SORTED DIAGONALS

The diagonals on the logically outfitted triple-cube naturally sort themselves into a set of 17 interior (non-surface) diagonals,

and a set of 20 exterior-interior surface diagonals,

which sets in relation to the entire 37 diagonals equal, respectively, the area of a circle whose diameter is the inverse 8th root of a hundred times the electromagnetic coupling constant (exact to the third place and the fourth rounded) and the area of a circle which is the 8th root of the fine structure constant (exact to the fourth place and the fifth rounded).

On September 12, 2007, Mr. J. DePompeo communicated the following with respect to the NIST CODATA 2006 Inverse Fine Structure Constant 137.035999679. . ..

Where 165888 (= 2 × 82944) is the total area of the interior and exterior faces of the logically outfitted absolute dead center triple-cube, 37 the total number of triple-cube diagonals, 25 = 37 − 12 (the total 37 triple-cube diagonals minus the 12 interior diagonals end-connected at the midpoint of a sub-cube edge), 32 = 37 − 5 (the 37 triple-cube diagonals minus the 1 interior not end-connected diagonal together with the 4 interior diagonals end-connected at the vertex of a sub-cube), and the exponent 3 = the number of sub-cubes,

25/321/2 × 8294425/37/165888 × π= 137.035999644. . . ,

accurate to 2.5 parts per 10 billion and thus within the 6.8 parts per 10 billion of the relative standard uncertainty of the NIST 2006 value, and accurate to at least 10 digits.

1.  On October 1, 2000, Mr. J. Iuliano called the writer's attention to the fact that 10-8 times the ratio of  the entire set of 37 diagonals to the 17 interior diagonals, 37/17 × 10-8, ~ 2.1767 × 10-8, the value in kilograms of the Planck mass.   On January 4, 2001, he further noted that, where gs, gw, a, and G are the values of the four fundamental force constants, they are related to that of the Planck mass by the ratio, 37/57, as follows: (37/57)(gs gw)/G)/1016 ~ 2.1767 × 10-8.

2.  On October 14, 2000, Mr. Iuliano noted, with respect to 17 and 37,  that (where the 1998 CODATA values of the electron and proton masses are, respectively, 9.10938188 × 10-31 kg and 1.67262158 × 10-27 kg) mp - me = [([37×17×5]2/2)-18]-4.

3.  On October 29, 2000, Mr. Iuliano called attention to the role of the ratio 17/37 in the following equation where G is the value of the gravitational constant, 6.673 × 10-11, e0 is the value of the electric constant, 8.854187817 × 10-12, e is the value of the elementary charge, 1.602176462 × 10-19, h is the value of the Planck constant, 6.62606876 × 10-34, 8π is the geometric constant in the Einstein field equation Gμν = 8πTμν, and c is value of the speed of light, 299792458:

(17/37)2 × 8π × ε02 × c5 × h × 102 ~ G.

Recognizing that the fine structure constant is defined by the identity (e2α-1)/(2ε0hc) = 1, Mr. Iuliano multiplied this identity into the left side of the above gravitational equation to produce the following alternative equation, where α-1 is the inverse fine structure constant, 137.03599976:

(17/37)2 × 4π × ε0 × c4 × e2 × α-1 × 102 ~ G.

Where Gμν is the space-time metric and Tμν is the energy-momentum tensor, converting Einstein's cosmological field equation Gμν = 8πTμν, the following substitutions may be made:

(17/37)2 × Gμν /Tμν × ε02 × c5 × h × 102 ~ G,

and

(17/37)2 × Gμν /2Tμν × ε0 × c4 × e2 × a-1 × 102 ~ G.

With respect to the Einstein field equation,

the writer notes that

G = (81/2/10)16/[1+(829440 × 9)-1/2].

Further for Einstein's field equation and the absolute dead center cube, see, on this web, The Magnitude of Being.

4.  On January 29, 2001, Mr. J. DePompeo called the writer's attention to the fact that the product of the odd prime factorial of 17 and the 37th prime equals, with a difference less than 1 meter, the value of the measure of the equatorial circumference of the earth, 40075035.5351 m, as estimated by the Year 2000 International Astronomical Union.  This equatorial circumference constitutes a bulge in the shape of the earth.  The gravitational attraction of the moon and sun on this bulge causes a precession of the earth's rotation axis.  On February 6, 2001, Mr. DePompeo communicated his discovery that when the natural numbers are mapped one to one with the odd primes, 6 corresponds to 17 and 36 to the 37th prime, and, analogously to the above product of the corresponding primes, the product of factorial 6 and 36 equals (rounding 71.6 years per degree to 72) the number of years of the precession of the earth's rotation axis, 25920.  He noted that 25920 = 144π radiansHe also noted that the rational product of Genesis 1:2, והארץ היתה תהו ובהו, "And the earth was without form and void," equals 25920.  For the definition of the term 'rational product', see, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product & Related Terms.   For the fundamental role of 6 and 36 in the logic of the thinking now occurring, cf. D.G. Leahy, Foundation: Matter the Body Itself (Albany, 1996), III.1, pp. 272ff., et passim.  Note also that the product of 6 and 36 equals the sum of 6, 36, 17, and the 37th prime.  For 2592 and the logic, cf. ibid., p. 277, n. 30, Section III.5, pp. 435ff., and Section III.7, p. 447, also, below, Addendum 11.   For 144π as unique value of the absolute dead center cube, cf., ibid., Section IV.2, p. 524, n. 98, and p. 525, n. 99.

7.  On April 4, 2001, Mr. DePompeo communicated the following observations: where the 2000 IAU polar circumference of the earth CearthPR = 39940671.42790 m, and one polar kilometer, defined as CearthPR/40000, = 998.51678 m, then the volume V of a sphere whose radius is this length in centimeters minus the equivalent in centimeters of 17(37/100) inches = 4π/3(100CearthPR/40000-17(37/100)2.54 cm)3 = 1.609344001 × 105 cm3 = 1 mi3, with a difference less than one in one million.  Since 17(37/100) inches = ft and the metric value of the Egyptian royal cubit ercm = .524 m, then the equivalent of the preceding formula is V = 4π/3[100CearthPR/40000-2.54(ercm/m ft) cm]3 = 1 mi3.  (When ercm = .52309 m, the equation is exact.)  Further, where φ is the golden section ratio, and the circumference of the sun in miles ~ 3×17×37×1439+j2 (cf., above, Addendum 6), the circumference of the sun in meters, ~ (3×17×37×1439+φ2)×1.609344001×103, =, with a difference less than one in a million, the edge of a cube whose volume Vcube = the volume of the sphere Vsphere = 4π/3[(CearthPR/40000-.0254[ercm/m ft])(3×17×37×1439+φ2) m]3 = 8.345377643 × 1028 m.  Finally, where the 2000 IAU polar radius of the earth rearthPR = 6356.755288 km, the volume Vsphere of the sphere whose radius this is = 4π/3(rearthPR)3 = the cube volume Vcube = 10247.03838 km3 = 6367.214453 mi3.  When the natural numbers are mapped to the odd primes (see, above, Addendum 4), 6367 is the odd prime corresponding to 829, while 829 in turn is the odd prime corresponding to 144.  The writer notes that the kilometric value of the polar radius, 6356.755288, with a difference less than one in one million, = 5280/[.82944(1+21/2×10-3)], and the related cube edge in miles, 6367.214453, with a similar difference, = 5280[1+(8.2944 × 10-30)1/8]/.82944.

8.  On April 12, 2001, Mr. Iuliano further observed (cf., above, Addendum 6) that ercin8/(3 × 1010) ~ 39.37/36, the meter/yard conversion factor in inches, and, with respect to the operative indifference of metric and English measures, that, where α is the fine structure constant and β is the critical exponent .37 (see, on this web, Rubidium (37), Lanthanum (57) & the Logically Outfitted Cube(s), notes 1 and 3), the numerical value of ercin = [5280(cmi-1/2β) ×1010/cm]1/8 (this, the writer notes, with a difference less than one in one million, when the 1998 CODATA values for the physical constants are used).  Mr. Iuliano also notes that α-1/2β ~ (-10 × COS ercin)2 ~ 102/p, where the last expression is the inverse of the factor common to the strong and weak force exponents of 82944 (see, on this web, 82944 & the Four Fundamental Forces & the God Particle, notes 9 and 17, et passim).

9.  The logical underpinning of the megalithic-Egyptian-English-metric indifference noted above, Addenda 5, 6, et passim, may be found in the infinite iterative sequential summing of the real trinary logic geometric series, which summing spontaneously produces the geometric series of ordinary mathematics in the form of an infinitely expanding triangle of logical digits, which expanding triangle is partially composed of a series of geometrically proportionate smaller triangles that share its descending edge (see, on this web, Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing).  At the infinite limit the ratio of the total arithmetic value of the logical digits in the major triangle to the number of digits = , and the ratio of the total arithmetic value of the digits in the edge-sharing triangles to the corresponding value of the major triangle = .  Let Thom's megalithic yard = 2.721999213 feet, and the ratio of centimeters to inches = 2.54 (exact), then, where MYft is the megalithic yard, MYft = [1//10-3/2.5423]23(223) - 22(3) feet.  When MYft = Thom's 2.722 feet, its metric equivalent MYmm = 829+/ mm, with a difference less than 1  in 10 million.  For the comparison of real trinary logic and the alternative duality-based ternary logic, see, on this web, Quantum Gravitational vs. Quantum Logic: Virtually Left-handed Real Trinary Logic.  Note that the real trinary logic geometric series unifies prehistoric measure with the most recent technologically enabled high-precision astronomical calculations.  For instance, with respect to the trinary logic edge-sharing/infinite triangle arithmetic value ratio, , and the total arithmetic value/number of digits ratio, , when the high-precision mean mass of the earth mearth = 5.97223 × 1024 kg, and that of the sun msun = 1.98843 × 1030 kg (cf., above, Addendum 6, Physics News Update 482, May 3, 2000), then, with a difference less than 1 in 1 million, 2 × (1+8/1000) = msun/mearth.  The difference less than one in a million, with a further difference that, with a yet further difference ~ (144π × 2808/3600 × 23 × 28)-1, = 1.0368-28, = (8.2944×10-23/784)1/4, where 784 is the only number other than 1 and 82944 that is its integral product (cf. Leahy, Foundation, III.6 and III.7).  Also, where the 2000 IAU polar circumference of the earth CearthPR = 39940.67142790 km (cf., above, Addendum 7), and one polar kilometer = CearthPR/40000 = .99851678 km, .99851678 km/1.5 = / km, with a difference less than one in a quarter million.  With a difference less than one and four tenths in a million / = (144π×103)-1/32Where 6 is the number of absolute dead center cube faces and 144π the circumference of the circle circumscribing each, 6(144π×103) =, with a difference ~ (8.5/3.5)-16 (cf., above, Addendum 6), 2715395.61769, the circumference of the sun in miles (cf., above, ibid. and Addendum 7).  Lastly, note that Petrie's mean height of the Great Pyramid (cf., above, Addendum 5, The Pyramids and Temples of Gizeh) is 5776 inches, which, where (×1094)-1/32 ~ the inverse square of the diagonal of one square Egyptian royal cubit in inches (cf., above, Addenda 5ff.), =, with a difference less than one and five tenths in a million, × (1+[×1094]-1/32) × 104 inches.  Where h is Petrie's mean height of the Great Pyramid in feet, 481, and b the mean base in feet, 755, then, with a difference less than 1 in 100,000,000, h/b(1+1/h.5) = /(1+[//1097]1/32).   Where p is the mean perimeter in feet, 3021, and φ the golden section ratio, then, with a difference less than one in a million, hp/[1+(h/b)32] = (8.2944×10-4-32)-2.   Further for , , and the ratio of one to the other, see, on this web, The Langerhans-Epidermal Cell Ratio & the Structure of the Epidermis and Theorem for a Flat Universe: The ρcrit = 3H02/8πG Building Block 1000.125.

10.  With regard to the lengths of the diagonal sets in the logically outfitted triple dead center cube, on January 5, 2003, Mr. DePompeo called attention to the following: the ratio of end-connected to non-end-connected diagonals, 36:1 (see above, Addendum 6) is, qua total lengths, 6753.593084288 : 144 ~ 46.9 (~ φ4+φ × π), where 46.9 ~ the sum of the range of acute angles in the obliquity of the ecliptic (the angle between the plane of the earth's orbit and that of the celestial equator).   Further, Mr. DePompeo points out that, where l37 (= 6897.593084288) is the total length of the total number of diagonals in the logically outfitted dead center cube, l36 (= 6753.593084288) that of the total number of end-connected diagonals, l20 (= 3591.975155039) that of the total number of yellow & green diagonals, l17 (= 3305.617929248) that of the total number of blue diagonals, l16 (= 3161.617929248) that of the total number of end-connected blue diagonals, and l1 (= 144) that of the non-end-connected blue diagonal, if one subtracts from one of these angles (which is within 1/2 minute of the tilt of the earth) the rational product of the lengths l37 , l36 , l20, l17 , l16 , and l1, each raised respectively to the inverse power of the corresponding number of diagonals, then one obtains the obliquity of the ecliptic accurate to .02 second of tilt––23.439298 vs. 23.439291:

(l36 : l1)/2 (l371/37/l361/36 × l201/20/l171/17 × l161/16/l11/1) = 23.439298.

The division 37/36/20/17/16/1 perfectly parallels the division of the actual lengths of the diagonals: where 1889 is the 289th (= 172) odd prime, 1.889 × 10-4 ~ 37/36/20/17/16/1 ~ 172/(1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17)/104 (the last divisor the "created 100"; see, above, Addendum 5), while 1.889 × 10-13 ~ 6897.593084/6753.593084/3591.975155/3305.617929/3161.617929/144.  Finally, Mr. DePompeo notes that 10-13/(37/36/20/17/16/1)/10 = 5.293 × 10-11 ~ 5.292 × 10-11, the radius of the Bohr hydrogen atom in meters.

11. For Mr. DePompeo's communication with the writer on November 1, 2005, see, on this web, Maxwell's Constant and the Trinary Logic Triple-cube.

12. On December 15, 2005, Mr. DePompeo called to the writer's attention the following: when the diagonal of the square face of the logically outfitted triple-cube = 1, the perimeter of each of the 8 triangles that lie on its non-square faces, as well as that of each of the 4 interior triangles that reflect these exterior surface diagonals, = φ2.  Where 12 is the number of overall edges of the triple-cube, the total length of the perimeters of these 8 exterior and 4 interior triangles = 12φ2 ~ 10π.  Where the diagonal of the square face of the triple-cube = 144 the diagonal of the face of the absolute dead center cube (cf. Leahy, Foundation IV.2) the total length of the perimeters of these 8 exterior and 4 interior triangles = 123 × φ2.  In this case the total area of these 8 exterior and 4 interior triangles = 62208 the body surface area  of the absolute dead center cube (ibid.).

13. On September 23, 2006, Mr. DePompeo called attention to the following: where 1/137.03599911 is the fraction of the velocity of light in vacuo at which a first generation lepton, or electron, orbits the nucleus of an atom in the first Bohr orbit, π/81/2 the ratio of the length of the circumference of a circle to the length of the square inscribed within it, 82944 the sum of the surface area + the area of the two interior planes of the absolute dead center cube, and 8.88024333 the NIST 2002 proton/muon mass ratio, where a muon is a second generation lepton,

100(π/81/2 × 829441/82944/8.88024333)3 = 137.03599947,

the NIST 2002 inverse fine structure constant accurate to within 3 parts in one billion.

14. For Mr. DePompeo's communication with the writer in September and October 2010, see, on this web, Mathematical & Physical Constants.

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