Experimental Mathematics: Finding Number Patterns

[Natus Videre]

The only thing I see at the moment is an anomaly where the only ternary prime ending with a zero is the decimal 3 which is probably simply explained

Yes, because the number 3 is the only prime which is divisible by 3. Numbers which are divisible by 3 always end in "0" in ternary (with zero).

3 = 10
6 = 20
9 = 100
12 = 110
15 = 120
...and so on.

 

[axj]

The mysterious sequence given by the C's is a good starting point, but it's not clear how it continues. The numbers 1/2, 1/3, and 121 muddy the waters in the sense that it is not obvious how they fit with (1,2,3), (11,22,33), (111,222,333), etc. We are left with more questions than answers.

I think @furryfrog found the explanation for this number sequence last year and it is actually quite simple:

The C's are talking about a Trinity spiritual based math system. Not base 10 as we normally use, but closer to a base 3 system but not exactly.

The C's said "WHO are YOUR prime numbers". Everyone is a Trinity composed of 3 lower centers, thought, body, and emotion, or in other words the number 3. We also hope to develop a soul which is also a Trinity, and composed of our 3 higher centers, thought, body, and emotion. All together we have 6 centers, 3 lower and 3 higher. You could also think of this as a 2Trinity, the number 33, or as I like to call it a Tritwoity. (This is why you have 33rd degree Freemasons for example.)

A: Yes, thank you Arkadiusz!!!!
..all of creation, of existence, is contained in 1, 2, 3!!! Look for this when you are trying to find the keys to the hidden secrets of all existence... They dwell within. 11, 22, 33, 1/2, 1/3, 1, 2, 3, 121, 11, 111, 222, 333, and so on! Get it?!?!

The C's say "They Dwell within".

The number 11 would mean one lower center along with its corresponding higher center, for example your lower physical body along with your soul body.
The number 22 would mean two lower centers along with 2 corresponding higher centers, for example your lower thought center, higher thought center, and lower emotional center, higher emotional center.

Fractions would apply to either out of 6 total centers or 3 centers.

The 111, 222, and 333 would apply to an individual who has merged their higher and lower centers, like the Toltec's call the third attention.


Graphical Representation.



More here: 2Trinity | Northern Orthodox Church

 

[John G]

To add these are the two-arm paths (1→0→5→3→2) & (3→0→5→1→2) that bear stricking similarity to the Svastika if represented in a 2D symbol. Also, notice the of the 10 possible node pairs (including the centroid)

Also, more detail to this progression, as a geometric topology, surfaces (faces) are distinct functions, and we could say, recursive patterns stemming from the first 4 points (0,1,3). This lead me to consider that 4-7 correspond to the faces and in Kabalistic terms - the 4 worlds. So if follow this through, symbolically at least, we have:

1. "Day 0": Point: Fire (0) - inhale
2. Day 1: Line: Air (1) - exhale
3. Day 2: Surface: Water (2) - inhale
4. Day 3: Volume: Earth (3) - exhale
5. Day 4: Face 1: Fire (0,1,2) - Atziluth (emmanation) - inhale
6. Day 5: Face 2: Air (0,1,3) - Beriah (creation) - exhale
7. Day 6: Face 3: Water (0,2,3) - Yetzirah (formation) - inhale
8. Day 7: Face 4: Earth (1,2,3) - Assiah (action) - exhale

View attachment 110552
Image above depicts how each "world" has a breath function too. This is further reiterated via next key events / function in Genesis: Tree of Knowledge and Tree of Life (inhale and exhale). So a decent into knowledge (1→0→5→1→0 fire and air) burns when not balanced with the exhale in life (3→0→5→1→2). We see that same metaphor in the Ark of the Covenant and the Holy Grail. In other words, truth without care, law without grace, structure without flow - is incomplete and casts us out of the garden of Eden.

Consider again the burning bush - it ignites without being extinguished, the breath YHWH that is fully "breathing". This is the complete 10 progressions (Decad / Tetraktys):

9. "Day 8": Arm 1 (Svastika): Fire and Air - via 4 connections - inhale (decent of strain to centroid - the 5th node / 5th element / Adam Kadman / Aether)
10: "Day 9": Arm 2 (Svastika): Water and Earth - via 4 connections - exhale (release of novelty from centroid)

This traces the 10 possible node pairs via this "breath cycle". Given both trees are fall on progression 9 and 10 (day 8 & 9) these "branches" might just be the 10 Sephira. But two of those remain "untraced". These are node pairs 1-3 (a double inhale) and 0-2 (a double exhale). Perhaps they are Keter - a crown (above head/body) - sometimes said to be "Ayin" - nothing. And Yesod - the foundation and transparency of transmission. Serving here more of "silent" function?

So back to the idea of the star tetrahedron. If the first tetrahedron is something like the most basic topology / consciousness unit / monad, then a mirror / shadow / polarity completes it (as above so below; STO/STS; action and equal and opposite reaction; peak is balanced by a trough, heartbeat and breath itself etc.). This provides then an additional 10 progressions across all nodes, faces and connections "sephira") (via 6 edges + 4 to centroid):

View attachment 110553

As you can see this foms the 3-4-3-4 cuboctahedron, an Archimedean solid, also known as ‘vector equilibrium’ coined by Buckmister Fuller. He called that the zero starting point for happenings or nonhappenings.. "it is the empty theater and empty circus and empty Universe ready to accommodate any act and any audience". This describes how the "centroid" (Aether, Adam Kadman etc.) is no actually a point but a non-local, 4th dimensional geometry. So the "breath cycle only ever moves towards the centre but never full entering it.

The star tetrahedron also forms a cube. This provides a cubic boundary of 6 new faces and x,y,z directional space + growth vectors (inhale and exhale): up, down, left, right, forward, back If we follow the inhale/exhale it transpires as follows:

...11-20 (day 10-19) the phases of the mirror tetrahedron: from Kaf (כ) to Resh (ר) that ends with an "exhale"
21. Shin (ש): Up
22. Tav (ת): Down
23. Kaf Sofit (ך): Left
24. Mem Sofit (ם): Right
25. Nun Sofit (ן): Forward
26. Pe Sofit (ף): Back
27. Tsade Sofit (ץ): Seed growth in 6 directions.

There's a bunch of other fascinating connections that seem to unravel from this too. Will add more.

Initially, what confused me is that this sequence was not ordered. 1/2 and 1/3 suddenly appear and then 121, followed by 11 again, which break the monotonously increasing logic I assumed. The C's suggested that using a number system based on primes is preferable, and that zero was an optional symbol that wasn't required to be explicitly part of the number system.


So here is a bijective base-3 system in comparison with our standard base-10 system.

Base-3 (no zero)	Base-10 (with zero)
1​	1
2	2
3	3
11	4​
12	5
13	6
21	7
22	8
23	9
31	10
32	11
33	12
111	13
112	14
113	15
121	16
122	17
123	18
131	19
132	20
133	21
211	22
212	23
213	24
221	25
222	26
223	27
231	28
232	29
233	30
311	31
312	32
313	33
321	34
322	35
323	36
331	37
332	38
333	39
1111	40

11 (one-one) is equal to 4 (four). 121 (one-two-one) is equal to 16 (sixteen). In our standard base-10, 11^2 (eleven squared) is equal to 121 (one hundred twenty-one), but 11 (eleven) isn't a square compared to 11 (one-one) in base-3.

11 (one-one, i.e. 4), 22 (two-two, i.e. 8), and 33 (three-three, i.e. 12) keep the same ratios if we translate them to base-10, i.e 11 (eleven), 22 (twenty-two), 33 (thirty-three). 4/12 = 11/33, 8/12 = 22/33.

Then, if we consider the transcripts, much is said about 11, 22, 33... and it seems to strangely apply to both bases until the C's say "both times 2 is your square, my dear. In other words, perfect balance," which is confusing. Did they mean "both times 2 is your square" as in "2 x 2" which equals 4, which translates to 11, a square number in base-3? Or did they mean 11 (eleven) times 2 which is 22 (twenty-two)... but how is 22 (twenty-two) a square?


Finally, the number 12 (twelve) is apparently also significant, and when we translate it to base-3, we get 33 (three-three)... talk about symbolism!

I think @furryfrog found the explanation for this number sequence last year and it is actually quite simple:

The C's are talking about a Trinity spiritual based math system. Not base 10 as we normally use, but closer to a base 3 system but not exactly.

The C's said "WHO are YOUR prime numbers". Everyone is a Trinity composed of 3 lower centers, thought, body, and emotion, or in other words the number 3. We also hope to develop a soul which is also a Trinity, and composed of our 3 higher centers, thought, body, and emotion. All together we have 6 centers, 3 lower and 3 higher. You could also think of this as a 2Trinity, the number 33, or as I like to call it a Tritwoity. (This is why you have 33rd degree Freemasons for example.)



The C's say "They Dwell within".

The number 11 would mean one lower center along with its corresponding higher center, for example your lower physical body along with your soul body.
The number 22 would mean two lower centers along with 2 corresponding higher centers, for example your lower thought center, higher thought center, and lower emotional center, higher emotional center.

Fractions would apply to either out of 6 total centers or 3 centers.

The 111, 222, and 333 would apply to an individual who has merged their higher and lower centers, like the Toltec's call the third attention.


Graphical Representation.

View attachment 101105

More here: 2Trinity | Northern Orthodox Church

This is also the head-heart-gut of the Enneagram 3-6-9 which can be doubled also via the cuboctahedron as in my avatar. The 12 vertex cuboctahedron is also the root lattice of the conformal group used by Ark for things like the antimatter universe and a degenerate metric to get to 4th density physics. The 12x12x12 Santilli matrix thing also started as 4th density physics question. Ark also related that 12 verified by the Cs to the hexagon metric signature (4 pluses, two minuses) plus 6 corresponding energies (a phase space thing maybe?). The three 12s could be a spacetime-matter-antimatter triality thing also relating to the 3-6-9 of the Enneagram. The Sefirot (and thus via Laura the densities) can also be represented via an Enneagram cuboctahedron plot.

 

[Sol Logos]

This is also the head-heart-gut of the Enneagram 3-6-9 which can be doubled also via the cuboctahedron as in my avatar. The 12 vertex cuboctahedron is also the root lattice of the conformal group used by Ark for things like the antimatter universe and a degenerate metric to get to 4th density physics. The 12x12x12 Santilli matrix thing also started as 4th density physics question. Ark also related that 12 verified by the Cs to the hexagon metric signature (4 pluses, two minuses) plus 6 corresponding energies (a phase space thing maybe?). The three 12s could be a spacetime-matter-antimatter triality thing also relating to the 3-6-9 of the Enneagram. The Sefirot (and thus via Laura the densities) can also be represented via an Enneagram cuboctahedron plot.

It's also what Bucky talked about it like the seed of the universe too (a.k.a. vector equilibrium - only geometric form wherein all of the vectors are of equal length). On 12 it has..

-12 identical vertices
- 12 axes around which it can be rotated to produce an identical configuration.
- 12 spheres closest-packed around a central sphere of the same size, 12 spheres fit perfectly.



3 sets of 4 spheres / as 4 sets of 3)? 1 set (3), 2 sets (6), 3 sets (9), 4 sets (12). There's a 3,6,9 progression.

Heart, head, gut, "hand" / action too?

For a geometric topology, given that the cuboctahedron may be a "seed". Could follow these expansive "cyclic" phase steps:

Cuboctahedron > Sphere > Icosahedron > Dodecahedron > Octahedron > Tetrahedron > Stella Octangula > Cube > Cuboctahedron


Bucky's notes I came across.

 

[Natus Videre]

Consider the following triangles:


These triangles are called isosceles right triangles, because they have two equal sides and a right angle. If you were to split a square along one of its main diagonals, you would get this type of triangle.

If we apply the Pythagorean theorem (a^2 + b^2 = c^2), we obtain:

1^2 + 1^2 = 2
2^2 + 2^2 = 8
3^2 + 3^2 = 18

What does the 2,8,18,... sequence represent?
Well, since the triangles are isoceles, the Pythagorean theorem has this form:

a^2 + a^2 = c^2
2*(a^2) = c^2

So the left-hand side of the equation can be expressed as:

a(n) = 2*n^2, for n ≥ 1.

It turns out that this formula generates the sequence A001105 in OEIS.

2, 8, 18, 32, 50, 72, 98, 128, 162, 200,...

This sequence corresponds to the maximum number of electrons in an atomic shell with total quantum number n.

a(n=1) = 2
a(n=2) = 8
a(n=3) = 18
a(n=4) = 32
a(n=5) = 50

Electron shell - Wikipedia

en.wikipedia.org

As you can see, there is also a sequence governing the maximum number of electrons in each subshell:

2, 6, 10, 14, 18,...

The formula is a(n) = 4*n+2, for n ≥ 0.

Interestingly, the sequence (A016825) can also be obtained by doubling each odd number.

1, 3, 5, 7, 9,...
2, 6, 10, 14, 18,...

Here is the order in which electron subshells are filled. Each square represents two electrons (spin +1/2 and spin -1/2).
Notice the triangular patterns!

Matter - Electron Shells and Orbitals

This article takes a detailed look at the concepts of electron shells and electron orbitals.

www.technologyuk.net

Hmm... all this doubling and squaring... there must be a reason!

Q: Okay, one interesting thing that we just discovered was that Hyakatuke and Hale Bopp both crossed the eye of Medusa, the star Algol, on April 11th exactly one year apart. What is the significance of this?

A: You must remember mosaic, matrix... When you are on the verge of quantum changes or discovery, the realities begin to reveal their perfectly squared nature to you.

Q: Well, do you have anything else to say about Andromeda? (It's VERY HOT in here!) Okay, Medusa 11. So, this was 11 of the 33, and assuming that you were not saying that there were 11 heads, but that Medusa was one of three heads, is that what we are getting at here, that there are three heads and Medusa was one?

A: Or both times 2.

Q: What do you mean? I don't understand.

A: Both times 2 is your square, my dear. In other words, perfect balance.

Q: (L) And the whole idea is to blend both pathways no matter which direction you come to it from?

A: Service to others provides the perfect balance of those two realities; service to self is the diametrical opposite closing the grand cycle in perfect balance.