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.

 

[Sol Logos]

Fascinating insights on the Trinity-based math system, the Enneagram’s 3-6-9 structure, and the Pythagorean connections to electron shells. And yes of course,’that makes a lot of sense - the 3 centres (thought, body, emotion) leading to 11/22/33 encodings.

The C’s reference to “WHO are YOUR prime numbers” and “They Dwell within” suggests primes as intrinsic to identity, perhaps as irreducible building blocks of consciousness. And then there’s that 27-point lattice (3^3, a trinity cubed), associating points with Hebrew’s 22 letters + 5 finals (sofit forms),

Both aligns with the Enneagram’s 3-6-9 (head-heart-gut) and that doubling via the cuboctahedron / vector equilibrium (12 vertices). I’m not sure but wonder if there’s a connection here with Ark’s work on conformal groups in contexts like antimatter universes and degenerate metrics for 4th-density transitions? 12 might represent a phase space with hexagon metric signatures (4 pluses, 2 minuses) plus 6 energies. The three 12s might symbolise a spacetime-matter-antimatter triality, tying into the Sefirot/densities as Laura’s interpretations where densities map to Enneagram-like structures in a cuboctahedral plot?

The Pythagorean link (1^2+1^2=2, 2^2+2^2=8, etc.) to electron shells (2,8,18,32 via 2*n^2) hints at universal structure from atoms to consciousness. The subshell sequence (2,6,10,14 via 4n+2 or doubled odds: 1→2,3→6,5→10) reinforces Cs “both times 2” as squaring+doubling. In quantum terms, shells reflect orbital capacities (s:2, p:6, d:10, f:14), building via angular momentum quanta. And that triangle split of squares geometrically derives it!

Recently, I did some modelling on EEG public datasets - 50 or so subjects. I suspected that 1,2,3 harmonic levels might just be 11/22/33.

It was close! But perhaps as significant was the doubling…

- 1st Harmonic (Alpha): 9.50 Hz (Baseline)
- 2nd Harmonic (Beta): 19.00 Hz - Perfect 2.000 ratio
- 3rd Harmonic (Gamma): 29.50 Hz - 3.105 ratio (3.5% error from perfect 3.000)

I wonder if taking a set of mediators’ EEG might just push even closer to 11/22/33?

As a side note, for some of the modelling I’m doing on these sort of geometric flows, I’ve looked to introduce p-adic maths. I’ve recently been using p=137 as the base. It’s the fine-structure α≈1/137 that governs electromagnetic interactions in shells , so using Q_{137} may prove significant.

Not the same modelling but one that looks at the same geometry and the laws of thermodynamics: tethraddynamics_clean (7).ipynb

 

[Sol Logos]

Fascinating insights on the Trinity-based math system, the Enneagram’s 3-6-9 structure, and the Pythagorean connections to electron shells. And yes of course,’that makes a lot of sense - the 3 centres (thought, body, emotion) leading to 11/22/33 encodings.

The C’s reference to “WHO are YOUR prime numbers” and “They Dwell within” suggests primes as intrinsic to identity, perhaps as irreducible building blocks of consciousness. And then there’s that 27-point lattice (3^3, a trinity cubed), associating points with Hebrew’s 22 letters + 5 finals (sofit forms),

Both aligns with the Enneagram’s 3-6-9 (head-heart-gut) and that doubling via the cuboctahedron / vector equilibrium (12 vertices). I’m not sure but wonder if there’s a connection here with Ark’s work on conformal groups in contexts like antimatter universes and degenerate metrics for 4th-density transitions? 12 might represent a phase space with hexagon metric signatures (4 pluses, 2 minuses) plus 6 energies. The three 12s might symbolise a spacetime-matter-antimatter triality, tying into the Sefirot/densities as Laura’s interpretations where densities map to Enneagram-like structures in a cuboctahedral plot?

The Pythagorean link (1^2+1^2=2, 2^2+2^2=8, etc.) to electron shells (2,8,18,32 via 2*n^2) hints at universal structure from atoms to consciousness. The subshell sequence (2,6,10,14 via 4n+2 or doubled odds: 1→2,3→6,5→10) reinforces Cs “both times 2” as squaring+doubling. In quantum terms, shells reflect orbital capacities (s:2, p:6, d:10, f:14), building via angular momentum quanta. And that triangle split of squares geometrically derives it!

Recently, I did some modelling on EEG public datasets - 50 or so subjects. I suspected that 1,2,3 harmonic levels might just be 11/22/33.

It was close! But perhaps as significant was the doubling…

- 1st Harmonic (Alpha): 9.50 Hz (Baseline)
- 2nd Harmonic (Beta): 19.00 Hz - Perfect 2.000 ratio
- 3rd Harmonic (Gamma): 29.50 Hz - 3.105 ratio (3.5% error from perfect 3.000)

I wonder if taking a set of mediators’ EEG might just push even closer to 11/22/33?

As a side note, for some of the modelling I’m doing on these sort of geometric flows, I’ve looked to introduce p-adic maths. I’ve recently been using p=137 as the base. It’s the fine-structure α≈1/137 that governs electromagnetic interactions in shells , so using Q_{137} may prove significant.

Not the same modelling but one that looks at the same geometry and the laws of thermodynamics: tethraddynamics_clean (7).ipynb

The attached complements above if of interest.

 

[Natus Videre]

There seems to be an inherent link between primes and elementary particles!

Primes, Geometry and Condensed Matter

Abstract
Fascination with primes dates back to the Greeks and before. Primes are named by some "the elementary particles of arithmetic" as every nonprime integer is made of a unique set of primes. In this article we point to new connections between primes, geometry and physics which show that primes could be called "the elementary particles of physics" too. This study considers the problem of closely packing similar circles/spheres in 2D/3D space. This is in effect a discretization process of space and the allowable number in a pack is found to lead to some unexpected cases of prime configurations which is independent of the size of the constituents. We next suggest that a non-prime can be considered geometrically as a symmetric collection that is separable (factorable) into similar parts- six is two threes or three twos for example. A collection that has no such symmetry is a prime. As a result, a physical prime aggregate is more difficult to split symmetrically resulting in an inherent stability. This "number/physical" stability idea applies to bigger collections made from smaller (prime) units leading to larger stable prime structures in a limitless scaling up process. The distribution of primes among numbers can be understood better using the packing ideas described here and we further suggest that differing numbers (and values) of distinct prime factors making a nonprime collection is an important factor in determining the probability and method of possible and subsequent disintegration. Disintegration is bound by energy conservation and is closely related to symmetry by Noether theorems. Thinking of condensed matter as the packing of identical elements, we examine plots of the masses of chemical elements of the periodic table, and also those of the elementary particles of physics, and show that prime packing rules seem to play a role in the make up of matter. The plots show convincingly that the growth of prime numbers and that of the masses of chemical elements and of elementary particles do follow the same trend indeed.

The plots of the mass growth (packing) of chemical elements and elementary particles (and hence all massive bodies), as shown here, follow very closely the rules of packing of spheres and also those of the prime numbers. Prime numbers or prime collections appear when it is not possible to divide a collection into symmetric (equal) parts and are hence more stable in structure. This makes the growth of primes to be naturally tied to the growth in the masses of condensed matter in its different phases. We also note that the prime character of a number is an independent property- more of an abstract physical property, and it is not a function of the base of the number system in use or the physical case that number might represent.

The eight fold rules frequently found in the behaviour pattern of chemical elements and elementary particles [4,8] may now be suspected to be a consequence of the packing rules of similar spheres in space. We might even suggest that the successes of the Bohr Theory for the atom, the Ballmer series formula for energy levels and indeed the Schrödinger equation itself in predicting discrete behaviour in atoms and other entities, might be mainly due to the discretization of space implied in their formulations.

Let's look at circle packing a bit closer.

In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap.

Generalisations can be made to higher dimensions – this is called sphere packing, which usually deals only with identical spheres.

The branch of mathematics generally known as "circle packing" is concerned with the geometry and combinatorics of packings of arbitrarily-sized circles: these give rise to discrete analogs of conformal mapping, Riemann surfaces and the like.

There are many "flavors" of circle packing:

• Circle packing in a circle
• Circle packing in a square
• Circle packing in a rectangle
• Circle packing in an equilateral triangle
• Circle packing in an isosceles right triangle

Now consider the "purest" form of packing: circle packing in a circle, where all circles have a radius equal to one.

Circle packing in a circle is a two-dimensional packing problem with the objective of packing unit circles into the smallest possible larger circle.

Circle packing in a circle - Wikipedia

en.wikipedia.org

Observations

• When there is only one circle, it fills the whole space. (law of One, all part of One)
• When there are 6 circles, there are two possible configurations.
• The radius of the outer circle is the same (r=3) whether there are 6 or 7 circles. So here is the connection between 3, 6, and 7!
• When an 8th circle is added, it is disconnected from the rest of the circles, i.e. it can move freely.
• The packing with 7 circles is denser than arrangements with 2, 3, 4, 5, 6, or 8 circles. So there is a peak at 7!
• Therefore, seven is the first minimal unit with the maximal number of connected circles!

Could circle packing be the simplest geometrical "proof" that there are only 7 densities?

Q: (Ark) Can physics describe densities?
A: Yes

Q: (Ark) How?
A: Algebra.

Q: (Ark) What kind of algebra?
A: Simple.

In Session 14 December 1996, the C's invited the group to analyze a crop circle related to the number 7.

A: All are lines you ought to follow. Now, look at the photos on the wall! [Referring to large photocopy of a number of crop circles we had pinned to the wall.]

Q: (L) OK, we're looking at them: point out something...
A: Count the large spheres in photo three.

Q: (L) There are seven.
A: Yes.

Q: (L) And what does that photo represent?
A: Not yet.

Q: (T) OK, there are seven large circles; a large central one, and then six outer ones that are smaller. Each of the six smaller circles is connected to the larger circle by a shaft, or a line, or a conduit of some kind.
A: Add large and small spheres.

Q: (L) OK, there's seven. Add the large to the small and there's seven; add the little teeny ones, there's thirteen; and then even the little teeny-teeny, the little knobs on the ends, there would be six more, so that would be nineteen.
A: Yes...

Q: (T) So, that's another prime: nineteen is a prime number. (L) OK, they're prime numbers. And... (T) Are they... just as an offshoot here, do the six circles, the first set surrounding the large circle, are those the sixth density attached to the seventh density?

A: No comment.

The Wave Chapter 69: The Whirlpool of Charybdis, the Sirens and the Navigator

I know that the very idea of being in an actual Matrix as depicted in the movie is a difficult pill to swallow. We have been taught so many things from so many sources throughout history that tend …

cassiopaea.org

In Session 17 June 1995, the C's talked about the geometrical representation of a perpendicular reality, i.e. an inlaid wheel formed by a circle within a circle and seven equally spaced partitions.

Q: (T) Several sessions back when we were discussing "Perpendicular Realities" you were talking about something that happened to me and that I had to look back over my life and analyze my relationships with other people from a certain point up until now and you said that this was a perpendicular reality. What is the definition of a perpendicular reality?

A: The perpendicular reality primarily, though not exclusively, refers to one's life path and how one's life path fits together in the cycle or in a wheel when connected with those of a similar life path. And, oddly enough, relates very closely to the previous question involving synchronicity. If you can picture an inlaid wheel formed by a circle within a circle, and adjoining partitions in a perfect balance, that would be the best representation of perpendicular reality for it does not completely involve one individual's experience, but rather a group of individual's experience for the progression of a greater purpose, if you understand what we mean. This is what we mean when we say: perpendicular reality. Picture again, a circle within a circle adjoined by equally spaced partitions in a perfect cycle. That is perpendicular reality.

Q: (T) You had us draw this symbol and put seven spokes or partitions between the two circles.
A: Correct.

Q: (T) Is seven the optimal number?
A: Seven is always the optimal number. There are seven levels of density. This reflects through all phases of reality.

 

[Natus Videre]

To follow up on circle packing, notice how the configuration with 19 circles is very similar to the one with 7.
12 circles "wrap" the core 7 circles to form 19 circles. Is this why there is a significance to the number 12?

The 12 additional outer circles help form 6 squares and 6 triangles.

Another interesting fact is that 19/7 gives 2.7142857143... which is very close to 2.7182818284... = e, Euler's number !
The ratio 19/7 echoes another famous one... 22/7, which gives 3.1428571429... which is very close 3.1415926535... = π, PI !

The deep connections between e and π are captured in Euler's identity, relating to the circle!

 

[Natus Videre]

Recently, I was interested in knowing how many "primary" planes (made of coordinate axes) are in each dimension.

In 0D, there are no axes, and no planes.
In 1D, there is 1 axis (x), and no planes.
In 2D, there are 2 axes (x,y) and there is only 1 plane (xy).
In 3D, there are 3 axes (x,y,z) and there are 3 planes (xy, xz, yz).
In 4D, there are 4 axes (x,y,z,w) and there are 6 planes, (xy, xz, xw, yz, yw, zw).
In 5D, there are 5 axes (x,y,z,w,u) and there are 10 planes (xy, xz, xw, xu, yz, yw, yu, zw, zu, wu).
In 6D, there are 6 axes (x,y,z,w,u,v) and there 15 planes (xy, xz, xw, xu, yz, yw, yu, zw, zu, wu, xv, yv, zv, wv, uv).
...and so on.

What does the sequence 1, 3, 6, 10, 15 represent?
The triangular numbers!

Now, let's see how many "primary" 3D hyperplanes are in each dimension.

In 0D, there are no axes, and no 3D hyperplanes.
In 1D, there is 1 axis (x), and no 3D hyperplanes.
In 2D, there are 2 axes (x,y) and no 3D hyperplanes.
In 3D, there are 3 axes (x,y,z) and there is only 1 3D hyperplane (xyz).
In 4D, there are 4 axes (x,y,z,w) and there are 4 3D hyperplanes (xyz, xzw, xyw, yzw).
In 5D, there are 5 axes (x,y,z,w,u) and there are 10 3D hyperplanes (xyz, xyw, xyu, xzw, xzu, xwu, yzw, yzu, ywu, wzu).
...and so on.

The sequence 1,4,10,... represents the tetrahedral numbers!

If we continue like this (finding the number of 4D-hyperplanes in n-dimensions, finding the number of 5D-hyperplanes in n-dimensions, etc.), we would obtain Pascal's Triangle! For example, if we want to know how many 4D-hyperplanes are in 6D, we have to calculate "6 choose 4", which means we are interested in knowing how many groups of 4 we can make out of 6 distinct items without considering the order of the elements in each group, i.e. "xyzw" is considered the same 4D-hyperplane as "yzwx", "zwyx", etc.

As you can see, even our Cartesian coordinate system has "triangular" properties!

 

[Natus Videre]

Another cool way to generate primes!

Step 1: Find the lowest common multiples incrementally.

lcm(1,2) = 2
lcm(1,2,3) = 6
lcm(1,2,3,4) = 12
lcm(1,2,3,4,5) = 60
lcm(1,2,3,4,5,6) = 60
lcm(1,2,3,4,5,6,7) = 420
lcm(1,2,3,4,5,6,7,8) = 840
lcm(1,2,3,4,5,6,7,8,9) = 2520
lcm(1,2,3,4,5,6,7,8,9,10) = 2520
lcm(1,2,3,4,5,6,7,8,9,10,11) = 27720
lcm(1,2,3,4,5,6,7,8,9,10,11,12) = 27720
lcm(1,2,3,4,5,6,7,8,9,10,11,12,13) = 360360
lcm(1,2,3,4,5,6,7,8,9,10,11,12,13,14) = 360360
lcm(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) = 360360
lcm(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16) = 720720
lcm(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17) = 12252240
...and so on.

Step 2: Divide each result by the preceding result.
For example, lcm(1,2,3,4) / lcm(1,2,3) = 12/6 = 2.

You will obtain a list containing only prime numbers! Can you see why this is always the case?
[3, 2, 5, 1, 7, 2, 3, 1, 11, 1, 13, 1, 1, 2, 17, 1, 19, 1, 1, 1, 23, 1, 5, 1, 3, 1, 29, 1, 31, 2, 1, 1, 1, 1, 37, 1, 1, 1, 41, 1, 43, 1, 1, 1, 47, 1, 7, 1, 1, 1, 53, 1, 1, 1, 1, 1, 59, 1, 61, 1, 1, 2, 1, 1, 67, 1, 1, 1, 71, 1, 73, 1, 1, 1, 1, 1, 79, 1, 3, 1, 83, 1, 1, 1, 1, 1, 89, 1, 1, 1, 1, 1, 1, 1, 97, 1, 1, 1, 101, 1, 103, 1, 1, 1, 107, 1, 109, 1, 1, 1, 113, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 5, 1, 127, 2, 1, 1, 131, 1, 1, 1, 1, 1, 137, 1, 139, 1, 1, 1, 1, 1, 1, 1, 1, 1, 149, 1, 151, 1, 1, 1, 1, 1, 157, 1, 1, 1, 1, 1, 163, 1, 1, 1, 167, 1, 13, 1, 1, 1, 173, 1, 1, 1, 1, 1, 179, 1, 181, 1, 1, 1, 1, 1, 1, 1, 1, 1, 191, 1, 193, 1, 1, 1, 197, 1, 199, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 211, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 223, 1, 1, 1, 227, 1, 229, 1, 1, 1, 233, 1, 1, 1, 1, 1, 239, 1, 241, 1, 3, 1, 1, 1, 1, 1, 1, 1, 251, 1, 1, 1, 1, 2, 257, 1, 1, 1, 1, 1, 263, 1, 1, 1, 1, 1, 269, 1, 271, 1, 1, 1, 1, 1, 277, 1, 1, 1, 281, 1, 283, 1, 1, 1, 1, 1, 17, 1, 1, 1, 293, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 307, 1, 1, 1, 311, 1, 313, 1, 1, 1, 317, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 331, 1, 1, 1, 1, 1, 337, 1, 1, 1, 1, 1, 7, 1, 1, 1, 347, 1, 349, 1, 1, 1, 353, 1, 1, 1, 1, 1, 359, 1, 19, 1, 1, 1, 1, 1, 367, 1, 1, 1, 1, 1, 373, 1, 1, 1, 1, 1, 379, 1, 1, 1, 383, 1, 1, 1, 1, 1, 389, 1, 1, 1, 1, 1, 1, 1, 397, 1, 1, 1, 401, 1, 1, 1, 1, 1, 1, 1, 409, 1, 1, 1, 1, 1, 1, 1, 1, 1, 419, 1, 421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 431, 1, 433, 1, 1, 1, 1, 1, 439, 1, 1, 1, 443, 1, 1, 1, 1, 1, 449, 1, 1, 1, 1, 1, 1, 1, 457, 1, 1, 1, 461, 1, 463, 1, 1, 1, 467, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 479, 1, 1, 1, 1, 1, 1, 1, 487, 1, 1, 1, 491, 1, 1, 1, 1, 1, 1, 1, 499,...]