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).

Addendum 2007

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,

25/32^{1/2 }
× 82944^{25/37/165888} × π^{3 }=
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.

Addenda 2000-2006

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 *g*_{s},* g*_{w}, 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)(*g*_{s} *g*_{w})/(α*G*)/10^{16}
~ 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) *m*_{p} - *m*_{e} = [([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}, e_{0}
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 *G*_{μν}
= 8π*T*_{μν}, and *c*
is value of the speed of light, 299792458:

(17/37)^{2} ×
8π
× ε_{0}^{2} × *c*^{5} × *h* × 10^{2}
~ *G.*

Recognizing that the fine structure constant is defined by the identity (*e*^{2}α^{-1})/(2ε_{0}*hc*)
= 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
× ε_{0} ×
*c*^{4} × *e*^{2} × α^{-1} × 10^{2}
~ *G.*

Where *G*_{μν} is the space-time
metric and *T*_{μν} is the energy-momentum
tensor, converting Einstein's cosmological field equation *G*_{μν} =
8π*T*_{μν}, the following substitutions may be made:

(17/37)^{2} × *G*_{μν} /*T*_{μν} × ε_{0}^{2} × *c*^{5} × *h*
× 10^{2} ~ *G*,

and

(17/37)^{2} × *G*_{μν} /2*T*_{μν} × ε_{0} ×
*c*^{4} × *e*^{2} × a^{-1}
× 10^{2} ~ *G.*

With respect to the Einstein field equation,

the writer notes that

8π*G* = (8^{1/2}/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 radians. He 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,

5. On February 14, 2001, Mr. DePompeo noted that the
equatorial radius of the earth to the nearest mile is 3963 miles, where 3963 is the
product of the odd primes 3 and 1321 (mapped, as above, to 1 and the product of 6 and 36),
and that, where *c* is the value of the velocity of light to the nearest kilometer,
the volume in cubic miles of the sphere whose radius exceeds 3963 miles by about a quarter
of an inch equals the sum of 144π*c* and 8.2944π × 10^{10}. Where the kilometric value of *c*
is exact the radius exceeds 3963 miles by less than a third of an inch. The writer
notes that since the volume of the sphere whose radius is the equatorial radius of the
earth V_{earthER} ~ 8.2944π × 10^{10} cubic
miles, its volumetric radius *r*^{3}_{earthER} ~ 6.2208 × 10^{10 }miles, where 82944 and 62208 are, respectively,
the area of the unitary structure of the dead center cube and that of its surface (cf. Leahy, *Foundation*, pp. 434ff.; for
the occurrence of the identical set of base factors in Genesis 1:3, see, on this web,
**82944 & the Four Fundamental Forces
& the God Particle, note 3**). Coincidentally, also on
February 14, Mr. Iuliano communicated his observation that, where *c* is the metric
value of the speed of light, and the Egyptian royal cubit ~ 20.63 inches (Newton's
estimate, corroborated by W.M. Flinders Petrie, *The Pyramids and Temples of Gizeh*
[London, 1883]. Online. Available:
http://www.ronaldbirdsall.com/gizeh/index.htm.
April 24, 2001), 82944^{2/20.63159148} × 10^{8
}=* c*. Likewise, where ε_{0} is the
metric value of the electric constant, 8.854187817 × 10^{-12},
(82944^{2/20.63159148} × 40πε_{0})^{-1}
=* c*. Based on a communication from Mr. R.N. Stoner, on March 23, 2001, the
writer notes that, where the Egyptian royal cubit *erc*_{in}
~ 20.63 inches, and the value of the equatorial circumference of the earth C_{earthER}
~ 24900 miles, and 1.0368 is the area of the face of the absolute dead center cube divided
by the 'created 100' (= 100^{2}; cf. Leahy, *Foundation*,
pp.
368ff., et passim), C_{earthER }× 8.2944 × 10^{-4} /1.0368^{1/32} ~ *erc*_{in}. In particular, the
observations of Mr. DePompeo and Mr. Iuliano highlight the role of the absolute dead
center number 82944 as centerpoint of identity uniting the metric and English systems of
measure. For this identical role of 82944 (and, in particular, that of its related
unique cube value, 144π), in relation to A. Thom's estimate of
the megalithic yard, cf. ibid., Section IV.2, p. 525, n. 99. The writer takes this
opportunity to note that, where *c* is the velocity of light and *h* is the
Planck constant, *c*/*h*, with a difference less than one in ten million,*
*equals 144π(1+8294.4^{-1}) ×
10^{39}. Also, he notes that, where 144 is the unit face-diagonal of the
absolute dead center cube, 176.3632615 its body-diagonal, *m*_{p} the value
in kilograms of the mass of the proton, 1.67262158 × 10^{-27},
and *m*_{e} that of the mass of the electron, 9.10938188 × 10^{-31}, then, with a difference ~ π^{-2}
× 10^{-8}, 144π/.82944^{-1/128}/10
×* m*_{p}/*m*_{e} -
(176.3632615/10^{2})^{-2 }equals 82944.

6. On March 2, 2001, Mr. DePompeo noted that, where the mean radius
of the sun is 695508 km (cf. T.M. Brown and J.Christensen-Dalsgaard, "Accurate
Determination of the Solar Photospheric Radius," *Astrophysical Journal *500,
Number 2, Part 2, Letters 1998 June 20, L195-198), 2715395.61769,
the circumference of the sun in miles ~ , where φ is the golden
section ratio, the sum of φ^{2 }and the product of the
odd primes, 3, 17, 37, and 1439. On March 6, 2001, he added that, where *G* =
6.6739 × 10^{-11}, the best calculated value of the
gravitational constant to date (Physics News Update 482, May 3, 2000. Online.
The American Institute of Physics. Available:
http://www.aip.org/enews/physnews/2000/physnews.482.htm.
March 5, 2001.), 3^{2}(17×37×1439)^{3}
× 10^{-29} ~ *G*, and, similarly, where V_{sun}
is the volume of the sun in miles, *G* ~ 2π^{2}V_{sun}/10^{29}.
The writer notes that, where *r*^{3}_{sun} is the volumetric
radius of the sun in miles, *r*^{3}_{sun }~ (8294400 × 2π^{2} ×
3^{.5})^{2}. On March 7, 2001, Mr. Iuliano called attention to the
Egyptian remen. The remen is equal to the radius of the circle in which is inscribed
a square whose area = 1 Egyptian royal cubit. The writer notes that the precision of
the last equation can be improved as follows: where the royal cubit = 52.4 cm and the
remen *Re* = 37.05239534 cm, and C_{sun} is the circumference of the sun in
miles calculated by Mr. DePompeo, then, with a difference less than 1 in 1 billion,
(8294400 × 2π^{2} × 3^{.5})^{2 }× (1+*Re*/10,000)
= *r*^{3}_{sun}/1+10/(C_{sun}+3×17×37).
The operative indifference of metric and English measures here is consistent with
the unifying role of 82944 and 144π, as noted above, and
further exemplified in the fact that the Egyptian remen in English measure ~ 14.58 inches
~ (10^{2}×144π)^{1/4}.
More directly & precisely, when *dc _{12E}*

7. On April 4, 2001, Mr. DePompeo communicated the following
observations: where the 2000 IAU polar circumference of the earth C_{earthPR} =
39940671.42790 m, and one polar kilometer, defined as C_{earthPR}/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(100C_{earthPR}/40000-17(37/100)2.54
cm)^{3} = 1.609344001 × 10^{5} cm^{3}
= 1 mi^{3}, with a difference less than one in one million. Since 17(37/100)
inches = ft and the metric value
of the Egyptian royal cubit *erc*_{m} =
.524 m, then the equivalent of the preceding formula is V = 4π/3[100C_{earthPR}/40000-2.54(*erc*_{m}/m ft) cm]^{3} = 1 mi^{3}.
(When *erc*_{m} = .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+j^{2} (cf., above, Addendum 6), the circumference of the sun
in meters, ~ (3×17×37×1439+φ^{2})×1.609344001×10^{3}, =, with a
difference less than one in a million, the edge of a cube whose volume V_{cube} =
the volume of the sphere V_{sphere} = 4π/3[(C_{earthPR}/40000-.0254[*erc*_{m}/m ft])(3×17×37×1439+φ^{2}) m]^{3} = 8.345377643 ×
10^{28} m. Finally, where the 2000 IAU polar radius of the earth *r*_{earthPR
}= 6356.755288 km, the volume V_{sphere} of the sphere whose radius this is =
4π/3(*r*_{earthPR})^{3} = the cube
volume V_{cube} = 10247.03838 km^{3} = 6367.214453 mi^{3}.
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+2^{1/2}×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 *erc*_{in}^{8}/(3 × 10^{10}) ~ 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 *erc*_{in
}= [5280(*c*_{mi}+α^{-1/2}β)
×10^{10}/*c*_{m}]^{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^{-1/2}β ~ (-10 × COS* erc*_{in})^{2} ~ 10^{2/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 *MY*_{ft} is the megalithic yard, *MY*_{ft} =
[1//10^{-3}/2.54^{23}]^{23(223)}
- ^{22(3)}
feet. When *MY*_{ft} = Thom's 2.722 feet, its metric equivalent *MY*_{mm}
= 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 *m*_{earth}
= 5.97223 × 10^{24} kg, and that of the sun *m*_{sun}
= 1.98843 × 10^{30} 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)
=* m*_{sun}/*m*_{earth}. The difference less than one
in a million, with a further difference that, with a yet further difference ~ (144π × 2808/3600 × 2^{3} × 2^{8})^{-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
C_{earthPR} = 39940.67142790 km (cf., above, Addendum 7), and one polar kilometer
= C_{earthPR}/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×10^{3})^{-1/32}. Where 6 is the number of absolute dead center
cube faces and 144π the circumference of the circle
circumscribing each, 6(144π×10^{3})
=, 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 (×10^{94})^{-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+[×10^{94}]^{-1/32}) × 10^{4}
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+[//10^{97}]^{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}-φ^{-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 ρ _{crit} = 3H_{0}^{2}/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 (~ φ^{4+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 *l*_{37} (=
6897.593084288) is the total length of the total number of diagonals in the logically
outfitted dead center cube, *l*_{36} (= 6753.593084288) that of the total
number of end-connected diagonals, *l*_{20} (= 3591.975155039) that of the
total number of yellow & green diagonals, *l*_{17} (= 3305.617929248)
that of the total number of blue diagonals, *l*_{16} (= 3161.617929248)
that of the total number of end-connected blue diagonals, and *l*_{1} (=
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 *l*_{37} , *l*_{36} ,* l*_{20}, *l*_{17}
, *l*_{16} , and *l*_{1}, 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:

(*l*_{36} : *l*_{1})/2* *–*
*(*l*_{37}^{1/37}/*l*_{36}^{1/36} × *l*_{20}^{1/20}/*l*_{17}^{1/17}
× *l*_{16}^{1/16}/*l*_{1}^{1/1}) =
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 (= 17^{2}) odd prime, 1.889 × 10^{-4}
~ 37/36/20/17/16/1 ~ 17^{2}/(1+2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17)/10^{4}
(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, = φ^{2}.
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 = 12φ^{2}
~ 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 = 12^{3 }× φ^{2}.
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/8^{1/2} 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/8^{1/2}
× 82944^{1/82944/8.88024333})^{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, The Diagonal Logic of the Triple Absolute Dead Center Cube & Certain Fundamental Mathematical & Physical Constants.