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The Music of Matter

Sunday, August 14th, 2011

The principles of harmonics which are well-known to the musician are the very same principles which govern the laws of physics, though this is very little realized at present. All octaves of energy, whether within the range of human investigation or beyond it, are the same. Each octave varies from the next only in scale and the amount of energy required to sustain the oscillation.

I have developed the chart shown below in Figure 1 to help you appreciate this more deeply. In it you can examining a full range of 100 octaves beginning from 1 cycle per second all the way up through 1,267,650,600,228,229,401,496,703,205,376 Hz (or a million billion billion cycles per second).

I have not tuned my chart to Standard Pitch where A = 440Hz. I have instead used Scientific Pitch (also known as Philosophical Pitch) where Middle C = 256 Hz and where A = 426.|6| Hz. (Note that some add the closed cycle 6 to the open cycle portion and call this, erroneously, A = 432Hz). I have shifted down from Standard pitch because the mathematics underlying Scientific Pitch are perfectly aligned with Foundational Mathematics while Standard Pitch produces many irrational values suggesting a kind of dissonance or lack of alignment with the fractality of the Conscious Field itself.

Figure 1. - 100 Octaves of Light

In high octaves of light, far higher then Gamma Rays, light condenses into the 10 Octaves of Matter. For more about this topic, see my earlier writing:

http://www.alexpetty.com/2011/07/20/the-periodic-table-of-the-light/

Beyond the Octaves of Matter begin the Octaves of Lower Astral Light. This point in the energy scale marks the borderline between human spirituality and western science.

“As above, so below”.

As the vibrations of light oscillate to ever higher rates, light extends further and further into subtler levels of the conscious field, which is to say that it extends into ever higher vibrations of consciousness.

–

In what follows, I will be working from 256 cycles per second as my root tone.

.:.

Posted in Foundational Analysis, General, Number Theory | 5 Comments »

Long Division and Euclid’s Lemma

Friday, May 20th, 2011

Do you remember the “long division” you once learned in grade school? Even though it seems now a rather quaint method to the microchip wielding man of today, long division is actually a very powerful procedure. It allows one to very efficiently determine the internal fractal structure of integers without introducing the cumbersome aspects of floating point arbitrary precision.

What we call Long Division is actually the Division Algorithm developed thousands of years ago in ancient Greece. The Division Algorithm is not really an algorithm as much as it is part of a theorem which assures the ability to divide integers and natural numbers known as Euclid’s Division Lemma.

But first, some basics. I will step you through the Division Algorithm.

Suppose you want to find out how many times a number “goes into” another number. For example, imagine that you have 128,654 grains of rice that you wish to distribute to 12 people. So then the question becomes, how many times can 12 be divided among 128,654.

In this case, we refer to 12 as the divisor, to 128,654 as the dividend and the number of times that 12 can be divided among 128,654 as the quotient. When the divisor can not be distributed equally to the dividend, the left over balance is referred to as the remainder.

Long Division (ie. the Euclid’s Division Algorithm) is a procedure that can be used to determine the quotient and remainder for a given divisor and dividend.

Here is how it works:

Arrange the divisor and dividend as shown below.

We address the dividend starting from its significant-most digit.

In other words, we start from the left side of the dividend and perform the long division procedure working from left to right.

We first ask ourselves whether 12 can be distributed evenly to 1.

When we realize it cannot and that 12 requires at least 12 in order to be distributed, we mark a zero in the quotient and we expand our consideration of the dividend one position to the right (which is 2). So now we are considering 12.

We now ask how many times can 12 be distributed to 12?

The answer is that 12 can be given to 12 exactly one time with nothing remaining. We express this by arranging our mathematical statement as shown below.

We then expand our consideration of the dividend out to the third digit (which is 8), asking how many times can 12 be distributed to 8? Since 8 is smaller then 12, 12 cannot be distributed among 8 and so we mark a zero in the quotient. Also, because we are considering the 8, we “bring it down” into the working area of the mathematical statement as shown below.

Since 12 cannot be distributed among 8, we expand our consideration of the dividend to include the fourth number (which is 6) by bringing it down into the working area of the mathematical statement. We then arrange our mathematical statement as shown below.

We now ask, how many times can 12 be distributed to 86?

The answer is that 12 can be given to 86 seven times with a remaining balance of 2. We express this by arranging our mathematical statement as shown below.

Next we ask how many times can 12 be distributed to 2?

Since 12 cannot be distributed among 2, we expand our consideration of the dividend to include the fifth number (which is 5) by bringing it down into the working area of the mathematical statement. We then arrange our mathematical statement as shown below.

Next we ask how many times can 12 be distributed to 25?

The answer is that 12 can be given to 25 two times with a remaining balance of 1. We express this by arranging our mathematical statement as shown below.

Next we ask how many times can 12 be distributed to 1?

Since 12 cannot be distributed among 1, we expand our consideration of the dividend to include the sixth number (which is 4) by bringing it down into the working area of the mathematical statement. We then arrange our mathematical statement as shown below.

Next we ask how many times can 12 be distributed to 14?

The answer is that 12 can be given to 14 one time with a remaining balance of 2. We express this by arranging our mathematical statement as shown below.

So now we have found out that if we have 128,654 grains of rice to distribute evenly to 12 people, each person would receive 10,721 grains of rice (our quotient) and that there would also be 2 grains of rice left over (our remainder).

If we wanted to be very precise, we would then determine how many portions to split the remaining 2 grains of rice up evenly amongst the 12.

Since we have run out of digits in our integer dividend, we now must add a decimal point and continue the long division procedure as shown below. Since we are free to add as many zeros to the right of the decimal point as we wish without changing the value, we can continue long division indefinitely. Therefore, it is said that we may continue producing numbers in the quotient to any arbitrary degree, a quality known as arbitrary precision.

So then, let’s continue.

Next we ask how many times can 12 be distributed to 20?

The answer is that 12 can be given to 20 one time with a remaining balance of 8. We express this by arranging our mathematical statement as shown below.

Since 12 cannot be given to 8, we add another zero and “drop it down” as shown below.

Next we ask how many times can 12 be distributed to 80?

The answer is that 12 can be given to 80 six times with a remaining balance of 8. We express this by arranging our mathematical statement as shown below.

With the remainder again being 8, we see that the solution will repeat 6 in the quotient forever.

Thus, the quotient for 128,654 / 12 = 10,721.1|6|

The above has been a particular example of long division. Now let’s consider the general procedure of long division.

Suppose we wish to divide an integer a by a positive integer d.

Let a = a_na_n-1…a₁a₀ in decimal representation.

Since a has n+1 digits, the long division will have n+1 steps as each digit of the dividend generates exactly one step and gives exactly one digit of the quotient.

At step i, we do this:
Look for the largest integer q_i so that d × q_i does not exceed B_i, where B_i is defined as follows:

B_n = a_n and

B_i = 10(B_i+1 – d × q_i+1) + a_i, 0 ≤ i < n.

Write q_i to the right of q_i+1in the quotient row.

After we reach i = 0, what remains is r, the remainder.

Why does long division work?

In the long division procedure, the dividend must equal the sum of the remainder and all the numbers that have been subtracted.

But the numbers subracted are d×q_i with place value 10ⁱ. So

a = (d × q_n)10ⁿ +(d × q_n-1)10^n-1 + … + (d × q₁)10¹ + (d × q₀) + r, where 0 ≤ r < d

= d × (q_n10ⁿ + q_n-110^n-1 + … + q₁10¹ + q₀) + r

= d × q_nq_n-1…q₁q₀ + r

= d × q + r, 0 ≤ r < d.

Here is where we need Euclid’s Lemma.

According to it, q and r must be unique. That is, the q we have found in the long division is indeed the one and only value possible, namely the quotient of a when divided by d.

Now that we know why long division works, it is easy to extend to dividends that are not integers.
Suppose a = 758.9 and d =5. Then a/5 = (1/10)(7589/5) so that we carry out the long division involving two integers and then divide the answer by 10 which is accomplished by moving the decimal point left.
Finally, noting that the Division Algorithm is valid in any base, we can extend these arguments to any base just as well as we can for base 10.

Below is an example of Euclid’s Division Algorithm in action.

Euclid's Division Algorithm in Animation

More about Euclid’s Division Lemma

The Division Lemma of Euclid assures us that each result obtained at every step of the Division Algorithm is absolutely the correct unique value for every possible dividend-divisor pair.

Here is how Euclid derived the Lemma:

For example, if you are given 13 and required to divide it among 4

Figure 15. - Euclid's Division Lemma, Step 1

Euclid thought about this in terms of 13 small spheres.

Figure 16. - Euclid's Division Lemma - Step 2

He then imagined the spheres being partitioned into 3 groups of 4 with 1 remaining ungrouped sphere left over.

Figure 17. - Euclid's Division Lemma - Step 3

Next Euclid arranged these spheres into the following mathematical relationships.

Figure 18. - Euclid's Division Lemma - Step 4

Then Euclid considered a wider set of data points.

In the examples included in figure 19 below, let’s refer to:

all of the dividends as ‘a‘

all of the divisors as ‘b‘

all of the quotients as ‘q‘

all of the remainders as ‘r‘

Figure 19. - Euclids Division Lemma - Step 5

Careful analysis begins to reveal that a pattern is emerging. We are getting unique combinations of quotients and remainders for any given pair of dividends and divisors. Furthermore, we see that in every case, remainder r is always greater then or equal to zero, but is always less then divisor b.

Figure 20. - Euclids Division Lemma - Step 6

Thus, Euclid’s Division Lemma may be stated as:

Given integers a, b with b > 0, there exist unique integers q, r
with 0 ≤ r < b such that a = bq + r.

To really be sure that the combinations are unique however, we need proof. So here it is:

Proof of Euclid’s Division Lemma is given as follows.

The proof consists of two parts — first, the proof of the existence of q and r, and second, the proof of the uniqueness of q and r.

Part 1: Existence of q and r

Consider the set:

We claim that S contains at least one non-negative integer. There are two cases to consider.

1) If a is non-negative, then choose n = 0.

2) If a is negative, then choose n = ab.

In both cases, a – nb is non-negative, and thus S always contains at least one non-negative integer. This means we can invoke the Well-Ordering Principle, and deduce that S contains a least non-negative integer r. By definition, r = a – nb for some n. Let q be this n. Then, by rearranging the equation, a = qb + r.

It only remains to show that 0 ≤ r < |b|. The first inequality holds because of the choice of r as a non-negative integer. To show the last (strict) inequality, suppose that r ≥ |b|. Since b ≠ 0, r > 0, and again b > 0 or b < 0.

If b > 0, then r ≥ b implies a-qb ≥ b. This implies that a-qb-b ≥0, further implying that a-(q+1)b ≥ 0. Therefore, a-(q+1)b is in S and, since a-(q+1)b=r-b with b>0 we know a-(q+1)b<r, contradicting the assumption that r was the least non-negative element of S.

If b < 0, then r ≥ -b implies that a-qb ≥ -b. This implies that a-qb+b ≥0, further implying that a-(q-1)b ≥ 0. Therefore, a-(q-1)b is in S and, since a-(q-1)b=r+b with b<0 we know a-(q-1)b<r, contradicting the assumption that r was the least non-negative element of S.

In either case, we have shown that r > 0 was not really the least non-negative integer in S, after all. This is a contradiction, and so we must have r < |b|. This completes the proof of the existence of q and r.

Part 2: Proof of the uniqueness of q and r

Suppose there exists q, q’ , r, r’ with 0 ≤ r, r’ < |b| such that a = bq + r and a = bq’ + r’ . Without loss of generality we may assume that q ≤ q’ .

Subtracting the two equations yields: b(q’ – q) = (r – r’ ).

If b > 0 then r’ ≤ r and r < b ≤ b + r’ , and so (r – r’ ) < b. Similarly, if b < 0 then r ≤ r’ and r’ < -b ≤ -b + r, and so -(r – r’ ) < -b. Combining these yields |r – r’ | < |b|.

The original equation implies that |b| divides |r – r’ |; therefore either |b| ≤ |r – r’ | or |r – r’ | = 0. Because we just established that |r – r’ | < |b|, by trichotomy we may conclude that the first possibility cannot hold. Thus, r = r’ .

Substituting this into the original two equations quickly yields bq = bq’ and, since we assumed b is not 0, it must be the case that q = q’ proving uniqueness.

–

The question then occurs to me. Is it possible to further generalize Euclid’s Division Lemma using Foundational Mathematics? Can we, when armed with the ordering clarity offered by The Foundational Mathematics, develop a method based on it to allow us, like a mathematical spider, to glide effortlessly and steadfast over the seemingly infinite labyrinthine web of numbers?

With this goal in mind, I have produced the following charts.

The first chart shows the unique euclidean lemma value matrix which I have generalized using Foundational Mathematics for ranges:

Dividend (a) = 1 for Divisors (b) = 0 through 9

Dividend (a) = 2 for Divisor (b) = 0 through 9

Dividend (a) = 3 for Divisor (b) = 0 through 9

Dividend (a) = 4 for Divisor (b) = 0 through 9

Dividend (a) = 5 for Divisor (b) = 0 through 9

Dividend (a) = 6 for Divisor (b) = 0 through 9

Dividend (a) = 7 for Divisor (b) = 0 through 9

Dividend (a) = 8 for Divisor (b) = 0 through 9

Dividend (a) = 9 for Divisor (b) = 0 through 9

Figure 21. - Petty Foundational Division Lemma for a equals 1 through 9

To see my work which led to these charts, review the follow document:

http://www.singularics.com/docs/petty-foundational-lemma-sequences_draft.pdf

Note that zero and nine are the Alpha and Omega. The end is the beginning. This is an expression of infinity.

9 is the intention of Mind to either constructively expand (multiply if the expansion in spatial) or to destructively contract (divide is the contraction is spatial).

3 and 6 are the force of Mind’s intention (extending thus from 9) exerted upon the conscious field as a vibratory pressure of light.

1, 2, 4, 8, 7 and 5 are the archetypal vibrations of form, shepherded through space and time, arranging themselves with respect to one another in accordance with the laws of polarized force and sacred geometry.

zero and nine are the alpha and the omega

Note also that every incrementing number that occurs where the spiral intersects with radials always compresses back to the radial’s initial value. For example, 546 is on the 6 radial because 5+4+6=15 and we then compress 15 as 1+5=6.

To develop a working understanding of this and other Foundational Mathematics concepts, please refer to my earlier writings:

http://www.alexpetty.com/2010/01/09/the-true-foundation-of-arithmetic/

Focus particularly on this rectangular table showing radial compression in base 10:

http://www.alexpetty.com/wp-content/uploads/2010/01/fae2016_small.png

You may also wish to review this circular version of the table:

http://www.alexpetty.com/wp-content/uploads/2010/01/fns-circle-chart-round-1×1.png

To see this principle at work in numerous other bases (indeed any arbitrary base), please review my earlier writing:

http://www.alexpetty.com/2010/08/23/the-effect-of-base-on-numeric-fields/

Can this characteristic of numbers help us more deeply understand what it means to divide?

Can we say that for any arbitrarily sized divisor, that “Foundational Radial Compression” assures that we can rely on the values listed in figure 21 for the general case?

.:.

Posted in Computer Science, Foundational Analysis, Number Theory, Prime Numbers | No Comments »

Vortex Math Based Computing

Saturday, September 11th, 2010

Consciousness Modeled Computational Paradigm

Conceived of by Alexander S. Petty

Creation, by its nature, operates the systems which arise within it in the most efficient manner possible. Only Creation is so economical in its every expression. Wisdom compels the thoughtful to observe and learn from nature’s miraculous ingenuity and employ these methods in the development of technology whenever it is possible to do so.

The system that Creation has devised for encoding and transmitting information is known as genetics whose base units of currency are RNA and DNA. I believe that the time has come to employ nature’s own information management system in the field of modern computer science.

It is the goal of this project to bring about this transformation in the state of art for computing machines.

Brief Summary of DNA Structure

DNA is a four bit system using the following nucleic acids as units for storing and conveying data.

These are:

Adenosine (A)

Formula C10H13N5O4

Figure 1. - Adenosine

Thyamine (T)

Formula C5H6N2O2

Figure 2. - Thyamine

Cytosine (C)

Formula C4H5N3O

Figure 3. - Cytosine

Guanine (G)

Formula C5H5N5O

Figure 4. - Guanine

By virtue of their geometry, which is governed by magnetic/numeric polarity, these nucleic acids will always pair such that Adenosine binds with Thyamine and Cytosine with Guanine. These units are referred to as base pairs.

Possibility of a DNA-Modeled Computing Paradigm

The basis of modern computer science is binary in nature, which is to say that it is based upon whether the voltage in a given transistor is off (zero) or on (one). These zeros and ones are rolled up into a base 2 number system which is operated to encode information “digitally”. By convention (e.g. standards like ASCII or UTF8), the letters of the English alphabet, for example, are digitally encoded by 8 bit phrases of zeros and ones called bytes.

DNA is also digital, but it is not limited to positions 0 and 1. DNA uses 4 possible bit positions and is therefore a quaternary digital system:

Figure 5. - Assigned numeric value for nucleic acids in DNA

By convention, a binary digital byte is 8 bits in length, providing 256 possible different values per byte.

A “byte” of DNA is called a codon. A codon is 3 bits long (e.g., AAA or GCT). Each bit can be one of 4 possible values (0, 1, 2 or 3) and so the DNA quaternary encoding system provides 64 possible values per codon.

Binary 8 Bit Byte

Figure 6. - Binary 8-bit byte

Any value between 0 and 255 can be expressed within this binary system.

For example, the number 73 would include 1 bit from 2^6, 1 bit from 2^3 and 1 bit from 2^0.

This binary value looks like:

01001001

and can be expressed in base 10 as:

64 + 8 + 1 = 73

Quaternary 3 Bit Byte (aka. Codon)

Figure 7. - Quat 3 bit byte, or Codon

Any value between 0 and 63 can be expressed within this quaternary system.

For example, the number 53 is represented by a 3 bit in the 4^2 ordinal, a 1 bit in the 4^1 ordinal and a 0 bit in the 4^0 ordinal. This is represented as 310 in base 4. Chemically, DNA would express this with the use of the following nucleic acids:

TGA which would pair with ACT.

Complete listing of codon bit pairs (All sum to 9)

The positive side (left) and negative side (right) sum to neutrality (9). This polar balance provides the underlying framework dictating DNA’s physical form. This is of course an expression of vortex mathematics in biology. It is the magnetic relationship between the base pairs which enables the compression of DNA in biological systems. The very same principles, however expressed by numeric polarity, provides the under-pinning for a DNA-modeled compression algorithm.

Figure 8. - Base pairs are magnetically neutral

Keyboard Encoding Standard

ASCII standard encoding with DNA translation

The long time standard for keyboard encoding is known as ASCII. Since ASCII is a base-2 8-bit standard, it supports 256 unique values. Just as a 16-bit binary standard requires 2 bytes to cover a range of 512 characters, a DNA-based keyboard encoding standard, with its limit of 64 values per codon, requires multiple codons to support the ASCII standard. This can be implemented as follows:

Codon 0

Figure 9. - ASCII CODON 0

Codon 1

Figure 10. - ASCII CODON 1

New Foundational Standards

It would be ideal to deprecate many of the legacy ASCII character codes such as ENQ, RS, EM, etc in a new DNA based standard.

In developing a new Foundational Encoding Standard, UTF8, UTF16, etc. should be considered. It may take as many as 8 codons to capture all of the needed keyboard codes for world-wide requirements.

Simple Encoding Example

Figure 11. - Textual "Genetics"

Only one half of the codon pair needs to be stored since the other half can be inferred and applied by an algorithm at compression runtime.

Figure 12. - Genetic modeled text encoding

The ability to persist only half the bit pair provides a fundamental efficiency quotient of 3 to 8 (62.5%) over the current binary paradigm before compression is even considered.

Numeric Polarity based Helical Compression Algorithm

Using these methods, data can be compressed in a manner identical to that of DNA. It is possible to utilize the principles of numeric polarity to compress information into a kind of “chromosomal superstructure” just as with physical DNA.

Furthermore, it appears that DNA modeled compression offers performance curves allowing large amounts of information to be more tightly packed then smaller amounts of information. This agrees with the observation that fruit fly DNA is physically much larger then human DNA and yet it contains many orders of magnitude LESS data.

DNA Runtime Environment

32 bit PC architecture allows the CPU to address four 8-bit bytes of data per clock cycle. Since a quaternary system is not compatible with today’s hardware, a translation layer is required which may be implemented as a runtime environment.

Figure 13. - Mapping Bytes to Codons in a VM

Bit position 9 is reserved for technical overhead in support the virtual run-time environment. The 9th bit can be reclaimed when implemented on quaternary based hardware.

I should also note that, while I am proposing a system of bits based on 0, 1, 2, 3 for the sake of feasibility of implementation. The system nature itself uses appears to rely on the following pairing system :

1 pairs with 8

2 pairs with 7

and 4 pairs with 5

Nature reserves 9, 3 and 6 for higher purposes.

9 is connected with the causal intention of Mind that underlies all observable force driven events.

3 and 6 are connected are expression of polarity aspect of force which we observe everywhere in our subject and object human experience.

CMYK Data Transmission

The subtractive CMYK four-state color description system appears to operate in a manner similar to that of how nature uses light within physical RNA/DNA. It stands to reason that the “technical access” to use of 1-8, 2-7 and 4-5 pairs may be achievable by using a light based system. It may be possible to input and output quaternary digital information to DNA-modeled machines using light harmonics in the form of a subtractive 4 color standard.

C = Cyan

M = Magenta

Y = Yellow

K = key (Black)

Figure 14. - CMYK

It is interesting to note that chromosomes (tightly packed DNA superstructures) naturally reflect these very colors (thus their name). See image of chromosome below.

Figure 15. - Chromosome exhibits CMYK subtractive color dynamics

Posted in Foundational Technology, Number Theory | 5 Comments »

Mapping the Terrains of Consciousness with Foundational Mathematics

Sunday, June 6th, 2010

I wanted to take a moment to thank everyone for the steady stream of kind and gracious feedback and the general show of support for my work. Please know how very thankful I am for that.

I’ve been away from the blog in recent months quite busily working on advancing my energy research and the physical implications of Foundational Mathematics. I am pleased to report that I am making steady and surprising progress in developing this area and have accomplished some very exciting results, yet, much work remains. Each answer I achieve begs many new questions!

My writing efforts in the last few months have been mostly directed into the completion of my first book, “The Nature of Numbers –Mapping the Terrains of the Conscious Field”. The book is almost completed now and I am very pleased with how it has unfolded both as a process and as a work.

.:.

Today I’d like to give you a taste of the main subject matter covered in my upcoming book which is at its heart an exploration of the archetypal terrains of the conscious field. I believe the method I have developed amounts to an important new way to visualize the underpinnings of consciousness. It is a potentially revolutionary channel for enabling an intuitive working understanding of how subject and object (particles) relate to one another on the illusory stage of light (or state of mind) into which humanity is born.

In the fractal glyph plates I have drawn below, I have used red lines to denote a positive or clockwise path and gray lines to denote a negative or counterclockwise path. Where positive and negative flows overlap, a color mixture of red and gray (a sort of light peach-brown) results. Darker brown colors correspond to a greater number of overlapping fractal paths. For prime sequences I have used yellow for positive and green for negative. In the same way, overlapping of prime fractal paths results in a kind of olive color. A darker olive color correspond to a greater number of overlapping prime fractal paths.

.:.

The Nature of Numbers

In my theory of Foundational Mathematics, numbers can be sorted into 3 classifications:

Structural Numbers (Primordial Numbers)

Structural numbers are few. These fields are the basic building blocks used to create the universe within the conscious field.

Polar Numbers (Closed Cycle Numbers)

Polar Numbers shepherd the world of form by imparting force. These include negative, positive and neutral polar number fields. These fields are in the world but not of it.

Form Numbers (Open Cycle Numbers)

Form Numbers obey the effects of polar forces and comprise the visible elements of the Creation.

I will explain these classes of numbers more in depth in what follows.

Structural Number Fields

1, 2 and 5 are archetypal. They are fully Foundational to consciousnesses in its plural aspect. When Unity, through an act of Mind falls into a plural state, the fields of 1, 2 and 5 appear automatically. Note that Unity and 1 are not the same. Unity is an expression of the absolute oneness of the universal whole, while 1 is an expression of “subject” in a “subject and object” world.

Zero

The zero field has no physical component. It represents the unmanifested Mind. It represents infinite potential existing apart from the physical Creation. The singularity is where the physical world and the zero field “touch”.

Unity

The field of one or the unity field represents the Universe in stillness. It is all that is without any distinction imposed upon it by the activity of Mind. Within the unified field there are no vibratory perturbations. Here, no effects have been caused.

foundational field of 1, unity

unity

Plurality

The field of 2 is Structural. This field is the archetypal expression of plurality.

The elective action of Mind to fall from unity into plurality spawns the basis of dualistic human experience. In this act, subject and object, the space between, the time to traverse and the power of force is born. The image below shows how the vector fields from two point particles form interference patterns on the stage of consciousness that is the magnetic dipole form itself. It is in this very manner that magnetic fields are born. It is when the mind elects to produces plurality that the stage of consciousness gives rise to the self evident experience of distance and force.

dipole interference patterns arising spontaneously from the intersection of offset point field. the cause of force in space and time is a manifestation of consciousness

See my earlier writing: http://www.alexpetty.com/2009/10/04/relative-motion-of-vectors-produces-fields/

When duality arises, consciousness sets up some basic “plumbing” to facilitate this state: 1, 2 and 5.

Whenever the field perturbation represented by 2 appears in the conscious field, the field perturbations represented by 5 automatically also appear. You must understand that numbers represent very specific energetic perturbations signatures and are not merely abstractions as is commonly supposed.

It should also be noted that in my theory of Foundational Mathematics, further vocabulary is needed to distinguish separate concepts of one. The “high concept” of Unity and the “lower concept” of a one as apart from two, ie. the first element of Unity divided.

foundational field table of 2

foundational field glyph of 2

So then, 2 gives rise to 1 and 5 and then spontaneously sets up the following ratios.

In the case of expansion into a higher enumeration, this relationship is constructed:

phi, the operator of enumerative expansion

In the case of contraction to a lower enumeration, this relationship is constructed:

phi, the operator of enumerative contraction

This expression provides the underpinnings for how the conscious field will use 1, 2 and 5 to enable any level of enumeration. The resulting magnetic interactions produce force and form which gives rise to the the universe we experience as human beings.

The conscious field possesses a kind of elasticity. Phi is the mechanism that consciousness uses to control the expansion and contraction of the conscious field into myriad parts. It allows the conscious field to experience itself in any number of divisions the Mind can conceive of. In this way, numbers themselves represent the elastic nature of the conscious field, that aspect which responds automatically to the constructive or destructive will of Mind.

The field of 5 is structural as it is an essential element for allowing the conscious field to experience itself dualisitically. Perhaps counter-intuitively, 5 appears in the field before 4 since it is a field perturbation that arises automatically whenever the primordial instance 2 occurs.

foundational field table of 5

foundational field glyph of 5

Polar Number Fields

Polarity is established automatically the moment that Unity falls into Plurality. There are three classes of Numeric Polarity: Positive, Negative and Neutral.

The chart below shows how polarity cycles around the circle of nine. Note that as the very first iteration of nine is cycled through (the Primordial Cycle), consciousness is bootstrapping the building blocks it uses to manifest The Physical Creation.

numeric polarity

For more on numeric polarity see my earlier entry: http://www.alexpetty.com/2009/11/15/on-numeric-magnetism-and-the-fundamentals-of-primality/

Postive Polarity

The field of three is the first example of what I have called “closed cycle fields”. Any field which is closed cycle is an element of polarity. This is as opposed to elements of physical expansion and contraction (ie. elements of form) which are all “open cycle fields”.

Polarity guides the direction and magnitude of the expansion and contraction of the world of form and it does so in accordance with the elective will mind, communicated by the field perturbations which the mind produces as a feature of its natural function. Consciousness itself is the long elusive underlying basis of electricity and magnetism.

foundational field table of 3

foundational field glyph of 3

The closed cycle field of 3 produce two circular paths |3| and |6| yielding clockwise flow 3-6-9 (or 3-6-0) and a counter-clockwise field 6-3-9 (or 6-3-0). The deflections of these flows form two overlapping, offset and opposed equilateral triangles. Much ancient symbology comes from this archetype including the Star of David.

Three is the positive pole and related directly to the concept of the proton.

The field of 12 is the second occurrence of a “layered cycle field” which means it simultaneously contains both closed and open cycles field elements (6 is the first such occurrence).

12 is the positive pole on the second iteration of the circle of 9. Twelve is also the field which marks the half way point for the the magnetic sequencing of the Fibonacci series. To see this sequencing refer to plates from my earlier writings:

http://www.alexpetty.com/wp-content/uploads/2009/11/fibonacci-in-mod9-with-polarity-mapping.png

http://www.alexpetty.com/wp-content/uploads/2009/12/fibonacci-converging-on-phi5.png

This topic is explored in depth in my coming book.

foundational field table of 12

foundational field glyph of 12

The field of 21 is a closed cycle field. The 21 field is the positive pole on the third iteration of the circle of 9.

foundational field table of 21

foundational field glyph of 21

Negative Polarity

The field of six is another very important “closed cycle field”, but it is also the first occurrence of a “layered cycle field” as it simultaneously contains both closed and open cycles. This means that within the field of six is both polarity and form all at once. Thus we see that it is the negative pole around which physical form arises. This is consistent with our experience. The electron is understood to be the carrier of negative charge flow. Since almost the whole of physical human experience is tared around net flows of negative charge, it is for this reason that physical interactions in our world are observed to center around electrons seeking equilibrium states. (In normal experience, we rarely observe positronic charge – so called cold current – except through exotic circuits which manifest non-linear effects.)

foundational field table of 6

foundational field glyph of 6

The closed cycle portion of the field of 6 produces two circular paths of |6| and |3| yielding counter clockwise flow 6-3-9 (or 6-3-0) and clockwise field of 3-6-9 (or 3-6-0). Just as with the field of 3, the deflections of these flows form two overlapping, offset and opposed equilateral triangles however the circular flows in 6 are opposite of 3.

Six is the negative pole and its nature gives rise to the concept of the electron. Human experience is very much about the perception of net flows of charge and the electron is understood to be the carrier of charge. It is for this reason that we interact with negative charge flows in the course of normal experience and rarely with positronic charge or cold current (which is related to 3).

It is also worth noting that within the field of 6, we see first hints at an entirely new method of computation. My book spends a lot of time on this topic.

The field of 15 is a mixed cycle field. The 15 field is the negative pole on the second iteration of the circle of 9.

foundational field table of 15

foundational field glyph of 15

The field of 24 is a mixed cycle field. The 24 field is the negative pole on the third iteration of the circle of 9.

24 is also the field which completes the magnetic sequencing for the Fibonacci series.

Again, you can review this sequencing in the plates included in my earlier writings:

http://www.alexpetty.com/wp-content/uploads/2009/11/fibonacci-in-mod9-with-polarity-mapping.png

http://www.alexpetty.com/wp-content/uploads/2009/12/fibonacci-converging-on-phi5.png

foundational field table of 24

foundational field glyph of 24

Neutral Polarity (The Number 9)

The field of 9 is the third polarity, it is the neutral pole which has no physical aspect in creation – and yet it gives rise to the elements which make up nature. Nine is both 0 and 9. It is the Alpha and the Omega. It is the beginning and end of the circle whose 9 divisions resonant with Nature.

The field of nine is composed of eight closed cycle fields.

foundational field table of 9

Interestingly, the harmonic overtone series for sound (an expression of energy traversing the medium of air as a compression wave) is definitely connected to this.

See: http://www.alexpetty.com/wp-content/uploads/2009/11/harmonics-of-each-harmonic-in-mod9.png

foundational field glyph of 9

Take a moment to review the beautiful degree of balance and symmetry which is contained by the field of nine, the neutral pole.

The field of 18 is a mixed cycle field. It is the neutral pole on the second iteration of the circle of nine.

foundational field table of 18

foundational field glyph of 18

Operators of Neutral Polarity (Primes)

The field of 7 indicates magnetic neutrality, 7 is therefore by the definition I have set forth in Foundational Mathematics, the first true prime field.

foundational field table of 7

foundational field glyph of 7

I here offer The Foundational Definition of Prime Numbers as follows:

A number is prime only when all elements of the number’s field possesses a closed cycle internal structure and when each of these closed cycles sum to zero (using clock math) for any arbitrary base.

All primes exhibit neutral numeric polarity!

I have discovered that primality is the result of magnetic neutrality in fields of numeric polarity.

The observation that primes are divisible only by themselves and one is purely an effect of their numeric polarity, since primes are internally balanced, they are magnetically inert and therefore unable to combine geometrically with other numeric fields. This is the actual basis for the “atomic” nature of primes. Furthermore, I hypothesize that the chemically inert nature of the Noble Gases is a directly related to this quality of prime numbers.

Primes fields possess an incredible and very beautiful degree of palindromic symmetry and cyclic resonances. It is my view and the view of Foundational Mathematics that all primes are made up of fractal series which possess the characteristic of closed cycle rotation, palindromic symmetry and net numeric neutrality.

It is my further view that the fields of 2 and 5 can not truly be considered prime as they are neither magnetically neutral nor do they possess a closed cycle nature. Instead I regard 2 and 5 as Structural numbers. I also do not regard 3 as prime since its closed cycle has a net positive polarity rather then a net neutral polarity.

All primes are closed cycle which means that despite their magnetic neutrality, they are indeed instruments of polarity, a kind of neutral “Divine” Polarity. The chart below shows the manner in which this neutral polarity is distributed and how the conscious field extends its tendrils of neutral polarity throughout the channels of expansion and contraction.

petty foundational chart of the primes

The field of 11 is the second magnetically neutral field and therefore it is considered in Foundational Mathematics as the second prime number. Again, in Foundational Mathematics, we consider that 2 and 5 are not Prime but Structural and that 3 is also not Primes but a Structural Archetype of Polarity.

foundational field table of 11

foundational field glyph of 11

Observe that the field of 11 it is composed of five separate sequences, each which sum to nine (again, a quality of the primes alone).

The first 5 sequences flow in a positive direction while the other 5 flow in a negative direction.

09, 18, 27, 36 and 45 are palindromic with the latter 54, 63, 72, 81 and 90.

These sequences are perfectly opposite flows. In my energy physics experiments, I have observed that numeric flow reversal relates directly to charge spin reversal in capacitors.

The field of 13 is the third magnetically neutral integer and therefore it is the third prime number.

foundational field table 13

foundational field glyph of 13

Observe that it is composed of two separate sequences, again, each sum to nine (a quality of the primes alone).

The first rotating closed sequence is 076923.

The second rotating closed sequence is 153846.

The field of 17 is the fourth magnetically neutral integer and therefore it is the fourth prime number.

foundational field table of 17

foundational field glyph of 17

This prime contains one rotating closed sequence of 0588235294117647 which again, sums to magnetic neutrality.

The field of 19 is the fifth magnetically neutral integer and therefore it is the fifth prime number.

foundational field table of 19

foundational field glyph of 19

The field of 23 is the sixth magnetically neutral integer and therefore it is the sixth prime number.

foundational field table of 23

foundational field glyph of 23

I have studied thousands of prime fields and without exception, they all possess neutral pole closed cycle rotation, (ie. all fractal elements of primes sum to nine.)

In this, I have arrived at a deeper, more causal definition for primes and the precise reason *why* these numbers, in a diamagnetic sense, can not be geometrically reduced by other numbers.

In my book, I have rigorously addressed the topic of primes as understood from the higher perspective of Foundational Mathematics.

Here are a few more Foundational Prime Tables for good measure. Watch in particular how the magnetic orientation in the fields become ever more elaborate and beautiful.

The perfect balance is always struck in the following manner:

0 (Neutral) pairs with 9 (Neutral)

1 (Neutral -) pairs with 8 (Neutral +) creating Form Neutrality around the neutral pole at 9

3 (Pole +) pairs with the 6 (Pole -)

7 (-) with 2 (+)

5 (-) with 4 (+)

Form Positivity is the result of 2 and 4 proximity to 3.

Form Negativity is the result of 5 and 7 proximity to 6.

As you consider this, you may find it helpful to study this plate:

http://www.alexpetty.com/wp-content/uploads/2010/04/numeric-polarity.png

foundational field table of 29

foundational field table of 31

foundational field table of 37

foundational field table of 43

foundational field table of 47

foundational field table of 51

The World of Form

The field of 4 is an open cycle field. Open cycle fields always represent the world of form in growth or contraction shepherded by the closed cycle forces of polarity.

foundational field table of 4

foundational field glyph of 4

Form always progresses in a doubling or halving pattern. Consider an embryo developing in the womb (the initiation of a consciousness being) follows this pattern during mitotic division:

1 cell doubles to 2

2 cells doubles to 4

4 cells doubles to 8

8 cells doubles to 16 (which compresses to 7 in mod9)

16 cells doubles to 32 (which compresses to 5 in mod9)

32 cells doubles to 64 (which compresses to 1 in mod9, completing the cycle around the circle of 9)

doubling and halving on the walls of the vortex

Within every process, from embryonic cellular division to complex fluid dynamics, the progression of all manifestations of form always obey this archetypal pattern.

In the second iteration of 9, one observes that:

64 double to 128 (mod9: 1 doubles to 2)

128 doubles to 256 ( mod9: 2 doubles to 4 )

256 doubles to 512 ( mod9: 4 doubles to 8 )

and so on.

doubling and halving on the walls of the vortex

The doubling and halving is governed by the closed cycle forces of positive, negative and neutral polarity ever guiding form in accordance with the will of conciousness which govern the shape and form of the Physical Creation.

The field of 8 is an open cycle field.

foundational field table of 8

foundational field table of 10

The field of 10 is an open cycle field.

foundational field table of 10

foundational field glyph of 10

The field of 14 is mixed cycle field:

foundational field table of 14

foundational field glyph of 14

The field of 16 is an open cycle field:

foundational field table of 16

foundational field glyph of 16

The field of 20 is an open cycle field:

foundational field table of 20

foundational field glyph of 20

The field of 22 is a mixed cycle field:

foundational field table of 22

foundational field glyph of 22

If you find that you are interested in my work, please keep an eye out for the release of my book this summer.

“The Nature of Numbers, Mapping the Terrains of the Conscious Field” in which I explore this subject in its depths.

Posted in Foundational Analysis, Foundational Technology, General, Number Theory, Prime Numbers | 8 Comments »

The Golden Ratio

Sunday, January 10th, 2010

The Golden Ratio, also known as Phi.

What is Phi?

Phi is a ratio of lengths which possesses many remarkable characteristics.

To me, perhaps the most telling unique feature of Phi is that if you add 1 to its value the result is Phi squared. Equally strange, if you subtract one from Phi the result is 1/Phi (ie. the reciprocal value of Phi).

Said another way, add 1 to this value and you produce an additional dimension. Subtract one from this value and you arrive at something very transcendental indeed which I will discuss later on. I do believe that this number, Phi, is a structural essential within the “operation” of the conscious field.

Want to see Phi? How can one construct it? It’s easy!

1) Draw a pentagon

2) Connect two of its corners as shown below

simple construction of phi

Phi is the ratio of lengths AB:BC

The diagram below shows some other key geometric constructs based upon the “Golden Ratio”.

the golden triangle

Some points of note:

1) The angles a, b and c all reduce to 9 on the circle of 9.

2) All three sides of the triangle formed by points A, B and C are divided by the various intersections of the circle and pentagon into the Phi ratio and other directly related lengths.

Below is another diagram showing some interesting Phi based geometric relationships.

phi based geometry

So, how may one compute the value of Phi with arbitrary precision?

This is also fairly easy to do provided you remember your basic Algebra. Start by taking the ratio of lengths given by AB and AC as shown below.

how to derive the value of phi using basic algebra

My own theory of mathematics (The Foundational Mathematics) is that the conscious field, of which all realms of existence are an inseparable part, in reality only makes provision for values between 0 and 1.

The distance between 0 and 1 may be divided and made vastly manifold to any arbitrary extent that the consciousness field of mind can conceive of. What we perceive as integers are just a play of the 5 human senses.

Think of it this way.

You look at your hand and you see 5 fingers. From the quantum mechanical perspective however, there is no differentiation between your body and the surrounding space. There are not five fingers there at all, only an undifferentiated field of energetic vibration. From this level one can see that the idea of fingers is a construct of the mind only – as is the business of counting them. The senses of our body provide a sort of lens through which we experience the conscious field. It is the case that the conscious field connects everything that is together. This is because everything in existence emanates as a manifestation of the One Conscious field. The lens of the physical body creates an apparently solid physical experience from this field but the dualistic reality of subject and object, however convincing it may be, is merely illusion.

Here is another way to see what I am trying to express. The integer 5 is really a universe of 1 being considered in 5 parts.

So then for counting integers one through five:

0 , 1 , 2 , 3 , 4 , 5

In reality we are counting fractional areas of one:

0.0

0.2

0.4

0.6

0.8

1.0

As another example let look at counting integers one through seven:

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

again, here we are counting fractional areas of one:

0.000000

0.142857

0.285714

0.428571

0.571428

0.714285

0.857142

1.000000

When one develops an understanding and working knowledge of this basis, then suddenly mysteries such as the nature of primes become much easier to comprehend and work with. This discovery enables for the first time in more then two millennium ground for traction in significant forward progress within the field of number theory. It will of course be no small challenge to convince the majority of scientists in the world of the unreality of integers and the illusory nature of sensual experience. Still, this important paradigm shift is the path down which our technological evolution lies.

So how does this relate to understanding Phi?

In Foundational Mathematics numbers do not represent ratios of lengths but rather they correspond directly to scale in terms of growth and contraction. The number 1 enjoys the distinction of representing the unified whole. The Uni – Verse, Creation’s One - Song. The number One is all that consciousness has manifested into apparent being while zero represents the dimensionless infinite potential of the unmanifested Mind.

Generally when people refer to the number 2, what they think they are referring to is the idea of 2 portions of 1. But in reality the act of counting as it is understood today, while certainly useful in ordinary experience, would see us manifest an entire second universe with the stroke of 2, and a third universe with the stroke of 3 and so on.

What is actually meant by counting two, is the consideration of a single universe of mind as two parts. When one counts to three, perhaps without knowing it they have just created 1 of 3 parts in the conscious field, NOT 3 portions of 1.

3 means 3 parts of 1 concious field

I know that many of you who read this will think that such an assertion is beyond absurd. I hear you saying, “Of course I can count to 3, it’s one of the first things I learned how to do!”. But I am telling you that from the perspective of Foundational Mathematics, you can not. The universe simply won’t cooperate with you on this forever. It will present you with such mischief as infinities beyond the range of comprehension and oddities such as Prime Numbers which defy all explanation and myriad other problems akin to chasing shadows. Nature will not tolerate people trying to count into being whole new Universes!

Perhaps Phi’s inverse value should be named after the Greek numeral 6, “Stigma” or “Ϛ” since it is itself an important value for the Foundational Mathematician. Stigma, Ϛ = ~ 0.618033

Phi is about controlling the growth of conscious fields into ever more divisions while Stigma is about controlling the contraction.

In the algebra Phi tells us more about itself.

the geometry of squaring phi by adding one

To get another perspective on this situation, we can re-arrange the equation a bit and think about it from a slightly different vantage.

the curious geometry of phi

I will elaborate more fully in a subsequent post.

Phi and the Fibonacci Series

In the late 1100′s, Fibonacci noticed something peculiar about the following series of numbers (now called the Fibonacci Series).

1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 , 144 , 233 … and so on

In case you don’t see it, the sequence builds upon itself by always adding the most recent result with the previous. For example, the next value in the sequence above would be found by adding 233 with 144 which is 377.

Fibonacci took a close look at the ratios produced by this series as given below:

1/1 = 1.000000

2/1 = 2.000000

3/2 = 1.500000

5/3 = 1.666666

8/5 = 1.600000

13/8 = 1.625000

21/13 = 1.615385

34/21 = 1.619048

55/34 = 1.617647

89/55 = 1.618026

144/89 = 1.617978

233/144 = 1.618056

and so on.

As is plain to see from the above listing, what Fibonacci found is that the higher you are in the series, the more the value of the ratio converges with Phi.

I have developed a chart showing this. I have color keyed this chart in accordance with my numeric polarity theory showing how the interlaced positive and negative magnetic polarity cycles in the Fibonacci series.

the fibonacci series converging on phi

For those interested, below is the error term of the convergence taken to 243 significant figures. Again, my Numeric Polarity is indicated.

chart showing diminishing error term as the fibonacci series approaches phi

The table below expresses the Fibonacci series values as points on the circle of 12 within the incrementation they appear. 12 is relevant here because the Fibonacci sequence itself naturally cycles from zero to zero through 12 completing the full cycle of polarity in 2Pi (ie. 24)

the fibonacci points on increments of 12 (granular view)

It is quite astounding to see that when you increase your “altitude” from this chart, you can see that the numbers have organized themselves into a very regular double helical pattern.

the fibonacci points on increments of 12 (high level view) showing the Fibonacci/Petty Double Helix

More to come!

Posted in Foundational Analysis, Foundational Technology, General, Number Theory, Prime Numbers | 9 Comments »

The True Foundation of Arithmetic

Saturday, January 9th, 2010

Breath.

Here is the process:

1) The in-flow of fresh air from outside space into the lungs. This is Yin flow.

2) With lungs fully expanded, the body is satiated and nourished with essential life force. When the breathing action reaches its in-flow boundary (ie. its upper limit or boundary interaction with positive polarity), the flow of breathe passes transiently through zero resulting in a reversal of directional flow. (ie. boundary crossing)

3) The out-flow of spent air back to space. This is Yang flow.

4) With lungs depleted, the body hungers desperately for more air. A physical being is at great peril of bodily death after only a few minutes of being deprived of air. When the breathing action reaches its out-flow boundary (ie. its lower limit or boundary interaction with negative polarity), the flow of breathe passes again transiently through zero resulting in another reversal of directional flow. (ie. boundary crossing).

5) The Yin flow begins again with a new breathe. The cycle of giving Yin to the Yang and Yang to the Yin by way of the polarity boundaries repeats itself so long as a being lives.

Through the course of my analysis, I have discovered that the principle of breathe is as much a reality for mathematical abstraction as it is for the evident physiology of respiration in biological systems.

The chart below is matrix style listing of how numbers 1 through 2016 are position on the radial of a circle divided into 9 parts.

Also indicated in the chart below is the numerological compression of each value listed. When ever a base10 sum changes by an order of magnitude, the color is alternated between blue and gold.

On each radial, the number represented by the radial oscillates in a palindromic manner. For example, in the 500 range of the One radial (values 505 through 595) you see the following patterns:

** yin palindrome **
505
514
523
–
532 (32 is the mirror of 23)
541 (41 is the mirror of 14)
550 (50 is the mirror of 05)

** yang palindrome **
559
568
–
577
–
586 (86 is the mirror of 68)
595 (95 is the mirror of 59)

These patterns are consistent everywhere in the system. The pattern that governs this behavior completes a full cycle after 223 incrementations around the circle of 9.

petty foundational table of arithmetic

When this matrix is plotted along a circular system, the result is the chart shown below.

Petty Chart for Foundational Arithmetic (1X)

The numbers set themselves up as interlaced spirals.

The Yin spiral expands the system. The system grows as large as it can in as it moves through 2 Pi along the spiral path.

When it reaches 2 Pi (at 9), the system begins a kind of contraction, not in size but in movement back towards the initial polarity. It seems the system that governs creation, which is reflected by mathematics, never looses size once gained.

This is where I was reminded of the principles of breath.

breath of the spiral

Below I have provided various magnified views of this chart.

2X Magnification

Petty Chart for Foundational Arithmetic (2X)

4X Magnification

Petty Chart for Foundational Arithmetic (4X)

6X Magnification

Petty Chart for Foundational Arithmetic (6X)

8X Magnification

Petty Chart for Foundational Arithmetic (8X)

16X Magnification

Petty Chart for Foundational Arithmetic (16X)

32X Magnification

Petty Chart for Foundational Arithmetic (32X)

Here is a 1X view with the Vortex Glyph superimposed.

Petty Chart for Foundational Arithmetic with Vortex Glyph

The prime numbers fall on this chart as follows:

Petty Chart for Foundational Arithmetic indicating Primes

See earlier post http://www.alexpetty.com/2009/11/15/on-numeric-magnetism-and-the-fundamentals-of-primality/ to better understand the color coding being applied to the primes in the chart above.

Here is a closer look

magnified view of the primes on petty foundational table of arithmetic

More to come!

Posted in Foundational Analysis, Foundational Technology, Number Theory, Prime Numbers, Singularity | 4 Comments »

The Foundational Tables of Multiplication

Sunday, January 3rd, 2010

I wanted to share with you some of the foundational basics of arithmetic I have uncovered through my research. Incredibly, it seems that numbers have been widely misunderstood throughout time. I have found evidence that certain ancient people knew of the true nature of numbers, but by and large the information has been kept shrouded in secrecy or otherwise concealed and/or forgotten.

The time has come to clarify this information and its implications, as the lack of this clear understanding has for too long contributed to the hindrance of humanities continued development as a truly advanced technological civilization.

One of the first “truths” children are taught about mathematics is that all numbers exist on a construct called the number line.

the number line

The idea of a number line easily supports the apparent reality that adding and subtracting numbers is an exercise in traversing distances to the right and left along the line. If you want to involve higher dimensions (multiplication, exponents, division and square roots) then you turn 90 degrees for each dimension, create some more distance between points and begin measuring areas and volumes and the unit integers contained by these spaces.

All of this geometry works out just fine when dealing with life within the normal range of our ordinary human experience, however there are layers of important meaning lost by not taking into account the underlying information which the numbers themselves indicate to us about how they are actually positioned in relation to one another.

The reality is that numbers position themselves on a circle divided into nine parts, and so we live in a MOD9 universe. To be even more precise, numbers are positioned on the circle along the lines of a spiral whose growth is governed by the Golden Ratio, Phi. This spiral gives form to the vortex which has shown itself to be so fundamental in the physical creation. It is at this very place that the “abstract” counting numbers can first be seen as what they really are – the fundamental granularity of the “physical” Creation.

numeric spiral

Let’s take a careful look at what it means to multiply numbers together.

Here are the multiplication tables which have been typically taught to students.

tables of multiplication

Although the above table is a clear expression of the geometric growth of numbers in two dimensions, it fails to offer a true representation of how the numbers relate to each other in the two dimensional geometry. When one accounts for the reality of the circle of nine, one sees that the tables of multiplication are in reality as given below.

foundational tables of multiplication

We can derive universal structure from the table above.

table of the universal structures underlying geometric growth

Each multiplication table for the foundational numbers 1 through 9 posses a specific geometric signature. The one’s table chart is given below in the diagram below.

the one's table foundational chart

The two’s table chart is given below in the diagram below.

the two's table foundational chart

The three’s table chart is given below in the diagram below.

the three's table foundational chart

The path of the Three Chart traces the enclosed equilateral triangle 3 times.

The four’s table chart is given below in the diagram below.

the four's table foundational chart

The five’s table chart is given below in the diagram below. Note that between 4 and 5 lies the path through the singularity (the zero point). Once crossed the direction of the flow is reversed.

the five's table foundational chart

The six’s table chart is given below in the diagram below. Here again, the triangle is lapped three times.

the six's table foundational chart

The seven’s table chart is given below in the diagram below.

the seven's table foundational chart

The eight’s table chart is given below in the diagram below.

the eight's table foundational chart

The path of nine has no component and is therefore always zero. It is a perfect unperturbed circle.

the nine's table foundational chart

Again, the shapes formed in the charts above are derived directly from the foundational multiplication tables

color coded table showing geometric growth structures

The sequences that arise from the foundational multiplication tables are perfectly summarized by the vortex glyph

vortex glyph

Simply by organizing the numbers 1 through 9 and 0, we can see the doubling pattern, it gives rise to a polarity driven system which provides the framework for the expansion and contraction of systems within Creation.

I think it was said best by Fibonacci himself in 1202 AD:

“These are the nine figures of the Indians: 9 8 7 6 5 4 3 2 1. With these nine figures, and with this sign 0 which in Arabic is called zephirum, any number can be written, as will be demonstrated.”

This statement, although seemingly straight forward, is actually quite significant and layered with meaning.

For those interested, I would like to announce that I will be releasing my first book on the subject of Foundational Mathematics.

The Nature of Numbers
Mapping the Terrains of Conciousness

by Alexander S. Petty

(to be released in Summer of 2010 – target release date: 08/15/2010)

Posted in Foundational Analysis, Foundational Technology, General, Number Theory | 8 Comments »

Energy Harmonics and the Balanced Square of Nine

Monday, November 30th, 2009

What is energy?

When a thermodynamic gradient is manifest such as a gravity field, an emf or temperature gradient, we say that there exists a well of energy potential.

When this well of potential is tapped and the energy is allowed to be transferred through space and time, we call this power.

Energy moving through the medium of air produces air pressure waves. Our ears sense these air pressure waves and translate them into the experience of sound in the space of our minds.

Energy moving through the medium of a conductive material lattice, such as through a copper conductor, produces charge pressure waves that we use to engineer electronic devices.

Energy moving through the medium of empty space produces pressure waves of light. Our eyes can perceive a certain range of these light waves and can translate them into the experience of human sight. Still many other frequencies of light exist which are beyond our ability to sense directly.

One very convenient way to intuit energy is through harmonics in sound. As a musician, I have spent my life working intimately with musical harmony and therefore find the transference of energy through air a very convenient avenue for intuiting this research.

Below is the harmonic series and then again each harmonic series for every overtone.

harmonics series showing harmonic series for each overtone

Below is the harmonic series table above reduced to mod9 values.

harmonics series showing the harmonic series for each overtone in mod9

This week I have encountered a very interesting book which I recommend reading called the Magic Square of Three Crystals by Arto Juhani Heino. http://artojheino.yolasite.com/resources/MagSqrThreeCrysBook1d.pdf. It’s a very good text which summarizes many of the areas of research I have been pursuing.

balanced square of nine

Posted in General, Harmonics, Number Theory, Prime Numbers | 1 Comment »

Geometric Derivations of the Transcendentals

Friday, November 27th, 2009

Over the last few months I have been analyzing the fractal lengths of the sacred geometry in the Vortex Glyph in a search for as many of the number theory transcendentals as I can find . I have discovered that if you are precise to 2 significant digits, you can find the transcendentals in the glyph, however when you take your measurements out to 3 or more significant figures you will discover a lack of precision in the result. So far, I can not find the source of the error. I believe there are two possible explanations:

1) The Vortex Glyph does not relate to the transcendentals of Number Theory

2) There is error associated with 2D planar analysis that can only be accounted for in higher dimensions.

deriving the transcendentals

Analysis in two dimensional analysis yields a very close approximation of “e” (Euler’s Number to within 0.51% error) and and “Phi” (the Golden Mean to within 0.0006%).

Posted in Foundational Analysis, Number Theory, Prime Numbers | No Comments »

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