Addition Zones (1-18)
Addition is the form of an expression set equal to zero as the additive identity which is common practice in several areas of mathematics.
This page is referring to child section which source is inherited by the prime spin with partitions listed below. Please return always to the Home Page to track the hierarchy.
- True Prime Pairs
- Primes Platform
- Δ(19 vs 18) Scenario
- Power of 168 vs 618
- The Scheme 13:9
- Prime Recycling ζ(s)
- The Implementation
By the Euler's identity this addition should form as one (1) unit of an object originated by the 18s structure. For further on let's call this unit as the base unit.
Prime Hexagon
Below is the list of primes spin along with their position, the polarity of the number, and the prime hexagon's overall rotation within 1000 numbers.
The Prime Hexagon is a mathematical structure developed by mathematician Tad Gallion. A Prime Hexagon is formed when integers are sequentially added to a field of tessellating equilateral triangles, where the path of the integers is changed whenever a prime number is encountered (GitHub: kaustubhcs/prime-hexagon).
5, 2, 1, 0
7, 3, 1, 0
11, 4, 1, 0
13, 5, 1, 0
17, 0, 1, 1
19, 1, 1, 1
23, 2, 1, 1
29, 2, -1, 1
31, 1, -1, 1
37, 1, 1, 1
41, 2, 1, 1
43, 3, 1, 1
47, 4, 1, 1
53, 4, -1, 1
59, 4, 1, 1
61, 5, 1, 1
67, 5, -1, 1
71, 4, -1, 1
73, 3, -1, 1
79, 3, 1, 1
83, 4, 1, 1
89, 4, -1, 1
97, 3, -1, 1
101, 2, -1, 1
103, 1, -1, 1
107, 0, -1, 1
109, 5, -1, 0
113, 4, -1, 0
127, 3, -1, 0
131, 2, -1, 0
137, 2, 1, 0
139, 3, 1, 0
149, 4, 1, 0
151, 5, 1, 0
157, 5, -1, 0
163, 5, 1, 0
167, 0, 1, 1
173, 0, -1, 1
179, 0, 1, 1
181, 1, 1, 1
191, 2, 1, 1
193, 3, 1, 1
197, 4, 1, 1
199, 5, 1, 1
211, 5, -1, 1
223, 5, 1, 1
227, 0, 1, 2
229, 1, 1, 2
233, 2, 1, 2
239, 2, -1, 2
241, 1, -1, 2
251, 0, -1, 2
257, 0, 1, 2
263, 0, -1, 2
269, 0, 1, 2
271, 1, 1, 2
277, 1, -1, 2
281, 0, -1, 2
283, 5, -1, 1
293, 4, -1, 1
307, 3, -1, 1
311, 2, -1, 1
313, 1, -1, 1
317, 0, -1, 1
331, 5, -1, 0
337, 5, 1, 0
347, 0, 1, 1
349, 1, 1, 1
353, 2, 1, 1
359, 2, -1, 1
367, 1, -1, 1
373, 1, 1, 1
379, 1, -1, 1
383, 0, -1, 1
389, 0, 1, 1
397, 1, 1, 1
401, 2, 1, 1
409, 3, 1, 1
419, 4, 1, 1
421, 5, 1, 1
431, 0, 1, 2
433, 1, 1, 2
439, 1, -1, 2
443, 0, -1, 2
449, 0, 1, 2
457, 1, 1, 2
461, 2, 1, 2
463, 3, 1, 2
467, 4, 1, 2
479, 4, -1, 2
487, 3, -1, 2
491, 2, -1, 2
499, 1, -1, 2
503, 0, -1, 2
509, 0, 1, 2
521, 0, -1, 2
523, 5, -1, 1
541, 5, 1, 1
547, 5, -1, 1
557, 4, -1, 1
563, 4, 1, 1
569, 4, -1, 1
571, 3, -1, 1
577, 3, 1, 1
587, 4, 1, 1
593, 4, -1, 1
599, 4, 1, 1
601, 5, 1, 1
607, 5, -1, 1
613, 5, 1, 1
617, 0, 1, 2
619, 1, 1, 2
631, 1, -1, 2
641, 0, -1, 2
643, 5, -1, 1
647, 4, -1, 1
653, 4, 1, 1
659, 4, -1, 1
661, 3, -1, 1
673, 3, 1, 1
677, 4, 1, 1
683, 4, -1, 1
691, 3, -1, 1
701, 2, -1, 1
709, 1, -1, 1
719, 0, -1, 1
727, 5, -1, 0
733, 5, 1, 0
739, 5, -1, 0
743, 4, -1, 0
751, 3, -1, 0
757, 3, 1, 0
761, 4, 1, 0
769, 5, 1, 0
773, 0, 1, 1
787, 1, 1, 1
797, 2, 1, 1
809, 2, -1, 1
811, 1, -1, 1
821, 0, -1, 1
823, 5, -1, 0
827, 4, -1, 0
829, 3, -1, 0
839, 2, -1, 0
853, 1, -1, 0
857, 0, -1, 0
859, 5, -1, -1
863, 4, -1, -1
877, 3, -1, -1
881, 2, -1, -1
883, 1, -1, -1
887, 0, -1, -1
907, 5, -1, -2
911, 4, -1, -2
919, 3, -1, -2
929, 2, -1, -2
937, 1, -1, -2
941, 0, -1, -2
947, 0, 1, -2
953, 0, -1, -2
967, 5, -1, -3
971, 4, -1, -3
977, 4, 1, -3
983, 4, -1, -3
991, 3, -1, -3
997, 3, 1, -3
Including the 1st (2) and 2nd prime (3) all together will have a total of 168 primes. The number of 168 it self is in between 39th (167) and 40th prime (173).
Each time a prime number is encountered, the spin or ‘wall preference’ is switched. So, from the first cell, exit from 2’s left side. This sets the spin to left and the next cell is 3, a prime, so switches to right. 4 is not prime and continues right. 5 is prime, so switch to left and so on. (HexSpin)
As the number line winds about toward infinity, bending around prime numbers, it never exits the 24 cells. and it is the fact that 168 divided by 24 is exactly seven (7).
The most obvious interesting feature of this prime hexagon is it confines all numbers of primes spin!
So there should be a tight connection between the prime within 1000 with the 24-cell hexagon. We found later that it is also correlated with 1000 primes.
Structure: True Prime Pairs
You may notice that there are twists and turns until 19 abuts 2 therefore this addition zone takes only the seven (7) primes out of the 18's structure of True Prime Pairs.
In linear algebra, there is vector is known as eigenvector, a nonzero vector that changes at most by a scalar factor when linear transformation is applied to it. The corresponding eigenvalues is the factor by which the eigenvector is scaled.
$True Prime Pairs:
(5,7), (11,13), (17,19)
layer | node | sub | i | f
------+------+-----+----------
| | | 1 | --------------------------
| | 1 +-----+ |
| 1 | | 2 | (5) |
| |-----+-----+ |
| | | 3 | |
1 +------+ 2 +-----+---- |
| | | 4 | |
| +-----+-----+ |
| 2 | | 5 | (7) |
| | 3 +-----+ |
| | | 6 | 11s
------+------+-----+-----+------ } (36) |
| | | 7 | |
| | 4 +-----+ |
| 3 | | 8 | (11) |
| +-----+-----+ |
| | | 9 | |
2 +------| 5 +-----+----- |
| | | 10 | |
| |-----+-----+ |
| 4 | | 11 | (13) ---------------------
| | 6 +-----+
| | | 12 |---------------------------
------+------+-----+-----+------------ |
| | | 13 | |
| | 7 +-----+ |
| 5 | | 14 | (17) |
| |-----+-----+ |
| | | 15 | 7s
3 +------+ 8 +-----+----- } (36) |
| | | 16 | |
| |-----+-----+ |
| 6 | | 17 | (19) |
| | 9 +-----+ |
| | | 18 | --------------------------
------|------|-----+-----+------
Since the transformation of the plane shall have symmetrical vector then this 7s will take the posistion on the last side of 18s structure.
A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them (Wikipedia).
The tessellating field of equilateral triangles fills with numbers, with spin orientation flipping with each prime number encountered, creating 3 minor hexagons.
A Prime Hexagon is formed when integers are sequentially added to a field of tessellating equilateral triangles, where the path of the integers is changed whenever a prime number is encountered. Since prime numbers are never multiples of two or three, all numbers from “2” to infinity are confined within a 24-cell hexagon (GitHub: kaustubhcs/prime-hexagon).
So there would be the empty spaces for 18 - 7 = 11 numbers. By our project these spaces will be filled by all of the eleven (11) members of identition zones.
A D0-brane is a single point, a D1-brane is a line (sometimes called a “D-string”), a D2-brane is a plane, and a D25-brane fills the highest-dimensional space considered in bosonic string theory. (Wikipedia)
The above seven (7) primes will act then as extended branes. This is what we mean by addition zones and it happens whenever a cycle is restarted.
Structure: Minor Hexagons
When we continue the spin within the discussed prime hexagon with the higher numbers there are the six (6) internal hexagons within the Prime Hexagon.
Cell types are interesting, but they simply reflect a modulo 6 view of numbers. More interesting are the six internal hexagons within the Prime Hexagon. Like the Prime Hexagon, they are newly discovered. The minor hexagons form solely from the order, and type, of primes along the number line (HexSpin).
So the fisrt thing that need to be investigated is why the prime spinned by module six (6). What is the special thing about this number six (6) in primes behaviour?
Similarly, I have a six colored dice in the form of the hexagon. If I take a known, logical sequence of numbers, say 10, 100, 1000, 10000, and look at their spins in the hexagon, the resulting colors associated with each number should appear random – unless the sequence I’m investigating is linked to the nature of the prime numbers.
Moreover there are view statements mentioned by the provider which also bring us in to an attention like the modulo 6 above. We put some of them below.
That is, if the powers of 10 all returned with blue spin, or as a series of rainbows, or evenly alternating colors or other non-random results, then I’d say prime numbers appear to have a linkage to 10. I may not know what the the linkage is, just that it appears to exist (HexSpin).
Another is that phi and its members have a pisano period if the resulting fractional numbers are truncated. It would mean that there should be some undiscovered things hidden within the residual of the decimal values.
I wondered if that property might hold for the incremental powers of phi as well. For this reason I chose to see numbers in the hexagon as quantum, and truncate off the decimal values to determine which integer cell they land in. That is what I found. Phi and its members have a pisano period if the resulting fractional numbers are truncated (HexSpin).
Going deeper there are many things raised up as questions. So in this project we are going to analyze it using a javascript library called Chevrotain.
It can be used to build parsers/compilers/interpreters for various use cases ranging from simple config files to full fledged programming languages.
You may find that the rest of topics will concern mainly to this matter.