136.1 Trinary Relativity Conclusions 

The Prime Number Cross courtesy of Peter Plichta (GIF)

Repeating Plichta::

"The Prime Number Cross offers the possibility of illustrating the six basic units of the universe" --

e, i , pi, 1, 0, and -1.

pi -- the number used in calculating circles.
i -- imaginary number which determines the cross structure of the zero shell.
e -- provides the order for the successive numbers. P.Plichta

We use a similar formulae:

Ref. 1. 0.22 The 7 / 5 Ratio

PHI = 7 / 5 PI / e.

PHI - the ratio that controls the formation of life.
PI - the ratio that controls the formation of circles.
e - the ratio that controls the formation of logarithmic spirals.

Ref. 2. 0.27 Number 9 24 cycles.
Ref. 3. 0.29 Hyperdimensional Physics 7/5 ratio and Music.

__________________________________

Referring To Plichta:

Notice the linear numbering of Plichta, above.

Below, I change the linear flow slightly.

Quoting Plichta again, "The method of determining the "prime number twin" series 1, 5, 7, 11, 13, 17, 19, and 23, etc., is to select the numbers to each side of 6 and 6's multiples, by the following rule and method: "Prime numbers or prime number twins will always occur around a number divisible by 6" -- [5], 6, [7], and [11], 12, [13], and [17], 18, [19], etc. The product of 0 * 6 = 0 must be found at six to the left of the number 6. The number 0 must also be surrounded by a number twin -- [-1], 0, [1]. The sequence of the first four number twins is therefore (-1;1) ---- (5;7) ---- (11;13) ---- (17;19)..."

"The Prime Number Cross offers the possibility of illustrating the six basic units of the universe" --

e, i , pi, 1, 0, and -1.
pi -- the number used in calculating circles.
i -- imaginary number which determines the cross structure of the zero shell.
e -- provides the order for the successive numbers. P.Plichta

Numbering the circles, Plichta uses 0-24 then simply jumps to the next increment on the next larger circle numbered 25, etc.

Locating primes, Plichta uses 6 and multiples of 6 to find all prime numbers. Arriving at 12 (6x2) we find 11 and 13 prime to each side. Arriving at 18 (6x3) we find 17 and 19 prime to each side. Very simple.

In the Anu Toroid, the 4 (quadratics of 4 toroidal positions) x 6 (directions of measurement) = 24 dimensions, at all levels of universe, and represents the same 24 numbers that represents a complete cycle in Peter Plichta's Prime Number Cross, wherein 8 prime numbers: 1, 5, 7, 11, 13, 17, 19, and 23 represent 8 rays going out from the first cycle. 24 sets of magnets is also used by Ed Leedskalnin's Perpetual Motion Holder and is also the integral number of faces and edges attributed to the dodecahedron/icosahedron.

Beginning at 0, I completed the first circle at 24, which was directly above the beginning 0, I noticed that 25 must be one increment to the right of 24. Continuing the numbering, I arrived at 48 directly over 24, then 72 and 96 directly over 24.

This was following Plichta's directions exactly.

Beginning at 1 and ending at 25, offered a slight twist. We know how Plichta arranged his numbers because of the graph, above (0-24). However, slightly diverging from instructions (1-25), if Plichta was indicating that 25 should be written over 0, arriving at 49 instead of 48, directly over 25, not 24. Then overwriting 73 over 49 and arriving at 97 directly over 73, etc., as if both beginning and ending, overwritten number were the same position, then assume the overwritten number is a jump to the next larger circle, it could be telling us that there is a wratcheting effect which is the same wratcheting found elsewhere, especially in the "wratcheted spirals" of certain crop circles. In this case, the spiral is not a smooth curve but has steps, or a jump, at certain points.

Is using primes, the correct way to construct a wratcheting spiral? I think so. I've seen no other method. Since this wratcheting spiral of 24 was noted to have a certain "Prime Number Cross" positions, could the wratcheting jumps be the same as the Prime Number positions? And could these jumps of Prime Numbers also represent jumps in a spiraling Table of Elements, or more likely, the timing mechanism for pulses of the spiraling, 24 phase rotating field, seeing that Plichta said there seems to be a musical, 8 Prime Number "rays" for the first 24 phase cycle, pulsing every 1 of 3 phases or 1/3 of a whole cycle.

This displacement each time around is the hallmark of a traveling wave, or as it is usually called, a rotating field. This is what we need. A rotating field can accelerate to velocities that becomes the gravifugal escape velocity of the rotating field of a levitating craft. Or perhaps a rotating field could have been converted from a rotating magnetic field, to a rotating electric field, by way of a black box tuner over large coral blocks, so that the electric field above the blocks could flip the electric field of coral blocks which aligns every magnetic atom.

The next step is to ask, "Can a control be fashioned upon the rotating field by phasing between "beginning with 0 to beginning with 1"?

One more 45 degree twist is necessary. The Prime Number Cross is upright. In previous examinations of many other systems, our email group found that all Pascal triangle energy paths, all ambidextrous music paths plus the arrangements of the I-Ching and Mayan Tzolkin indicate that the paths of energy is always at the 45 degree angle, across the board. Should the Prime Cross also be tipped to the side so the "Cross" becomes two 45 degree angles in relation to the vertical?

Diverting to another site:

Below, this diagram of Primes is called the...

The Prime Numbers in Table Format (GIF).

It is also called the...

Abarim Pillar

...in which all prime numbers larger than 3 occur along two perfectly straight lines:

 
 2  3  .  5  .  7
 .  .  .  11  .  13
 .  .  .  17  .  19
 .  .  .  23  .  *
 .  .  .  29  .  31
 .  .  .  *  .  37
 .  .  .  41  .  43
 .  .  .  47  .  49
 .  .  .  53  .  *
 .  .  .  59  .  61
 .  .  .  *  .  67
 .  .  .  71  .  73
 .  .  .  etc  .  etc
 As you can see, in the Amazing Abarim Pillar, all primes after 3 are neatly sorted in two columns. There's never one out. All primes mankind is ever going to find are in one of these two columns: 6a+1 or 6a-1 (a being any number).

How about that?

But why are some numbers in the two pillars crossed out and turned into a little star? It's when a so-called iso-factor crosses them out. Do yourself a favor. Print this page and then write in the Amazing Abarim Pillar all numbers that are divisible by 5. You will find that they too are neatly lined up, except that the line is not vertical but diagonal.

Imagine the Amazing Abarim Pillar to be rolled up to a tube so that the number sequence becomes a spiral swirling down that tube. You will find that all iso-factors (lines that connect all numbers divisible by some other number) are spirals too, and swirl along the same tube.



A rose without thorns

Let's do another clever trick. Let's pretend that the two prime columns of the Amazing Abarim Pilar are not interrupted by iso-factor victims, but perfectly solid with prime numbers. We can do that because for as long as the number sequence runs, its possible prime numbers occur in the same rhythmic cadence, namely "number, number, number, possible-prime number, possible-prime". Like so ("P" is a possible prime):

1 2 3 * P * P * * * P * P * * * P * P * * * P * P * * * P * P

It's the basic, underlying pattern of the number sequence. Now watch what happens if we coil this basic pattern of the number sequence up like Ulam did:

(Note: Now we find the 45 degree pattern in Prime-Number-matrices, found elsewhere in the toroidal-Tzolkin, the toroidal-I-Ching, the toroidal-ambidexterous-music-scale, the toroidal-Pascal-triangle-matrice-patterns, counter-windings upon a torus, all other underlying matrices of space, the above Plichta-Prime-Number-Cross, including the basic building blocks of the smallest measure, the toroidal-Anu.)

 .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .
 P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P
 .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .
 .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .
 .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  P  .
 P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  .  .  P
 .  .  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .
 P  .  P  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  P  .  P  .  P
 .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  .  .  P  .  .  .
 .  .  P  .  .  .  .  .  P  P  .  .  .  P  .  P  .  .  .  .  P  .  P  .  .
 .  P  .  P  .  P  .  P  .  .  .  .  P  .  P  .  .  P  .  P  .  P  .  P  .
 P  .  .  .  P  .  P  .  .  P  .  P  .  .  .  .  .  .  P  .  .  .  .  .  P
 .  .  .  .  .  P  .  .  .  .  P  .  1  2  P  .  P  P  .  P  .  .  .  P  .
 P  .  P  .  P  .  P  .  P  .  .  P  .  3  .  P  .  .  P  .  P  .  P  .  P
 .  P  .  P  .  .  .  P  .  .  P  .  .  .  P  .  P  .  .  .  .  P  .  .  .
 .  .  P  .  .  .  .  .  P  P  .  P  .  .  .  P  .  .  .  .  P  .  P  .  .
 .  P  .  P  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  P  .  P  .
 P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  .  .  P
 .  .  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .
 P  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  P
 .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .
 .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .
 .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .
 P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P
 .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .  P  .  .  .  P  .

Now realize that although the prime number density thins out gradually, they still appear at more or less regular intervals, et viola: you have yourself a magical hoax.

The number sequence cycles. The most basic cycle is obviously the odd-even-odd-even cycle, which lasts 2 numbers and repeats ad infinitum. All prime numbers (except 2) occur in the odd bunch.

The next cycle is the one we've established above, which lasts 6 numbers, and repeats add infinitum. All prime numbers after 3 occur in the fourth or sixth column.

The next cycle is 2x3x5=30 long repeats for ever. Next comes 2x3x5x7, after that 2x3x5x7x11, then 2x3x5x7x11x13, and so on until the cows come home and recite Shakespeare.

Numbers: a timeless, spaceless and awesome stage for fine fiction.

Most of us like to believe that the number sequence begins with 1 and then produces 2 and then goes on onto infinity, and also that the number sequence is a natural thing, which was created along with everything else in the Beginning when God created the heavens and the earth. Well, no. The number sequence is a human invention and came to be at once and complete when Euclid wrote down its few driving axioms.

Just remember that the number sequence does not depend on time and when its axioms where established, the whole sequence burst forth in all its splendor and utter predictability:

There are many ways to represent the number sequence, but we will never be able to convey it fully in any way. The number sequence is infinite and works only when infinity is observed. There are some beautiful and stunning infinite series that converge upon some finite value. Mind boggling and fascinating, these series illustrate that infinity comes in different sizes, and this glorious phenomenon defies any definition of it. Infinity lives in the number sequence like a mechanical soul, utterly incomprehensible and always fleeting the limitations posed by numbers, and many have noted that math is rather an art form than a science. Maybe it's both and math too gives us another reason to be awed about the things we can not fathom.

This is a copy to insure its web presence. This section describes how 2D prime tables generate spirals. These diagrams of prime numbers spiral outward from a center. Prime spirals form automatically from the primes. The spirals fade in and out of visibility according to the scale that the observer is using, but each set is always existing. The first spirals form as 2 arms. Moving out to larger scale, we find 10 spirals forming on each of the previous 2 spiral arms. From the two groups of 10, we find spirals increasing in this progression: 2, 20, 44, 333, 710, 103993 etc.

A strange discovery about primes numbers

Since this is a copy, you can click on any image to view an enlarged version, only on the original page

I present in this page the result of personal research. I've not been able to see any counterpart in the math literature. I present this research, on graphical form, with several images. The first five images do not lead to any new or unknown discovery, but the last ones...are very astonishing.

By the way, Renaud Lifchitz discovered the same thing. See his page

1) The facts

Recently, I wanted to plot all numbers "in spiral", that is for every integer n, I plot a point in polar coordinates (r, theta) with 

r = n, and

theta = n mod 2pi = n - 2pi [n / 2pi]
where the brackets means "take the integer part".

So for instance the 1st numbers are : 

n   r   theta (radians)
1   1   1
2   2   2
3   3   3
4   4   4
5   5   5
6   6   6
7   7   0.716815
8   8   1.716815

For my amusement, I marked the prime numbers on the plot. This gave this plot :

Integers up to 50(See the original to view an enlarged version)

Fig 1. Numbers up to 50 are in red, primes are in blue. Circle of radius 10 is in green.

The spirals formation is not astonishing : after all, each time you add one to n, you add one to the radius r and one radian to theta (about 57.29 degrees). So when you add 6 to n, you add 6 radians to theta, which is 2pi - 0.283 radians or 16.23 degrees, so a small angle ;  hence the six righthand spirals in which numbers go from six to six.

What is more curious, but can be easily explained, is the fact that prime numbers belong all to TWO of these six spirals, the one starting with 5 (5,11,17...) and the one starting with 1 (1,7,13,19...). This is because all primes are of the form 6k+1 or 6k+5, just because

This is elementary. But let us continue a bit, plotting integers from 1 to 200.

Primes up to 200(See the original to view an enlarged version)

Fig 2. Primes up to 200 are big red dots. Other integers are blue points. circle of radius 50 is in green.

We continue to see our righthand spirals, and we just remark that there are holes in the two prime spirals. Again, this is elementary : not all 6k+1 and 6k+5 are primes !

So let us plot integers from 1 to 1000:

numbers up to 1000, with primes in red(See the original to view an enlarged version)

Fig 3. Primes up to 200 are big red dots. Other integers are blue points

Here, the surprising fact is that we see 44 new, lefthand spirals, in which primes belong to 20 spirals only ! Where are our 6 previous righthand spirals ? They are still there, but our eye tries to see the most 'prominent' figure, which is the 44 spirals.

This number of 44 is one of the "convergent" of the reduced fraction for 2pi :

You may know from your college years that pi can be approximated with 22/7, so 2pi ~ 44/7.  This is the best 3 digits approximation.

You obtain these numbers like this : 

2pi = 6.283185307179586... = 6 + 0.283185307179586...  best one-digit approx
  2pi ~ 6
1/0.283185307... = 3.531256652... = 3 + 0.531256652... 
  2pi ~ 6 + 1/3 = 19/3
1/0.531256652... = 1.882329368... = 1 + 0.882329368...
  2pi ~ 6 + 1/(3 + 1/1) = 25/4
1/0.882329368... = 1.133363612... = 1 + 0.133363612...
  2pi ~ 6 + 1/(3 + 1/(1+1/1)) = 44/7  : best 3 digit approx
1/0.133363612... = 7.498297203... = 7 + 0.498297203...
  2pi ~ 6 + 1/(3 + 1/(1+1/(1+1/7))) = 333/53
1/0.498297203... = 2.006834462... = 2 + 0.006834462...
  2pi ~ 6 + 1/(3 + 1/(1+1/(1+1/(7+1/2)))) = 710/113  : best 6 digits approx
1/0.006834462... = 146.3172953... = 146 + 0.31729537598442...
  2pi ~ 6 + 1/(3 + 1/(1+1/(1+1/(7+1/(2+1/146))))) = 103993/16551
and so on.

Again, it's an easy task to see that on those 44 new spirals, in each of which integers are of the form n0+44k, with n0 ranging from 1 to 44, only 20 can hold primes (for instance 44k+1 can be prime, but 44k+2 is always divisible by 2, 44k+4 is always divisible by 4 and so forth).

If we look carefully (at) the above table, we might ask why (do we) not see 19, 25, 333, 710 spirals ? The answer is : these spirals are here, but again, our eye sees only the most "visible" thing : Look again at the plot 1-200. You will see 25 spirals about n~200 : but these are vanishing quickly (in our eyes and mainly, because the approximation 44/7 is so much more precise).

So let us plot integers between 1 and 5000 :

Primes up to 5000(See the original to view an enlarged version)

Fig 4. Numbers up to 5000, with primes in red.

Our 44 spirals, 20 of which holds primes, are still here.

Why just 20 ? See above.

We now expect to see 333 or 710 new spirals..

Primes from 1 to 20000(See the original to view an enlarged version)

Fig 5. Primes from 1 to 20,000

Still nothing...

Primes from 1 to 100,000(See the original to view an enlarged version)

Fig 6. Primes from 1 to 100,000

Here we are, some alignments are emerging.

... but we have to go up to 600,000 to see them.

(On the below image, do not trust "holes" in the pattern : See the original to click on the image to see it at the normal resolution).

Primes up to 600,000(See the original to view an enlarged version)

fig 7. Primes up to 600,000

We have 280 lines (below you will see these are spirals) of primes, but they belong to a total of 710 that you would have seen if I had plotted all integers and not only primes (but unfortunately you would not be able to see anything because the screen would be covered with too many points).

Let's go a little further, up to 3 millions :

Primes up to 3,000,000(See the original to view an enlarged version)

Fig 8. Primes up to 3 million.

Nice figure, isn't it ? On each spiral number have the form n0 + 710k 

I cannot resist to give you a plot of the primes numbers up to 100,000,000. To avoid filling the image with points, I plotted only 3% of them, chosen at random (I used a special random generator designed to avoid bias, which resets its seed at random time, depending on the current cpu time and load). The red part of the image is the previous one (primes less than 3 million).

It took two days to perform the computation of this image with a Digital (now Compaq) alphastation 500 Mhz workstation ...

Primes up to 100,000,000(See the original to view an enlarged version)

Fig 9. Primes up to 100,000,000

You see that the expected 103993 new spirals are still invisible... I can understand that !

The theory

The fact, why integers (and primes among them), are so well distributed on spirals, is well explained by the theory of continued fractions.

What is interesting is the way the distributions of prime numbers creates new spirals.

Look at fig 2 again :

The very essence of that is there : holes in the distribution of primes in spirals are needed to accommodate for existence of "higher levels" spirals.

More precisely, we know from the continued fraction of 2pi that we will have six, then 19, 25, and 44 spirals of integers. Some of these cannot be "occupied" by primes (see above). Namely the spirals that hold number 6k, 2+6k, 3+6k, 4 + 6k. This explains why primes are found only on two spirals on that six possible : 1+6k and 5+6k

But at next stage we know we will have 19 spirals : primes could be on any of them, because 19 is prime, so this is not of great value.

So we look at (the) next stage : 25 spirals : here primes can be found on any but 0+25m and 5+25m. This means, for instance, taking the spiral 1+6k, we will have a "hole" for every number n such as n = 1+6k = 25m or 5+25m. This can never hold, because 6 and 25 are primes together.

So go (to the) next stage : 44 spirals. Here only 20 can be "occupied" by primes : 44k+p, where p is one of 1,3,5,7,9,13,15,17,19,21,23,25,27,29,31,35,37,39,41,43 (these numbers are relative primes with 44).

Impossible Correspondence Index

© Copyright. Robert Grace. 2004