By Richard C. Hoagland
and Erol O. Torun
Following these discoveries, the authors (this paper) began the
current systematic inquiry into whether there was indeed a "message"
at Cydonia: encoded geometrically in terms of specific
placement of specific objects, by means of redundant mathematical
ratios derived by dividing the observed angular relationships into
one another. Over the last century or so, several prominent
proposals have been made for encoding "CETI" messages by means of
mathematical constants (Cocconi and Morrison, 1959;
Sagan, 1973; Rubtsov and Ursal, 1984), and even
physical geometric relationships on planetary surfaces (Gauss, et
al., -- see Crowe, 1986).
In particular, the authors were attempting to determine if e/pi =
0.865 [as opposed to the more fundamental ratio (sqrt 3)/2 =
0.866] was the ratio specifically intended at Cydonia.
Others (notably Davies) had already raised key questions
regarding this potential ambiguity.
Other constants demonstrated at Cydonia by Hoagland and
Torun being "sqrt 2," "3" and "sqrt 3" (1988), this confusion
regarding which constant was "really" represented by the observed,
redundant angle ratios, trig functions, and radian measure was
considered an important question to resolve. Since "3" and "sqrt 3"
are numbers essential to calculating "areas" and "volumes," Torun
decided to explore their geometric implications first, following on
Gauss (op cit).
He began by investigating geometrical relationships among several
fundamental "Platonic solids": the tetrahedron,
cube, octahedron, icosohedron, and
dodecahedron. In pursuing these explorations, Torun
examined the mathematical properties of "circumscribed polyhedra"
-- the Platonic solids embedded in a sphere.
Almost immediately, he discovered something quite astonishing (to a
non-specialist): the surface area of a tetrahedron (the
"lowest order," simplest Platonic form), inscribed
inside a "higher-order" form -- a sphere-- results in a surface
ratio (sphere/tetrahedron) almost precisely equivalent to "e",
the base of natural logarithms:
e = 2.718282
surface of sphere
---------------------------------------------------- =
2.720699
surface of circumscribed tetrahedron
Difference = 0.002417
The derivation of the above is as follows: (expressions are
written in FORTRAN notation)
Let A(t) = surface area of tetrahedron
A(s) = surface area of circumscribing sphere
R = radius of circumscribing sphere
For a regular tetrahedron of edge a:
A(t) = a**2 * sqrt(3) and R = a * sqrt(6)/4
For the circumscribing sphere:
A(s) = 4*pi*R**2 = 4*pi * (a*sqrt(6)/4)**2 = (3/2)*pi*a**2
Area of sphere/area of circumscribed tetrahedron
A(s)/A(t) = (3/2)*pi*a**2/(a**2 * sqrt(3)) = 3*pi/(2*sqrt (3))
A(s)/A(t) = 2.720699 - an approximation of e = 2.718282
When Torun
substituted this "close approximation of e", termed e', in the
equation most approximated at Cydonia:
e/pi = 0.865
He discovered that:
e'/pi =
2.720699/3.141593 = 0.866025 = (sqrt 3)/2
Or . . . precisely the
observed "e/pi" ratio discovered at Cydonia!.
The fact that e'/pi equals (sqrt 3)/2 can
be demonstrated algebraically:
Since e' was defined
as 3*pi/(2*sqrt (3)),
e'/pi = 3*pi/(2*sqrt (3)) / pi = 3/(2*sqrt (3)) = sqrt(3)/2
To place the above math in
simple terms:
The values of e/pi and
(sqrt 3)/2 are precisely equal when e/pi is evaluated using the
approximation of e that is generated by the geometry of a
circumscribed tetrahedron.
This simple fact
completely resolves the ambiguity regarding which ratio -- e/pi or
(sqrt 3)/2 -- was intended at Cydonia
(see
Fig. 4):
Apparently, both were!
Since the most redundantly observed Cydonia ratio is
0.866 and not 0.865 (the true ratio of the base of natural
logarithms, divided by Pi -- to three significant-figures), it must
now be clear, however, that the *primary* meaning of the "geometry
of Cydonia" was in all likelihood intended to memorialize
the (sphere)/(circumscribed tetrahedron) ratio [which is also (sqrt
3/2)], and not "e/pi".
Further examples of "e/pi" at Cydonia -- appearing in
connection with the ArcTan of 50.6 degrees (present at least twice
in association with the Face) -- when examined by
Hoagland, confirm that Torun's "circumscribed tetrahedral
ratio" -- e' = 2.72069 -- and NOT the base of natural
logarithms (e = 2.718282) provides a closer fit to the observed
number.
Thus strongly implying that "tetrahedral geometry" (and
NOT the usual association of "e" with "growth equations") is the
predominant meaning of "e/(sqrt 5)" and "(sqrt 5)/e" -- two other
specific ratios found redundantly throughout the complex:
e/(sqrt 5) = 1.215652
e'/(sqrt 5) = 1.216734
Cydonia ratio = 1.217 = ArcTan 50.6 degrees
(The detailed
implications of this association -- e' and (sqrt 5) -- will be
examined in a subsequent paper.)
These results, combined with other examples in the Complex
(D&M Pyramid angles 60 degrees/ 69.4 degrees = 0.865 )
are what lead us to the conclusion that in fact *both* constants --
e and e' -- are deliberately encoded at Cydonia.
In particular:
D&M Pyramid apex =
40.868 deg N = ArcTan 0.865256 = e/pi
But another feature on the
D&M -- the wedge-shaped projection on the front
-- defines the Pyramid's bilateral symmetry and orientation directly
toward the Face. This feature also now seems to mark
an equally important latitude:
D&M "wedge" = 40.893
deg N = ArcTan 0.866025 = e'/pi = (sqrt 3)2
Torun identifies a
conspicuous "knob," lying at the end of this wedge, as the
"benchmark" designed to mark precisely the correct "e'/pi" latitude
--
40.893 degrees, approx. 1/40th degree North of the true apex
of the Pyramid. The terminus of this wedge, together with the NW
corner of the pyramid, are the only two points on the pyramid that,
when connected, denote a line of latitude
(see
Fig. 5).
Again, putting this in simple terms:
The geometry of a
circumscribed tetrahedron is not only suggested by the alignments in
Cydonia, but also by the sitting latitude, size, shape, and
orientation of the D&M Pyramid itself.
This discovery only
underscores the importance apparently attached to "circumscribed
tetrahedral geometry" in the construction of Cydonia
-- raising the important question: Why?
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