Multiplication Zones (18-30)

In this one system, represented as an icon, we can see the distribution profile of the prime numbersas well as their products via a chessboard-like model in Fig. 4. This fundamental chewing

• We show the connection in the MEC 30 mathematically and precisely in the table Fig. 13. The organization of this table is based on the well-known idea of ​​Christian Goldbach.
• That every even number from the should be the sum of two prime numbers. From now on we call all pairs of prime numbers without “1”, 2, 3, 5 Goldbach couples.

The MEC 30 transforms this idea from Christian Goldbach into the structure of a numerical double strand, into an opposite link of the MEC 30 scale. (MEC 30 - pdf)

A square system of coupled nonlinear equations can be solved iteratively by Newton’s method. This method uses the Jacobian matrix of the system of equations. (Wikipedia)

Mathematically, this type of system requires 27 letters (1-9, 10–90, 100–900). In practice, the last letter, tav (which has the value 400), is used in combination with itself or other letters from qof (100) onwards to generate numbers from 500 and above. Alternatively, the 22-letter Hebrew numeral set is sometimes extended to 27 by using 5 sofit (final) forms of the Hebrew letters. (Wikipedia)

He designed a new approach to predicting market behavior using several disciplines, including geometry, astrology, astronomy, and ancient mathematics. They say that not long before his death, Gann developed a unique trading system. However, he preferred not to make his invention public or share it with anyone. (PipBear)

Integration of ordinary and stochastic master equations is performed on density operators parametrized by 𝑑² real numbers, where 𝑑 is the dimension of the system Hilbert space.

• These are the components of the density operator as a vector in a basis that is Hermitian and, excepting the identity, traceless.
• Since the ordinary and stochastic master equations - pdf under consideration are trace preserving, one could neglect the basis element corresponding to the identity.

But as the module currently stands it is included to simplify some expressions and provide a simple test to make sure calculations are proceeding as they ought to. (PySME-pdf)

In this short review, we have briefly described the structure of exceptional field theories (ExFT’s), which provide a (T)U-duality covariant approach to supergravity. These are based on symmetries of toroidally reduced supergravity; however are defined on a general background.

• From the point of view of ExFT the toroidal background is a maximally symmetric solution preserving all U-duality symmetries. In this sense the approach is similar to the embedding tensor technique, which is used to define gauge supergravity in a covariant and supersymmetry invariant form. Although any particular choice of gauging breaks certain amount of supersymmetry, the formalism itself is completely invariant. Similarly the U-duality covariant approach is transferred to dynamics of branes in both string and M-theory, whose construction has not been covered here.
• In the text, we described construction of the field content of exceptional field theories from fields of dimensionally reduced 11-dimensional supergravity, and local and global symmetries of the theories. Various solutions of the section constraint giving Type IIA/B, 11D and lower-dimensional gauged supergravities have been discussed without going deep into technical details. For readers’ convenience references for the original works are present.
• As a formalism exceptional field theory has found essential number of application, some of which have been described in this review in more details. In particular, we have covered generalized twist reductions of ExFTs, which reproduce lower-dimensional gauged supergravities, description of non-geometric brane backgrounds and an algorithm for generating deformations of supergravity backgrounds based on frame change inside DFT. However, many fascinating applications of the DFT and ExFT formalisms have been left aside.

Among these are non-abelian T-dualities in terms of Poisson-Lie transformations inside DFT [100,101]; generating supersymmetric vacua and consistent truncations of supergravity into lower dimensions [102,103,104] (for review see [105]); compactifications on non-geometric (Calabi-Yau) backgrounds and construction of cosmological models [54,55,106,107]. (U-Dualities in Type II and M-Theory)

During the last few years of the 12th century, Fibonacci undertook a series of travels around the Mediterranean. At this time, the world’s most prominent mathematicians were Arabs, and he spent much time studying with them. His work, whose title translates as the Book of Calculation, was extremely influential in that it popularized the use of the Arabic numerals in Europe, thereby revolutionizing arithmetic and allowing scientific experiment and discovery to progress more quickly. (Famous Mathematicians)

Simply stated, the Golden Ratio establishes that the small is to the large as the large is to the whole. This is usually applied to proportions between segments.

• In the special case of a unit segment, the Golden Ratio provides the only way to divide unity in two parts that are in a geometric progression:
• The side of a pentagon-pentagram can clearly be seen as in relation to its diagonal as 1: (√5 +1)/2 or 1:φ, the Golden Section:
• When you draw all the diagonals in the pentagon you end up with the pentagram. The pentagram shows that the Golden Gnomon, and therefore Golden Ratio, are iteratively contained inside the pentagon:
• There are set of sequence known as Fibonacci retracement. For unknown reasons, these Fibonacci ratios seem to play a role in the stock market, just as they do in nature. The Fibonacci retracement levels are 0.236, 0.382, 0.618, and 0.786.
  • The key Fibonacci ratio of 61.8% is found by dividing one number in the series by the number that follows it. For example, 21 divided by 34 equals 0.6176, and 55 divided by 89 equals about 0.61798.
  • The 38.2% ratio is discovered by dividing a number in the series by the number located two spots to the right. For instance, 55 divided by 144 equals approximately 0.38194.
  • The 23.6% ratio is found by dividing one number in the series by the number that is three places to the right. For example, 8 divided by 34 equals about 0.23529.
  • The 78.6% level is given by the square root of 61.8%
• While not officially a Fibonacci ratio, 0.5 is also commonly referenced (50% is derived not from the Fibonacci sequence but rather from the idea that on average stocks retrace half their earlier movements).

This study cascade culminating in the Fibonacci digital root sequence (also period-24). (Golden Ratio - Articles)

The Hexagon chart begins with a 0 in the center, surrounded by the numbers 1 through 6. Each additional layer adds 6 more numbers as we move out, and these numbers are arranged into a Hexagon formation. This is pretty much as far as Gann went in his descriptions. He basically said, “This works, but you have to figure out how.”One method that I’ve found that works well on all these kinds of charts is plotting planetary longitude values on them, and looking for patterns. On the chart above, each dot represents the location of a particular planet. The red one at the bottom is the Sun, and up from it is Mars. These are marked on the chart. Notice that the Sun and Mars are connected along a pink line running through the center of the chart. The idea is that when two planets line up along a similar line, we have a signal event similar to a conjunction in the sky. Any market vibrating to the Hexagon arrangement should show some kind of response to this situation. (Wave59)

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).

The first 1000 prime numbers are silently screaming: “Pay attention to us, for we hold the secret to the distribution of all primes!” We heard the call, and with ‘strange coincidences’ leading the way have discovered compelling evidence that the 1000th prime number, 7919, is the perfectly positioned cornerstone of a mathematical object with highly organized substructures and stunning reflectional symmetries. (PrimesDemystified)

There is a fascinating connection between prime numbers and the Golden ratio.

• The Golden ratio is an irrational number, which means that it cannot be expressed as a ratio of two integers. However, it can be approximated by dividing consecutive Fibonacci numbers.
• Additionally, it has been observed that the frequency of prime numbers in certain sequences related to the Golden ratio (such as the continued fraction expansion of the Golden ratio) appears to be higher than in other sequences.
• Interestingly, the Fibonacci sequence is closely related to prime numbers, as any two consecutive Fibonacci numbers are always coprime.

However, the exact nature of the relationship between primes and the Golden ratio is still an active area of research.

Another fascinating feature of this array is that any even number of–not necessarily contiguous–factors drawn from any one of the 32 angles in this modulo 120 configuration distribute products to 1(mod 120) or 49 (mod 120), along with the squares.

• We see from the graphic above that the digital roots of the Fibonacci numbers indexed to our domain (Numbers ≌ to {1,7,11,13,17,19,23,29} modulo 30) repeat palindromically every 32 digits (or 4 thirts) consisting of 16 pairs of bilateral 9 sums.

• The digital root sequence of our domain, on the other hand, repeats every 24 digits (or 3 thirts) and possesses 12 pairs of bilateral 9 sums. The entire Prime Root sequence end-to-end covering 360° has 48 pairs of bilateral 9 sums.
• And finally, the Prime Root elements themselves within the Cirque, consisting of 96 elements, has 48 pairs of bilateral sums totaling 360. Essentially, the prime number highway consists of infinitely telescoping circles …
• Also note, the digital roots of the Prime Root Set as well as the digital roots of Fibonnaci numbers and Lucas numbers (the latter not shown above) indexed to it all sum to 432 (48x9) in 360° cycles.
• The sequence involving Fibonacci digital roots repeats every 120°, and has been documented by the author on the On-Line Encyclopedia of Integer Sequences: Digital root of Fibonacci numbers indexed by natural numbers not divisible by 2, 3 or 5 (A227896).
• The four faces of our pyramid additively cascade 32 four-times triangular numbers (Note that 4 x 32 = 128 = the perimeter of the square base which has an area of 32^2 = 1024 = 2^10).
• These include Fibo1-3 equivalent 112 (rooted in T7 = 28; 28 x 4 = 112), which creates a pyramidion or capstone in our model, and 2112 (rooted in T32 = 528; 528 x 4 = 2112), which is the index number of the 1000th prime within our domain, and equals the total number of ‘elements’ used to construct the pyramid.

A thirt, in case you’re wondering, is a useful unit of measure when discussing intervals in natural numbers not divisible by 2, 3 or 5. A thirt, equivalent to one rotation around the Prime Spiral Sieve is like a mile marker on the prime number highway. If we take the Modulo 30 Prime Spiral Sieve and expand it to Modulo 360, we see that there are 12 thirts in one complete circle, or ‘cirque’ as we’ve dubbed it. Each thirt consists of 8 elements. (PrimesDemystified)

When the digital root of perfect squares is sequenced within a modulo 30 x 3 = modulo 90 horizon, beautiful symmetries in the form of period-24 palindromes are revealed, which the author has documented on the On-Line Encyclopedia of Integer Sequences as Digital root of squares of numbers not divisible by 2, 3 or 5 (A24092):

1, 4, 4, 7, 1, 1, 7, 4, 7, 1, 7, 4, 4, 7, 1, 7, 4, 7, 1, 1, 7, 4, 4, 1

In the matrix pictured below, we list the first 24 elements of our domain, take their squares, calculate the modulo 90 congruence and digital roots of each square, and display the digital root factorization dyad for each square (and map their collective bilateral 9 sum symmetry). (PrimesDemystified)

In the matrix pictured below, we list the first 24 elements of our domain, take their squares, calculate the modulo 90 congruence and digital roots of each square, and display the digital root factorization dyad for each square (and map their collective bilateral 9 sum symmetry). (PrimesDemystified)

The 13 circles of the Metatron’s cube can be seen as a diagonal axis projection of a 3-dimensional cube, as 8 corner spheres and 6 face-centered spheres. Two spheres are projected into the center from a 3-fold symmetry axis. The face-centered points represent an octahedron. Combined these 14 points represent the face-centered cubic lattice cell. (Wikipedia)

Notice that the divisions in the original square have been done according to some Fibonacci numbers: 5, 8 and 13=5+8; therefore the sides of the transformed rectangle are also Fibonacci numbers because it has been constructed additively. Now, do you guess how could we correct the dimensions of the initial square so that the above transformation into a rectangle was area-preserving? Yes, as it could not be another way round, we need to introduce the Golden Ratio! If the pieces of the square are constructed according to Golden proportions, then the area of the resulting rectangle will coincide with the area of the square. (Phi particle physics)

The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively.[ (Wikipedia)

Every repository on GitHub.com comes equipped with a section for hosting documentation, called a wiki. You can use your repository’s wiki to share long-form content about your project, such as how to use it, how you designed it, or its core principles. (GitHub)

In sum, we’re positing that Palindromagon + {9/3} Star Polygon = Regular Enneazetton.

• The significance of this ‘chain-of-events’ is that we can state with deterministic certainty that cycling the period-24 digital root dyads of both twin primes and the modulo 90 factorization sequences of numbers not divisible by 2, 3, or 5 generates an infinite progression of these complex polygons possessing stunning reflectional and translational symmetries.
• Lastly, let’s compare the above-pictured ‘enneazetton’ to an 18-gon 9-point star generated by the first three primes; 2, 3 and 5 (pictured below), and we see that they are identical, save for the number of sides (9 vs. 18). They are essentially convex and concave versions of each other.

This is geometric confirmation of the deep if not profound connection between the three twin prime distribution channels (which remember have 2, 3, and 5 encoded in their Prime Spiral Sieve angles) and the first three primes, 2, 3, and 5. (PrimesDemystified)

The terminating digits of the prime root angles (24,264,868; see illustration of Prime Spiral Sieve) when added to their reversal (86,846,242) = 111,111,110, not to mention this sequence possesses symmetries that dovetail perfectly with the prime root and Fibo sequences.

• And when you combine the terminating digit symmetries described above, capturing three rotations around the sieve in their actual sequences, you produce the ultimate combinatorial symmetry:
• The pattern of 9’s created by decomposing and summing either the digits of Fibonacci numbers indexed to the first two rotations of the spiral (a palindromic pattern {1393717997173931} that repeats every 16 Fibo index numbers) or, similarly, decomposing and summing the prime root angles.
• The decomposition works as follows (in digit sum arithmetic this would be termed summing to the digital root) of F17 (the 17th Fibonacci number) = 1597 = 1 + 5 + 9 + 7 = 22 = 2 + 2 = 4:Parsing the squares by their mod 90 congruence reveals that there are 96 perfect squares generated with each 4 * 90 = 360 degree cycle, which distribute 16 squares to each of 6 mod 90 congruence sub-sets defined as n congruent to {1, 19, 31, 49, 61, 79} forming 4 bilateral 80 sums. (PrimesDemystified)

The vortex theory of the atom was a 19th-century attempt by William Thomson (later Lord Kelvin) to explain why the atoms recently discovered by chemists came in only relatively few varieties but in very great numbers of each kind. Based on the idea of stable, knotted vortices in the ether or aether, it contributed an important mathematical legacy.

• The vortex theory of the atom was based on the observation that a stable vortex can be created in a fluid by making it into a ring with no ends. Such vortices could be sustained in the luminiferous aether, a hypothetical fluid thought at the time to pervade all of space. In the vortex theory of the atom, a chemical atom is modelled by such a vortex in the aether.
• Knots can be tied in the core of such a vortex, leading to the hypothesis that each chemical element corresponds to a different kind of knot. The simple toroidal vortex, represented by the circular “unknot” 01, was thought to represent hydrogen. Many elements had yet to be discovered, so the next knot, the trefoil knot 31, was thought to represent carbon.

However, as more elements were discovered and the periodicity of their characteristics established in the periodic table of the elements, it became clear that this could not be explained by any rational classification of knots. This, together with the discovery of subatomic particles such as the electron, led to the theory being abandoned. (Wikipedia)

This web enabled demonstration shows a polar plot of the first 20 non-trivial Riemann zeta function zeros (including Gram points) along the critical line Zeta(1/2+it) for real values of t running from 0 to 50. The consecutively labeled zeros have 50 red plot points between each, with zeros identified by concentric magenta rings scaled to show the relative distance between their values of t. Gram’s law states that the curve usually crosses the real axis once between zeros. (TheoryOfEverything)

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

Mathematically, they specify the decomposition of the tensor product of two irreducible representations into a direct sum of irreducible representations, where the type and the multiplicities of these irreducible representations are known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912), who encountered an equivalent problem in invariant theory.

Generalization to SU(3) of Clebsch–Gordan coefficients is useful because of their utility in characterizing hadronic decays, where a flavor-SU(3) symmetry exists (the eightfold way) that connects the three light quarks: up, down, and strange. (Wikipedia)

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

From what we learned above about segregating twin prime candidates, we can demonstrate that they compile additively in perfect progression, completing an infinite sequence of circles (multiples of 30 and 360)

The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner’s Mathematical Games column in Scientific American a short time later.

Both Ulam and Gardner noted that the existence of such prominent lines is not unexpected, as lines in the spiral correspond to quadratic polynomials, and certain such polynomials, such as Euler’s prime-generating polynomial x²-x+41, are believed to produce a high density of prime numbers. Nevertheless, the Ulam spiral is connected with major unsolved problems in number theory such as Landau’s problems (Wikipedia).

The published diagram by Feynman helped scientists track particle movements in illustrations and visual equations rather than verbose explanations. What seemed almost improbable at the time is now one of the greatest explanations of particle physics — the squiggly lines, diagrams, arrows, quarks, and cartoonish figures are now the established nomenclature and visual story that students, scientists, and readers will see when they learn about this field of science. (medium.com)