# Imaginary Square

``````---+-----+-----
1 | {1} | {2}
---+-----+-----
2 |  3  | 20
---+-----+-----
3 | 21  | 46
---+-----+-----
4 |{47} | 58
---+-----+-----
5 | 59  | 70
---+-----+-----
6 |{71} |{93}
---+-----+-----
7 | 94  | 103
---+-----+-----
8 | 104 |{109}
---+-----+-----
``````

The tetractys is a triangular figure consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number (Wikipedia).

However, we can conclude that the distribution of prime numbers must have a static base structure, which is also confirmed logically in the further course

Φ(11,13) = Φ(1,2,3) + Φ(4,2) = 123 + 42 = 165

``````---+-----+-----+-----+-----+
1 |  72 | 1   | 73  |  74 |-----------------
---+-----+-----+-----+-----+                 |
2 | {20}| 74  | 94  | 168 |-----------      |
---+-----+-----+-----+-----+           |     |
3 | {18}| 95  | 113 | 208 |-----      |     |
---+-----+-----+-----+-----+     |     |     |
4 |  {7}| 114 | 121 | 235 |- {7}| 5   |  1  |{61} = 18th prime
---+-----+-----+-----+-----+     |     |     |
5 |  13 | 122 | 135 | 257 |-----      |     |
---+-----+-----+-----+-----+           |     |
6 |  19 | 136 | 155 | 291 |-----------      |
---+-----+-----+-----+-----+                 |
7 |   9 | 156 | 165 | 321 |----------------
---+-----+-----+-----+-----+
``````

300 - 2x11x13 = 300-286 = 14

``````layer | node | sub |    i     |   f
------+------+-----+----------+-----+-----+-----+                                    ---
|      |     |    1,2:1 |   1 |  30 |  40 | 71 (2,3) ‹-------------------       |
|      |  1  +----------+-----+-----+-----+                              |      |
|  1   |     |        2 |                                                |      5¨  encapsulation
|      |-----+----------+            -----------------------------       |      |
|      |     |        3 |           |                             |      |      |
1   +------+  2  +----------+----       |       LAGGING SCHEME        |      |     ---
|      |     |        4 |           |    (Exponentiation Zone)    |      |      |
|      +-----+----------+           |                             |      |      |
|  2   |     |        5 |           ------------------------------       |      7¨  abstraction
289   |      |  3  +----------+                                                |      |
|     |      |     |        6 |  ‹---------------------------- Φ               | {6®} |
------+------+-----+----------+-----+-----                                     |     ---
|      |     |     11:7 |   5 |   9 |  14 (20) --------› ¤               |      |
|      |  4  +----------+-----+-----+-----+                              |      |
|  3   |     |     12:8 |   9 |  60 |  40 | 109 (26) «------------       |     11¨  polymorphism
|      +-----+----------+-----+-----+-----+                       |      |      |
|      |     |     13:9 |   9 |  60 |  69 (27) «-- Δ19 (Rep Fork) | {2®} |      |
2   +------|  5  +----------+-----+-----+-----+                       |      |     ---
|      |     |    14:19 |   9 |  60 |  40 | 109 (28) -------------       |      |
|      |-----+----------+-----+-----+-----+                              |      |
|  4   |     | 15,18:11 |   1 |  30 |  40 | 71 (29,30,31,32) ------------      13¨  inheritance
329   |      |  6  +----------+-----+-----+-----+                                     |
|     |      |     |    19:12 |  10 |  60 | {70} (36) -------› Φ                      |
------+------+-----+----------+-----+-----+                                          ---
``````

Proceeding, the number line begins to coil upon itself; 20 lands on 2's cell, 21 on 3's cell. Prime number 23 sends the number line left to form the fourth hexagon, purple. As it is not a twin, the clockwise progression (rotation) reverses itself.

Twin primes 29 and 31 define the fifth hexagon, cyan. Finally, 37, again not a twin, reverses the rotation of the system, so 47 can define the yellow hexagon (Prime-Hexagon)

7th collumn ◄- 65 + 12 = 77

``````|         1st (Form)          |         2nd (Route)         |         3rd (Channel)       |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
150 | 151| 152| 153| 154| 155| 156| 157| 158| 159| 160| 161| 162| 163| 164| 165| 166| 167| 168|
=====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+
Δ1 | 19 |  - | 31 | 37 |  - |  - |  - |  - |  - |  - |  - |  - |  - |  - | 103|  - |  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ2 | 20 | 26 |  - | 38 |  - |  - |  - |  - |  - | 74 |  - |  - |  - | 98 | 104|  - |  - |  - |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ3 | 21 | 27 |  - | 39 |  - |  - |  - |  - |  - | 75 |  - |  - |  - | 99 | 105|  - |  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ4 | 22 | 28 |  - | 40 |  - |  - |  - |  - |  - | 76 |  - |  - |  - |100 |  - |  - |  - |  - |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ5 | 23 | 29 |  - | 41 |  - |  - |  - |  - |  - | 77 |  - |  - |  - |101 |  - |  - |  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ6 | 24 |  - |  - | 42 |  - | 54 |  - |  - | 72 | 78 |  - | 90 | 96 |  - |  - |  - |  - | 114|
=====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+
Δ7 | 25 |  - |  - | 43 |  - | 55 |  - |  - | 73 | 79 |  - | 91 | 97 |  - |  - |  - |  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ8 |  - |  - |  - | 44 |  - | 56 |  - |  - |  - | 80 |  - | 92 |  - |  - |  - |  - |  - |  - |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ9 |  - |  - |  - | 45 |  - | 57 |  - |  - |  - | 81 |  - | 93 |  - |  - |  - |  - |  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ10 |  - |  - |  - | 46 | 52 | 58 |  - | 70 |  - | 82 | 88 | 94 |  - |  - |  - |  - | 112|  - |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ11 |  - |  - |  - | 47 | 53 | 59 |  - | 71 |  - | 83 | 89 | 95 |  - |  - |  - |  - | 113|  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   T
Δ12 |  - |  - |  - | 48 |  - | 60 | 66 |  - |  - | 84 |  - |  - |  - |  - |  - | 108|  - |  - |   H
=====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+   E
Δ13 |  - |  - |  - | 49 |  - | 61 | 67 |  - |  - | 85 |  - |  - |  - |  - |  - | 109|  - |  - |
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   P
Δ14 |  - |  - | 32 | 50 |  - | 62 | 68 |  - |  - | 86 |  - |  - |  - |  - |  - | 110|  - |  - |   O
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   W
Δ15 |  - |  - | 33 | 51 |  - | 63 | 69 |  - |  - | 87 |  - |  - |  - |  - |  - | 111|  - |  - |   E
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   R
Δ16 |  - |  - | 34 |  - |  - | 64 |  - |  - |  - |  - |  - |  - |  - | -  | 106|  - |  - |  - |
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   O
Δ17 |  - |  - | 35 |  - |  - | 65 |  - |  - |  - |  - |  - |  - |  - | -  | 107|  - |  - |  - |   F
+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+
Δ18 |  - | 30 | 36 |  - |  - |  - |  - |  - |  - |  - |  - |  - | -  |102 |   -|  - |  - |  - | ∑=168
=====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+====+
1  |  2 |  3 |  4 |  5 |  6 |  7 |  8 |  9 | 10 | 11 | 12 | 13 | 14 | 15 |  16|  17| 18 | 19 |   V
-----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+----+   S
|       Δ    Δ                |                     Φ12     |       Δ                   Δ |
113  150                                   ≜114-25          557                619 = 1+618
``````

See that by this 6 (six) spines, the number of 19, 43 and 71. The 71 as the third cycles would land on the 2's cell which are the same direction as the 3's, so both of them got this 71.

Below are the spins for the first 80 numbers; 2 (0 and 1) spiral blue, meet prime 5, spiral purple briefly, meet 7 and spiral red. The bands are pretty short because primes are common in the early numbers, but the grow rare as numbers grow large and the bands broaden.

This 71 is a conformation that it has the same result as we have explained on the residual objects of 571 turn to a vektor of 71 while the rest of 500 turn to 200 objects of 3's identity and the last objects of 300 goes to the next cycles.

For now we will be coy and state that the secret of the pyramid lies with two 4-times triangular numbers: The first, 4 x triangular number 528=2112, equates to the total number of elements used to construct the pyramid, which is subdivided into its four lateral faces comprised of 528 elements each in the form of 32 cascading triangular numbers where 2112/96 = 242.

242 = number of Mirror Prime Pairs and 96 is the 360 degree digital root Fibonacci periodicity when indexed to n not divisible by 2, 3, or 5, i.e., period 32 every 120 degrees (Primesdemystified).

## Construct the pyramid

This polarity is happened per six (6) cycles by the polar of six (6) to one (1) and six (6) to seven (7) by which we finally found if this behaviour is cascaded bilaterally within the correlation between 61 as the 18th prime and 67 as the 19th prime.

The most obvious interesting feature of this 24-cell hexagon is it confines all numbers! That is, as the number line winds about toward infinity, bending around prime numbers, it never exits the 24 cells.