Triple Cube Diagonal Logic

The following is an edited selection of material forwarded by Mr. John DePompeo during September and October 2010. Here and there relevant illustrations have been added.

“Gauss is said to have stated that whoever did not immediately see the truth of Euler's identity did not have the makings of a great mathematician. Coincidentally, over the last year and in the weeks immediately preceding Steven Hoath's email,1 I engaged in an examination of Euler's identity in the form

and its near identity in relation to a number of fundamental mathematical and physical constants, and the mathematics of real trinary logic. I will start with the remarkable near identity

accurate to nearly 10 digits, where K = Catalan's constant, α_F is Feigenbaum's α constant, and δ_F is Feigenbaum's δ constant. Catalan's constant equals Dirichlet Beta 2, the infinite alternating sum of the inverse square odd numbers, i.e., the infinite alternating sum of the inverse square centered numbers, corresponding to the alternating sum of the squares in the following figure:2

[where 17, the number of interior diagonals in the real trinary logic triple-cube whose end points are logical 0’s (see illustration below, the blue diagonals), is the number of places constituting the edge of the ninefold square here illustrated (analogous to the ninefold structure of the real trinary logic Cornerstone that distributes the threefold repetition of the three logical digits, 0,

, 1) the area of which ninefold square = 17² = 1 + 82944^1/2 — editor’s insert.]3

“Catalan exists in a near identity with the golden ratio, the Delian constant of the classical problem of doubling the cube (cube root of 2), and the "golden ratio" of exponentials, the Omega constant, which is the infinite power tower of the inverse of e, the base of the natural logarithms,

“Even more directly related to the 37/144 diagonal logic of the triple absolute dead center cube,4

Euler's identity derives the near identity of the fourth power of 37 through the Feigenbaum δ and Graham's biggest little hexagon area Α

which calculates Feigenbaum δ accurate to 10 digits when 37 is exact, where A, is the area of the pentagram inscribed hexagon of polygon diameter one:5

“37⁴ is the denominator of the 4th power of the ratio of the highest and lowest eigenvalues of the least prime magic square:6

The matrix of the least prime magic square excludes the only even prime to preserve parity, includes unity in its place, its center number equals 37, the number of diagonals of the trinary logic triple absolute dead center cube, its "magic sum" equals 111, analogous to the 111 diagonal of the "cornerstone" of real trinary logic, the arithmetic mean of any row, column or diagonal equals 37, and the total sum of the square equals 333 = 9 times 37.

What is particularly striking about the square matrix is that its corresponding eigenvalues equal respectively

where 36√2 equals the inradius of the "absolute dead center cube," and the fourth power of the ratio of the highest and lowest of these eigenvalues equals

[Note that where x = edge length of the absolute dead center cube, (37[36√2/111])⁴ = ([37x/2]/111)⁴ = (x/6)⁴ = 82944 — editor’s insert.]7

“The fourth power of the ratio of the highest and lowest eigenvalues of the least prime square is the ratio of the 82944 V/BSA hypercube [(x/6)⁴] and the fourth power of 37—which is to say 37 can be put in a perfect ratio with the edge of the "absolute dead center" cube through the eigenvalues of the least prime square.

The determinant of the matrix = λ₁ × λ₂ × λ₃ = -287712 = √82944 - √82944000000.

where 15984 equals 3 × 37 × 144, and every row, column and diagonal of the factored inverse matrix equals 144, neatly identifying 144 and 37 through the least square equal essence of the prime numbers. This is the only exact identification of 37 and 144 of which I am aware.

“Observe the re-appearance of the number 144 through the ratio of interior to total diagonals of the trinary logic triple absolute dead center cube, and the mapping of the natural numbers to the prime numbers.

P₁₄₄ is the 144th odd prime number.8 This and the above least prime square 37/144 relation confirms indirectly through reciprocity and natural/prime mapping the relation of 37 and 144.”

Near Identities of the 37/144 Relations to Important Mathematical & Physical Constants

“The NIST 2006 CODATA value 137.03599967 is in error due to a miscalculation on the part of the researches, and has since been readjusted to 137.035999084 accurate to .37 ppb. The NIST value will not be updated until the end of 2010 or the beginning of 2011, and will presumably take account of the corrected value.

“Of all relationships between the fine structure constant and real trinary logic one of the most striking in simplicity although not accuracy shows the fine structure constant to be a function of 144 and the ratio of the π and e roots of e:

The identical equation in the following form reflects the 12 fermions of matter of the standard model and their 1/2 integer spin

This value can be compared to a more complex exponential version which incorporates the fine structure constant, the proton/electron mass ratio, the neutron/electron mass ratio, and e, π, and Euler γ:

where 1836.1526725 and 1838.683661 are respectively the NIST 2006 proton/electron and neutron/electron mass ratios. Setting α exact the Harvard value yields 144 accurate to 9 digits

The accuracy value of 137.0360008 is close to the cosine 137.03600099 equation incorporating π and 82944,9

as well as to an analogous equation incorporating 82944, the total/interior/surface 37/17/20 diagonal structure of the trinary logic triple absolute dead center cube, and Feigenbaum α

The last two equations approximate a value of the fine structure constant accurate to each other to 1/2 ppb. They also equal to comparable accuracy another equation a bit more complex which incorporates the number of the 20 amino acids and 24 chromosomes of the human genome

The ratio of chromosomes to amino acids 24/20 = 1.2 ~ 1.19998 = one half the golden angle in radians = π/φ². Note also the relation between e and the golden ratio

There exists an identity accurate to 5 ppt between Euler's identity, the plastic or radiant constant, and the number of the 64 codons and the 24 chromosomes of the human genome, as well as the number of the 23 chromosomes unique to each human organism:

and thus appears to possess unique relations to cubicity and hypercubicity. The ratio of the plastic constant P and π equals the twin prime constant raised to the power of log 8 accurate to 9 digits

and the twin prime constant and π are related to 82944 raised to the total/interior/surface diagonal ratio of the trinary logic triple absolute dead center cube above

To return to the fine structure constant there is an elegant equation incorporating 37, Euler's identity, e, π, and Catalan's constant which approximates the Rydberg value of the second fine structure constant above to high degree of accuracy

Amazingly, 37, π, and e estimate to 9 digit accuracy through two analogous equations of great simplicity, the g/2 electron magnetic moment

[Where the sum of the three unique natural numbers that are their integral products 1+784+82944 = 83729, (2+8.3729e10^.125/1e4)/2 = 1.00115965 = g/2 accurate to 4.6 ppb. — editor’s insert.]10

Finally, using the complete sum of the least prime magic square, and the exponential integral taking the value of unity, it is possible to estimate the anomalous magnetic moment g-2 accurate to nearly 11 digits

In terms of the sum of the rows and columns of that square where cube root of 2 is the Delian constant

It is also possible to estimate the fine structure constant using the NIST 2006 proton/electron mass ratio, 37, π, and the square root of e and 2

“I have been able to estimate the fine structure constant to within the relative standard uncertainty of .37 ppb of the Harvard 2008 value using 144 and 37.”

4-D Analogue of the Volume of the Trinary Logic Triple Absolute Dead Center Cube.

“I discovered a new trinary logic triple absolute dead center hypercube fine structure identity, which accurately reflects in value the limit of the high precision QED state of the art measurement of the researchers at Harvard University. Whether the equation possesses scientific merit I cannot say. Nonetheless, it is an immensely satisfying discovery, culminating years of inquiry into the matter.

“The estimate below uses the volume of 3 absolute dead center hypercubes as a 4-D analogue of the volume of the trinary logic triple absolute dead center cube.11

“The novelty of the discovery is the appearance of a simple cubic lattice sum over infinitely expanding cubes, the so-called Madelung constant, which independent of its purely mathematical significance, is used to determine the electrostatic potential at the origin of a cubic crystal produced by unit charges at all nonzero lattice points, in particular the sodium chloride crystal, a unit cell of atoms of which corresponds in number to the number of logical digits of the logically outfitted absolute dead center cube (1/3^rd the triple cube):12

“Sodium Chloride is a face-centered cubic crystal. The sum over the lattice equals -M or M depending upon whether one is summing over Sodium or Chloride ions and it equals = −1.747564594633182 ... = M =

“The prime mark indicates summation over (0, 0, 0) is excluded. It is an infinite alternating sum from negative to positive infinity excluding zero, over expanding cubes. The equation incorporating it which derives the fine structure constant, where π equals the area of the unit disk and the circumference/diameter ratio, 4π equals the surface area of the unit sphere, and eⁱ is Euler's identity in the form (−1)^1/^π,

calculates the most precise estimate of the fine structure constant to date, the Harvard 2008 value exact to 12 digits, and 3 parts per trillion. Since the value of the sum of the series is a negative value, it is necessary to preface M with a negative sign to avoid inserting the imaginary unit outside the square root sign.

The following form where M is squared avoids the necessity of taking the square root of a negative number

The value of the sum of the series is essentially the alternating sum of the inverses of the lengths of the body/space diagonals of the cubes of the whole numbers from minus infinity to plus infinity, with the square root of 3—the body diagonal of the unit cube—the archetype.”

“Using Feigenbaum's α and δ constants, the prime zeta function of inverse squares P(2),13 and the Backhouse constant C_B,14 it is possible to estimate the arithmetic mean length of the 37 diagonals of the trinary logic triple absolute dead center cube accurate to nearly 10 places

“Where Δ is the arithmetic mean length of the 37 diagonals of the triple logic cube; β equals the inverse prime coefficient power series Backhouse constant; and α equals Feigenbaum α, δ equals Feigenbaum δ accurate 10 digits, and 75 parts per trillion

“P(2) is the simplest converging sum of the primes. There are a number of other constants derived from prime polynomials, the Backhouse constant appears to be the simplest power series. In it the natural numbers and the prime numbers are mapped one to one through the coefficients and powers of x. The appearance of the prime numbers here is certainly consistent with the 37/144 least prime square/inverse identity.

“What is striking is that the discreteness/discontinuity of the minimum square equal prime number essence identity of 37 and 144 when mapped to the logic of the triple logic cube derives the two fundamental constants of chaos, and two fundamental discrete/continuous constants of the prime numbers.

“The distribution of prime numbers also possesses fractal properties, and there have been a number of papers and books published in the last two decades exploring those properties, although I am unaware of any which have been able to show a high correlation between the constants of traditional (Feigenbaum) chaos, and the distribution of the prime numbers. The above equation demonstrates that the structure of the triple logic cube is a key to understanding a possible relation. It would require some pretty advanced tools of analysis to show step by step how to arrive at the above equation. It is necessary to either to take the absolute value of Feigenbaum α

“There is another equation which estimates Feigenbaum α through a Euler product of the prime numbers and the diagonals accurate to 9 places rounded, where C_A is Artin’s constant,15

“Artin's constant A,16 and Feigenbaum δ appear in an estimate of the value of the inverse L series accurate to 8 places

The Absolute Dead Center V/BSA Hypercube and Quantum Chaos

In December 2010, Mr. DePompeo forwarded the following additional material [edited as above].

“I have had the opportunity to give a little more thought to the relationship between the structure of the equation of the integral product founded absolute dead center triple logic hypercube, quantum chaos, and the least prime polynomial inverse square exponential identity of the prime numbers.

Note that, where Pn is the nth odd prime, e₄ = , and δ^|α|+1 = δδ^|α| = ΔP^P(2), the discreteness of the unique numbers, imaginary i, integer 137 of the fine structure constant, the triple logic cube diagonal/interior prime numbers 37 and 17, indexing the prime numbers, and even prime number 2, derive to an accuracy of 12 ppt the product of the length of the edge of the absolute dead center V/BSA hypercube (= ) and quantum chaos, the latter here defined as the product of fine structure constant α (= the ratio of the velocity of the electron orbit of the proton of the Bohr model of the hydrogen atom to the velocity of light), the mass ratio of the proton and electron of that atom μ, and the Feigenbaum δ and α universal constants of mathematical chaos.¹⁸

that is,

where the left side of the quantum equation is a power of the circular function secant π/274 which in its imaginary form identifies the maximal quadratic polynomial prime producing discriminant of imaginary quadratic field of class number one Q( ), and the one to one mapping of the natural and prime numbers through the inner/diagonal logic of the triple cube.

The polynomial of Q( ) produces the maximum number of consecutive primes, which neatly calculates the inverse prime coefficient polynomial power series of infinite degree, the inverse squares of the primes, and the arithmetic mean diagonal of the triple logic cube with the maximal prime producing quadratic polynomial

“In the equation above when μ = 1836.15267247 = the current CODATA recommended value of the proton/electron mass ratio, α = 1/137.03599908152 accurate 12 ppt to 1/137.035999084 = the most accurate estimate of the fine structure constant to date. Since

δδ^α = ΔP^P⁽²⁾

are accurate each to the other to within 1/3 ppb, and the fine structure and mu measured to an accuracy of around 0.5 ppb, the right side of the equation in which Δ is the arithmetic mean length of the 37 diagonals and P = the inverse prime coefficient power series Backhouse constant, and P(2) is the Prime Zeta function of the sum of the inverse prime squares, is also within the error uncertainty of the adopted constant values.^"19

The Unique Natural Number/Integral Product Ratio and the Fine Structure Constant

“Where CNR is the nested radical constant

DeP0101208a ,

G_Ga is Gauss' constant, the arithmetic-geometric mean of unity and the square root of two, 7 and 9 are, respectively, the number of logical planes, and logical digits on each plane, which determine the direction of the diagonal vectors of the triple logic cube, and 82944/784 = the ratio of the volume of the absolute dead center hypercube to its integral product, 107495424/1016064,²⁰

accurate to 50 ppq (parts per quadrillion).”

1 Hoath's email dated August 29, 2010 to Leahy, in which, referring to Euler's equation eⁱ^π + 1 = 0, he asked, ". . . [t]he form of the equation, examined according to the rules of [real] trinary logic, suggests that the quantity eⁱ^π must equal . Right?" For the writer's response, see, on this web, Real Trinary Logic, the Imaginary Number i, & the Cyclic Series of Complex Numbers.

2 See, on this web, The v_maxwell^1/16 × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe.

3 See, on this web, Measure Beyond Beyond Reach. Cf. D.G. Leahy, The Cube Unlike All Others (CreateSpace, 2010), Section I.2.

4 For the images following, cf. ibid., Section II.1. For the edge length of the absolute dead center cube x = 10368^1/2, assumed here and throughout, cf. ibid., Section I.1. See, on this web, Logically Outfitted Cube(s) and Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube.

5 Cf. Wolfram Mathworld: “Graham's biggest little hexagon is the largest possible (not necessarily regular) convex hexagon with polygon diameter 1 (i.e., for which no two of the vertices are more than unit distance apart). It is therefore the biggest little polygon for the case n = 6.” Available online at: http://mathworld. wolfram.com. October 2010.

6 Cf. ibid.: “the 3 × 3 prime magic square (containing a 1) having the smallest possible magic constant . . . discovered by Dudeney in 1917.”

7 See, on this web, Measure Beyond Beyond Reach. Cf. Leahy, The Cube Unlike All Others, Section I.2. For the inradius of the absolute dead center cube, see, on this web, Epitome or Food for Thought.

8 The 144^th odd prime is 829. Qua rational products 829 = 82944. 8/2 × 9 = 8/2 × 9/4 × 4 = 36. See, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product & Related Terms. Qua integral products 829 × 82944 = 72² × 288² = 5184 × 82944 = 144⁴ = 429981696 the volume of the hypercube whose integral product 107495424 is the volume of the absolute dead center hypercube. Qua linear products 829 × 82944 = (8 × 2 × 9) × (8 × 2 × 9 × 4 × 4) = 144 × 2304 = 331776 at once the integral product of—and the V/BSA hypercube volume of—the volume 2985984 (= 36 × 82944) of the base cube of the hypercube whose volume = 144⁴ = 429981696 and whose hypercube boundary volume = 82944 × √82844 = 23887872. Where 3.41667 is the condition number of the least prime magic square,

See, on this web, Measure Beyond Beyond Reach. Cf. Leahy, The Cube Unlike All Others, Section I.2.

9 See, on this web, 82944 & the Four Fundamental Forces & the God Particle.

10 For 1+784+82944 = 83729, cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), pp. 525f. For g/2 = 1.00115965218073, cf. G. Gabrielse et al., "New Measurement of the Electron Magnetic Moment and the Fine Structure Constant." Available at http://arxiv.org/PS_cache/arxiv/pdf/0801/0801.1134v2.pdf. October 2010.

11 For the absolute dead center hypercube, see, on this web, Measure Beyond Beyond Reach. Cf. Leahy, The Cube Unlike All Others, Section I.2.

12 For the first image following, cf. ibid., Section II.1. See, on this web, Logically Outfitted Cube(s). For the second image, cf. Interactive Learning Paradigms Incorporated. Available online at http://www.ilpi.com. October 2010.

13 The infinite sum of the inverse square prime number

14 The Backhouse constant is derived from the prime coefficient power series P(x) =

Letting Q(x) = 1/P(x) =

the limit of the absolute value of the ratio of the coefficients of Q(x) as the power of x approaches infinity equals the Backhouse constant.

15 Artin's constant is

¹⁶ Ibid.

¹⁷ Cf. Wolfram Alpha. Available online at: http://www.wolframalpha.com. October 2010. See also, on this web, The Magnitude of Being. Cf. Leahy, The Cube Unlike All Others, Section I.3.

¹⁸ Note when α appears in the exponent it equals the absolute value of Feigenbaum α, otherwise it equals the fine structure constant. For the edge of the absolute dead center V/BSA hypercube = , see, on this web, Measure Beyond Beyond Reach. Cf., also, Leahy, The Cube Unlike All Others, Section I.2. 1/μ is the ratio at which the rotational, vibrational, and electronic levels of an atom scale. With regard to α and μ, E. J. Salumbides writes in Laser precision metrology for probing variation of fundamental constants (2009): "As has been pointed out originally by Max Born, the fine structure constant and the proton-to-electron mass ratio μ = M_p/m_e are the only two dimensionless parameters that are needed for the description of the gross structure of atomic and molecular systems."

¹⁹ See, above, "The Arithmetic Mean Length of the 37 Diagonals." Note in the preceding equation the occurrence of the imaginary unit exponent of the 37^th odd prime = 163 = the 9th and final Heegner number corresponding to the imaginary quadratic field

.
of maximal discriminant. Compare to , on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube. Also, Leahy, The Cube Unlike All Others, Section I.1. Note also the relationship between Feigenbaum δ and 163

DeP0101208f

Cf. http://marvin.sn.schule.de/~inftreff/modul33/task33.htm (equation 49).

²⁰ For the logically outfitted triple dead center cube, see, above, 2^nd illustration, and note 4. For the unique natural number/integral product ratio of the absolute dead center hypercube, 107495424/1016064 = 82944/784, see, on this web, Measure Beyond Beyond Reach. Cf., also, Leahy, The Cube Unlike All Others, Section I.2. Cf., also, above, note 8.

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