Addition Zones (1-18)
This section is referring to wiki page-2 of zone section-2 that is inherited from the zone section-zones by prime spin-19 and span-addition with the partitions as below.
/World
Prime Hexagon
The Prime Hexagon is a mathematical structure developed by mathematician T. 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: 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)
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).
Structure: True Prime Pairs
$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 |
------+------+-----+-----+------ } (36)
| | | 7 |
| | 4 +-----+
| 3 | | 8 | (11)
| +-----+-----+
| | | 9 |
2 +------| 5 +-----+-----
| | | 10 |
| |-----+-----+
| 4 | | 11 | (13)
| | 6 +-----+
| | | 12 |
------+------+-----+-----+------------------
| | | 13 |
| | 7 +-----+
| 5 | | 14 | (17)
| |-----+-----+
| | | 15 |
3 +------+ 8 +-----+----- } (36)
| | | 16 |
| |-----+-----+
| 6 | | 17 | (19)
| | 9 +-----+
| | | 18 |
------|------|-----+-----+------
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: prime-hexagon).
By having the total of 168, the 102 and the 30+36=66 will take the 1st and 2nd prime on The Primes Platform. This leads to 168 - 29 - 96 = 139 - 96 = 43 primes on the last of 7th row. That what and why 18+13+12=43 by the last 9 cells is standing for!
-----------------+----+----+----+----+----+----+----+----+----+-----
The last 9 cells | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Sum
=================+====+====+====+====+====+====+====+====+====+=====
3,2→18+13+12→43 | 73 | 79 | 83 | 89 | 97 | 101| 103| 107| 109|29th →13th→ 2,3
-----------------+----+----+----+----+----+----+----+----+----+-----
- This 43 is 18+13+12 in bilateral on perfect square of 9 goes to 89 → π(89²) = 1000
Structure: Minor Hexagons
Within a cycle this scheme would generate the prime platform which is performing the rank of 10 shapes starting with the primes 2,3,5,7. Via the 11 partitions as the lexer and 13 frames as the parser we do a recombination to build the grammar in 6 periods.
We color-code the six hexagons, identifying patterns in key number sequences, including the Fibonacci sequence, powers of two and three, and power of pi. For the series of consecutive powers of pi, we have found that no two fall within the same six-cell hexagon. We have computed this for pi^32, which has less than a 1/400 chance of occurring randomly (GitHub: prime-hexagon).
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).
