# Addition Zones (0-18)

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 section is referring to *wiki page-1* of *zone section-1* that is *inherited * from *the zone section-1* by *prime spin-* and *span-* with *the partitions* as below.

- True Prime Pairs
- Primes Platform
- Pairwise Scenario
- Power of Magnitude
- The Pairwise Disjoint
- The Prime Recycling ζ(s)
- Implementation in Physics

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**.

## The 24 Cells 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).

The most obvious interesting feature of proceeding this prime hexagon, the number line begins to coil upon itself, is it confines all numbers of primes spin!

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)*Surprisingly, the 24-cell hexagon confines all natural numbers. The reason: no prime numbers occupy a cell with a right or left wall on the t-hexagon’s outer boundary, other than 2 and 3, the initial primes that forced the number line into this complex coil. Without a prime number in the outer set of triangles, the number line does not change to an outward course and ** remains forever contained in the 24 cells**.

*(HexSpin)*

So there should be a tight connection between 168 primes within 1000 with the *24-cell hexagon*. Indeed it is also correlated with 1000 prime numbers.

## 18s Structure

You may learn that sets of algebraic objects has a multilinear relationship related to a vector space called *tensor*.

Tensors may map between different objects such as vectors, scalars, even other tensors contained in a group of *partitions*.

Since the transformation of the plane shall have symmetrical vector then this 7s will take the posistion on the last side of 18s structure.

The number of primes less than or equal to a thousand (π(1000) = 168) equals the number of hours in a week (7 * 24 = 168).

```
$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 = f(1000) √
3 +------+ 8 +-----+----- } (36) |
| | | 16 | |
| |-----+-----+ |
| 6 | | 17 | (19) |
| | 9 +-----+ |
| | | 18 | --------------------------
------|------|-----+-----+------
```

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 eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them *(Wikipedia)*.

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*.

The tessellating field of equilateral triangles fills with numbers, with spin orientation flipping with each prime number encountered, creating ** 3 minor hexagons**.

*π(6+11) = π(17) = 7*

So there would be the empty spaces for `18 - 7 = 11`

numbers. By our project these spaces will be *unified* by all of the eleven (11) members of ** identition zones**.

Prime numbers are numbers that have only 2 factors: 1 and themselves.

- For example, the first 5 prime numbers are 2, 3, 5, 7, and 11. By contrast, numbers with more than 2 factors are call composite numbers.
- 1 is not a prime number because it can not be divided by any other integer except for 1 and itself. The only factor of 1 is 1.
- On the other hand, 1 is also not a composite number because it can not be divided by any other integer except for 1 and itself.

In conclusion, the number 1 is neither prime nor composite.

*(11x7) + (29+11) + (25+6) + (11+7) + (4+1) = 77+40+31+18+5 = 171*

The above seven (7) primes will act then as ** extended branes**. This is what we mean by

**and it happens whenever a cycle is restarted.**

*addition zones*## Undiscovered Features

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 most important 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.

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)*.

It would mean that there should be undiscovered things hidden within the residual of this decimal values. In fact it is the case that happen with *3-forms in 7D*.

In our approach a 3-form is not an object that exist in addition to the metric, it is the only object that exist and in particular *the 4D metric, is defined by the 3-form*.

- We would like to say that our present use of G2 structures (3-forms in 7D) is different from whatone can find in the literature on Kaluza–Klein compactifications of supergravity.
- We show that the resulting 4D theory is (Riemannian) General Relativity (GR) in Plebanski formulation, modulo corrections that are negligible for curvatures smaller than Planckian.
- Possibly the most interesting point of this construction is that the dimensionally reduced theory is GR with a non-zero cosmological constant, and the value of the cosmological constant is directly related to the size of . Realistic values of Λ correspond to of Planck size.

Also, in the supergravity context a 7D manifold with a G2 structure is used for compactifying the 11D supergravity down to 4D. In contrast, we compactify from 7D to 4D. *(General relativity from three-forms in seven dimensions - pdf)*

The complete theory was obtained by dimensional reduction of the 11D supergravity on a seven (7) torus and realizing the exceptional symmetry group E7(7)