other , forming either a dipole or a hydrogen atom or even a neutron as in the K-capture, depending on the energy of the bonding' - Talal Ghannam

This in my mind is the spark of creation and is supportive of the idea that the Number 5 is central to creation and the first Prime in physical reality.

Proton - Value 2 + Electron Value 3 = 5

A meeting between a Particle and its Mirror results in a Photon - 1 and 8, 2 and 7, 3 and 6, 4 and 5 all add to 9 making this the number of the Photon, or Light.

#### Number 9 - The Number of Energy

''It is interesting to see how similar the effect Number 9 has on the first eight numbers to the effect energy has on particles, which we explain as follows:

Firstly: when 9 is added to one of the 8 numbers it raises it numerically higher, e.g. 9 + 2 = 11, which is just what energy does to particles, it raises them into higher energetic states.

Secondly: adding 9 to a number will not change the digital root of the number (Talal's way of doing Mod 9), in the same way that energy will not change the nature of the particle; an electron stays an electron, it just gains more energy. (Nevertheless, in the extreme case when energy is too much, it can break some particles apart like in the B+ decay. But this is because these particles are composite to start with, made from so called quarks.)

Thirdly: adding 9 to these numbers will change their parity from even to odd and vice versa, e.g. 9 + 2 = 11 (odd). Amazingly, this is very similar to what happens in atoms, as when an electron absorbs a photon, the electron will want to go up into a higher state. However, the parity of this new state must be opposite to that of the original state; odd if it is even and vice versa, otherwise the transition is considered prohibited with zero probability. It is like saying that when an electron absorbs energy, its parity is reversed. This parity reversal is not restricted to electrons only, but applies to all other particles.

Hence, it seems that the correspondence between our nine numbers and the subatomic world is almost exact''. - The Mystery of Numbers - Talal Ghannam.

## Chapter 8: Canonical Numbers & Mod 99

In light of Talal's work, it is proposed that where the Number Pairs are undivided, so 27 as opposed to 297 for example, they are at their most potent at the atomic level.  When they come together, 'undiluted' so to speak, they produce a photon.

As examples, 3663, 4545, 5544, 3681 and 7227 are all canonical numbers as they are balanced in their number pairings.

As examples, 3143, 7203, 4288, 8093 and 7117 would not be canonical numbers as they are not balanced in their number pairings.

However, where 0 or 9 appear with an otherwise number pair balanced number that too would fall within the classification for a canonical number.

As examples, 639, 396, 270, 279 and 207 would all be considered canonical numbers.

After canonical two digit numbers like 27, numbers which include a 9 or 0 would be the next most pure interaction.

When looking at numbers in general, canonical or not, we can now translate these into particle interactions.

As an example 217 indicates a combination of Proton, Neutron and Anti Proton.

### 2 Digit Numbers & Mod 99

Given that Talal's findings at the particle physics level show the number pairs and how when they interact they produce physical reactions, it played into something that revealed itself to me very slowly as I perused a large wall containing all the various number sequences looking for more patterns that might reveal themselves.

Earlier I showed the quick way to get the Mod 9 result for any given number. What I didn't realise initially, but later discovered through experimentation, was that if we wanted to find the Mod 99 result and therefore finish with a two digit number, we could do something similar by taking the digits from any given number, adding them in pairs, working backwards.

For example:-

1234 would be 34 + 12 = 46 Mod 99 or 12345 would be 45 + 23 + 1 = 69 Mod 99

On discovery of this, initially in the Hexagram, or 6 pointed star, sequence, it showed a repeating pattern as we shall see overleaf.

I then applied Mod 99 to all the important number sequences.

I am going to show that for example, the number 234567 is 'related' fractally to the number 36 in a way I think everyone might think crazy were it not for the irrefutable proof of its congruence which follows.

This is an aspect of the decimal system of numbers that is probably under-appreciated.

### Hexagram Number Sequence

The Hexagram number sequence is a key number sequence for my Integer Partition Theory, which occupies the third and final section of the book, and as such it is one which had been on my wall for a long time.

I applied the Mod 99 technique to this sequence first of all and was instantly rewarded with a geometric proof for what I was looking for. Text from Image in Italics

#### The Hexagram Sequence

On the left we see the Hexagram sequence of numbers.

We see the repeating Mod 9  sequence in the third column showing 1 4 1 every three terms.

Then we have the first iteration of applying the Mod 99 technique and where the total is more than 99 the second iteration shows the resulting number giving the final Mod 99 result.

For example, on the 12th Term of the sequence 793, the first iteration gives a total of 93 + 7 =  100 and then applying the rule again gives us 00 + 1 = 1.

The analysis shows a clear 33 number palindromic sequence that repeats indefinitely.

The 33 number repeating sequence is centred on the 17th term by the number 49.

1 13 37 73 22 82 55 40 37 46 67 1 46 4 73 55

49

55 73 4 46 1 67 46 37 40 55 82 22 73 37 13 1

The number 49 is of particular interest appearing at this central space for this particular geometric sequence. Clearly 49 = 7 x 7 or 7^2 reinforcing this idea that so much about the natural world focuses on the interaction of the numbers 2 and 3 with the number 7. What is even more interesting and intriguing however is that the human eye can sense only one octave of the electro-magnetic spectrum, the 49th.

The Electro-magnetic (EM) Spectrum is the range of all possible electro-magnetic radiation. Electro-magnetic radiation can be divided into octaves, as sound waves are, adding up to a total of 81 octaves. It is interesting to note that there are only 81 stable chemical elements. Food for thought when considering Walter Russell's Octave Wave Theory. Light is Sound at a higher oscillation. ### Platonic Solids - Mod 99 None of the Platonic Solid Number sequences under the Mod 99 microscope exhibit moduli, with the exception of the Cube where we see a 33 Modulus, where the 2 digit sequence repeats or wraps around and begins again.

This is of critical importance and something that sets the Cube apart from its Platonic brothers and sisters.

1 8 27 64 26 18 46 17 36 10 44 45 19 71 9 37 62 90 28 80 54 55 89 63 82 53 81 73 35 72 91 98 99

However, I noticed that if we exclude the final 99 that punctuates the sequences, we see the number pairs, each two digit pair adding to 99 to show a unique type of numerical symmetry as we move towards the centre pairing.

01 08 27 64 26 18 46 17 36 10 44 45 19 71 09 37

98 91 72 35 73 81 53 82 63 89 55 54 80 28 90 62

### Number Groups Analysis - Mod 99

I then applied the 2 Digit Technique to these specific combinations of numbers to see:-

3 6 9(0)

369 + 396 + 963 + 936 + 639 + 693 = 3996 = 135 = 36 Mod 99

72 + 99 + 72 + 45 + 45 + 99 = 432 = 36 Mod 99

360 + 306 + 63 + 36 + 630 + 603 = 1998 = 117 = 18 Mod 99

63 + 9 + 63 + 36 + 36 + 9 = 216 = 18 Mod 99

1 4 7

147 + 174 + 741 + 714 + 471 + 417 = 2664 = 90 Mod 99

48 + 75 + 48 + 21 + 75 + 21 = 288 = 90 Mod 99

2 5 8

258 + 285 + 528 + 582 + 825 + 852 = 3330 = 63 Mod 99

60 + 87 + 33 + 87 + 33 + 60 = 360 = 63 Mod 99

The sums of each of the Number Group combinations in their raw form reduce to 36, 63 and 90 which is really elegant being the magnetic infrastructure which contains the electric aspect.

When Mod 99 is applied to each of the combinations then we see:-

3 6 9 Number Group sums to 432, the frequency for Pythagorean Tuning A of Middle C.

3 6 0 Number Group sums to 216, 1/10th the mean diameter of the Moon and an octave below the 3 6 9 Number Group.

2 5 8 Number Group sums to 360, the degrees contained in any circle or quadrilateral.

1 4 7 Number Group sums to 288, the dimensions of New Jerusalem, a number whose importance will be fully revealed a little later on in the book.

## Chapter 9: Recurring Decimals

During an early investigation into the reciprocals of Prime Numbers I was inspired to produce an analysis of the numbers 0 - 99 and their divisive effect on all other numbers. I strongly suspect that recurring decimals are evidence and indicative of geometry and this analysis was essential in light of what is to come later on in the book. Therefore, when we see a 3 digit recurring decimal, I am thinking that represents a triangle. A 4 digit recurring decimal would represent a square, 5, a pentagon, 6 a Hexagon etc.

Below is the table of the analysis of the first 99 numbers and how many digits their recurring decimals cause. There are some surprising observations as you will see. Text from Image in Italics

27 & 37 are unique root numbers for triangles - 3 digit recurring decimals.

7 & 13 are unique root numbers for Hexagons - 6 digit recurring decimals with two dimensional numerical symmetry.

21 stands out as the only producer of a 6 digit recurring decimal which lacks two dimensional numerical symmetry.

41 uniquely produces a 5 digit recurring decimal - a pentagon.

73 uniquely produces a 8 digit recurring decimal with two dimensional numerical symmetry - an octagon.

81 uniquely produces a 9 digit recurring decimal with three dimensional numerical symmetry - a nonagon.

77 produces is a 6 digit recurring decimal with two dimensional numerical symmetry - a hexagon. Ordinarily we might expect a Line, being a multiple of 11.

49 produces a 42 digit recurring decimal with two dimensional numerical symmetry when all other multiples of 7, except 7 x 7 produce hexagons.

n/101 = 4 digit recurring decimals - Squares.

n/137 = 8 digit recurring decimals - Octagons.

All the numbers that produce 6 digit recurring decimals exhibit the two dimensional numerical symmetry showing the number pairs.

Number pairs always add up to 9 so the total of the two halves placed one above the other will always be 999 for a six digit recurring decimal. For example:-

1/7 = 0.142857 Recurring

1 4 2

8 5 7

9 9 9

All that is, except one.

Numbers divided by the number 21 produce 6 digit recurring decimals which uniquely lack this two dimensional numerical symmetry.

1/ 21 = 0.047619 Recurring

0 4 7

6 1 9

BUT THERE IS NO 2 OR 3 DIMENSIONAL SYMMETRY HERE!

The two dimensional numerical symmetry always totals 999 but here we see:-

0 4 7

6 1 9

6 6 6

666 - A well-known number with devilish connotations!

## End of First Section

Much of what has been set out in this section will, I hope, help to make sense and give a deeper appreciation of the remaining two parts of the book. I apologise if it has been a bit dry, the later sections of the book, I hope, will be much more interesting for those not obsessed with numbers and patterns.

The main purpose of this section has been to show the patient reader how all natural integer and geometric sequences, essential in the universe, operate in pairs and groups when we use modular arithmetic to show this order that cannot otherwise be appreciated from the seeming chaos of numbers.

I have demystified the Prime Number puzzle and shown how musical octaves behave and introduced a numerical structure of all things that exist in physical reality.

In addition, through the work of Talal, I have shown the reader how these Number Pairs can be applied to the world of particle physics and armed with this understanding have realised the importance of Mod 99.