Mar 19, 2006 09:41 PM
by Cass Silva
Anyone care to check this site out
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 memoralize 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 predominent 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 siting 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:
The "Message of Cydonia"
Verification of a highly-specific and redundant communication of "circumscribed tetrahedral geometry" -- including its obviously *deliberate* extension to the siting of the Cydonia Complex on the planet -- would be deemed a phenomenonal discovery. If this is indeed "the message of Cydonia," crafted by what Mars' hostile environment strongly implies was a visiting interstellar culture (Hoagland, 1987), then what could have been its purpose?
To communicate the "importance" of tetrahedral geometry itself!
If this is the successful "decoding of the Message" -- its existence (if not the sheer effort expended in its communication) must in turn raise obvious questions regarding "hitherto unrecognized properties" of circumscribed tetrahedra.
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