Introduction For the past 13 years -- ever since an unmanned NASA Viking spacecraft successfully photographed the surface of the planet Mars in 1976 -- a mystery has loomed . . . a mile-long, 1500-ft high humanoid "face" discovered in a northern Martian desert called "Cydonia." In its immediate vicinity have been identified other "anthropomorphic objects": most notable, several "pyramids" (see Fig. 1). Various investigators [Owen, 1976 -- see Hoagland (1987); DiPietro and Molenaar (1980); Hoagland (1986);Pozos (1986); Hoagland (1987); and Carlotto (1988)] have examined this collection of objects over the past 13 years, and have reached widely varying conclusions. The essence of the controversey -- its potential importance or non-importance as a "scientific" problem -- is perhaps summed up best by Hoagland (1987):
| "Either these features on Mars are natural and this investigation is a complete waste of time, or they are artificial and this is one of the most important discoveries of our entire existence on Earth. If they are artificial it is imperative that we figure them out, because they 'do not belong there.' There presence may be trying very hard to tell us something extraordinary." |
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Background to This Study The initial purpose of this study was an examination of the "Cydonia mathematics," which at first glance emphasize the importance of two "dimensionless constants": "e" (the base of natural logarithms = 2.718282);and "pi" [the ratio of the circumference of a circle (or sphere) to its diameter = 3.14159]. These two constants appear, both separately (as "e" and "pi"), and apparantly together (as "e/pi"), redundantly encoded in the fundamental geometry of the layout of the "anomalous features" at Cydonia.
------------------------------------------------------------------------* The Mars Mission: Box 981, Wytheville, VA 24382; (703)-228-7822@ The Defense Mapping Agency: Reston, VA 22090-3207------------------------------------------------------------------------ Following Hoagland's proposal (op cit) that a mathematical "relationship model" would be the key to validating the basic reality of Cydonia as an architectural construction, and that "e/pi" might be one significant relationship related to the Complex, Torun (1988) made key mathematical discoveries within a major geometric "Rosetta Stone" located at Cydonia -- a unique, five-sided, symmetrical "pyramid": the so-called "D&M." He elegantly "decoded" a series of internal angles found within the pyramid, and discovered the two mathematical constants, "e" and "pi," encoded several times and in several different ways (via angle-ratios, trignometric functions, and radian measure) -- to three significant-figure accuracy (see Fig. 2).
Hoagland (1988), in remeasuring the "complex" he initially proposed, promptly verified the existence of identical constants, encoded via identical "dimensionless ratios," in geometry linking *all* the previously identified key objects at Cydonia -- and to at least the same measurement accuracy as Torun (see Fig. 3a, "Cydonia Geometric Relationship Model").
Subsequently, using geodetic data from "The 1982 Control Network of Mars" (Davies and Katayama, 1983), up-dated by Davies for Cydonia (1988), Hoagland discovered (op cit) that the critical object Torun had "decoded" -- the Pyramid -- lies precisely astride the key geodetic Martian latitude expressive of the ArcTangent equivalent of "e/pi": 40.87 degrees = ArcTan 0.865!
The Significance of "e/pi" 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 tetrahedronA(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)/2To 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. |
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This discovery only underscores the importance apparently attached to "circumscribed tetrahedral geometry" in the construction of Cydonia -- raising the important question: Why?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?
Apparently: 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.
Understanding the Meaning of "the Message": Verification in Hitherto Unrecognized Geophysical Phenomena Since the latitude of the entire Cydonia Complex seems to have been carefully chosen to reflect the ArcTANGENT of this circumscribed tetrahedral "message, it occurred to the authors that "something important might lie at the LATITUDE represented by the vertices of a circumscribed tetrahedron -- placed 'inside a planet'." This would represent the most elegent expression of the ArcTANGENT trignometric function emphasized repeatedly within the Complex -- especially the choice (out of more than 18,000 possible other choices, to equal numerical precision) of the specific Martian latitude: 40.87 N.
In working out the several possible implications of such geometry, Torun promptly discovered the following: if a circumscribed tetrahedron is placed inside a globe representing a gridded planetary surface, with one vertex located either on the geographical "North" or "South" polar axis, the resulting latitude TANGENT to the other three vertices will lie at 19.5degrees N. or S. -- 120 degrees of longitude apart.
Torun (1989) immediately noted on Earth the existence of several significant Meso-American ceremonial complexes at this specific northern latitude -- raising intriguing cultural and scientific possibilities for lost or forgotten "ancient knowledge of the significance of circumscribed tetrahedral geometry" (Becker and Hagen, 1987). Unfortunately, these implications are too extensive for inclusion here.
Hoagland noted something more physically significant: the largest shield volcanic complex on Earth -- the Hawaiian Caldera -- is located very close to 19.5 North! He then realized that a similar latitude marks the location of the largest shield volcano currently known in the entire solar system: Olympus Mons, at 19 N. -- on Mars.
Subsequent survey of solar system geodetic maps -- made from spacecraft photography of the past thirty years, encompassing planetary surfaces from Lunar Orbiter images of the Farside of the Moon, to Voyager 2 close-ups of Uranus, its satellites, and now (at this writing) the planet Neptune -- revealed a remarkable (and currently inexplicable) geophysical phenomenon (see Table 1):
The majority of "active centers" on these objects -- from the greatest shield volcanos on the "terrestrial planets" (including equivalent features on their most anomalously active satellites!), to the enormous atmospheric disturbances seen on some "gas giants" ("The Great Red Spots" of Jupiter and, now, of Neptune) seem preferentially to occur very close to 19.5 degrees N. or S., irrespective of other planetary factors -- mass, rotation rate, obliquity to their respective orbits, etc. (see Fig. 6)!
There was some indication, however, that the polarity of the dipole magnetic field, offset from the spin-axis, determined in *which hemisphere* the phenomenon appeared; Jupiter's GRS, at approximately 20 degrees S., is consistent (in this model) with its opposite (from terrestrial convention) dipole field polarity. [This raises the interesting possibility of a "magnetic field prediction" vis a vis Neptune, before the up-coming Voyager Encounter (Aug 25, 1989) -- based on observation that its "Great Red Spot" is at the same latitude, and in the same hemisphere, as Jupiter's . . .]
"Embedded Tetrahedral Latitude" Discovered At Cydonia Following this striking, system-wide geophysical confirmation of a predictive (if baffling) "embedded tetrahedral model," the authors made another significant geometrical discovery at Cydonia itself:
The critical 19.5-degree tangential latitude of the "embedded tetrahedron" is specifically associated with a massive, *tetrahedral pyramid* located TANGENTIALLY, on the circular rim, of a 2-mile impact crater; in turn, this "pyramid" is connected TANGENTIALLY (via a line denoting the exact North/South meridian) to a circular (planet-like?) feature termed "the Tholus"; which, in turn, is connected to a third, linear feature ("the Cliff") positioned TANGENTIAL to the same crater (see Fig. 3b).
This highly-specific geometric "statement" -- a 19.5-degree angle offset to the local meridian, connecting three objects (one of them a tetrahedron!) in a way that reinforces the TANGENTIAL importance of that relationship -- seems to explicitly establish a "geometric connection" between "a tetrahedron" (the pyramid), a circular, "planet-like" construction (the Tholus), and the linear "Cliff' (the 19.5-degree offset reference), a relationship also known to be coded elsewhere in the Complex, in terms of derived mathematical constants: specifically, "e'/pi."
This explicit geometric statement also uniquely establishes an *identical* 19.5-degree angle offset between the D&M (at the other end of the Complex) and the resulting "map grid" -- further underscoring the significance of the D&M's unique latitude relationships (see Fig. 5).
Conclusion:
| These interlocking, extremely meaningful, and highly predictive relationships -- coded now in both the mathematical and blatant geometric aspects of the Complex -- can only be interpretated with extraordinary effort as anything other than the result of a deliberate and systematic plan -- designed to underscore the importance of "tetrahedral geometry." |
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That the anomalies predicted by this "geometry" encompass a range of demonstrable solar system phenomena -- from deeply-buried planetary mantle "hot spots," to associated shield volcanoes, to atmospheric thermal "upwellings," etc. -- is also now readily apparent--
Even if the reason for their specific "latitude-dependence" is not!The Significance of the "Cydonia Message" Lest there be any confusion, the authors are NOT claiming there is "a tetrahedron buried inside each planet!" Rather, it is suggested that the "tetrahedral geometry" explicitly designated by "Cydonia" is revealing an equivalent higher-order mathematical topology: i.e., a vorticular "two-torus" energy flow and internal fluid dynamics, equivalent to tetrahedral mathematics. That such an internal "vorticular pattern" could be explicitly modeled by an "embedded tetrahedral topology" is mathematically well-known (Porteous, I. R., 1981). That such a "geometric short-hand" -- directing us specifically to some underlying physical manifestation of tetrahedral mathematics -- was left specifically for us at Cydonia, seems now almost inescapable . . . if not inescapably significant.
A quantitative treatment of the physics underlying this phenomenon would appear likely to advance our understanding of energy transfer inside planets considerably -- a not unexpected outcome, if this indeed is "Mankind's first successfully-decoded extraterrestrial message." Additional observations suggest, however, that the significance of these predictions could extend far beyond "the simple siting of active volcanic centers on the surfaces of near-by worlds . . ."
"Cydonia Tetrahedral Model" Extended to the Giant Planets We have already alluded to the surprising conformance of the planet Neptune to this mysterious "embedded tetrahedral model." Its newly-discovered "Great Red Spot" (as imaged by the Voyager 2 spacecraft) now strikingly coincides with the "19.5-degree latitude predictions" communicated by Cydonia. It is the growing suspicion of the authors, however, that the imminent Voyager studies of Neptune, coupled with a re-analysis of those studies it conducted of Uranus, may provide vital evidence that the "Cydonia equations" are trying to tell us about more than just energy transfer . . .
Based on the evidence detailed below, it is our suggestion that these observations may relate to actual energy generation.
For many years there have been observed "energy excesses" in the overall energy balances exhibited by the four major planets of the outer solar system: Jupiter, Saturn, Uranus and Neptune. These planets, inexplicably, all radiate significantly more energy into space than they receive from the Sun at their respective distances (Hubbard, 1980).
Jupiter's positive energy balance (1.67 -- compared to solar input) apparently derives mainly from primordial heat retained during it's "collapse phase" from the original solar nebula, 4.5 billion years ago. A secondary contribution is calculated as created by the internal separation of helium from hydrogen ("helium drip"), with the resulting release of additional gravitational potential energy (Smoluchowski, 1967; Graboske et al., 1975).
Saturn, far less massive than Jupiter, is considered too small to have retained significant primordial heat. Thus, it's observed "excess" (1.78 solar input) is wholly ascribed to the gravitational separation of helium from hydrogen, tentatively verified by the 1980 and 1981 Voyager infrared observations of Saturn (Hanel, et al., 1983).
Ground-based data prior to Voyager's 1986 Uranus Encounter indicated that Uranus and Neptune, similar telescopically from Earth, differ dramatically in their observed "energy excesses" (Pollack, et al., 1986). Uranus from ground-based studies seemed to possess only a marginal (if any) heat source, compared to Jupiter and Saturn; Voyager's January, 1986 fly-byenabled investigators to lower even this minimal estimate (Pearl, et al., 1989). The new upper-limit on the ratio of internal Uranian heat to solar input is 1.14 -- almost non-existent compared to Jupiter and Saturn, and dramatically lower (by comparison) than current ground-based measurements of Neptune's radiated excess (2.7) over solar input.
Conventional sources for explaining Uranus' internal energy, as slight as it is, encounter difficulties. Based on Voyager data interpreted as evidence of a non-depleted helium/hydrogen ratio in the Uranian atmosphere, the "helium drip" model (so successful for Saturn) is not thought applicable (Conrath, et al., 1987). And, for assumed early solar system nebula compositions, resulting in a "rocky core" for Uranus equal to between one and three earth masses, only 15-50% of the formal excess can beaccounted for in terms of "radiogenic heating" (decay of radioactive elements -- Williams and von Herzen, 1974). This leaves "exotic elemental compositions" (more than 6 earth masses of "rocky materials") and novel energy transport mechanisms ("suppressed deep-atmosphere convection") as remaining "conventional" possibilities (Stevenson, 1987). These difficulties in accounting for the source of Uranus' internal heat are only made more difficult when the planet is compared to Neptune -- it's supposed "twin" in terms of size and (presumably) composition.
Because of these potential compositional problems, and the great disparity in internal energy-generation between these two otherwise so-similar planets, the authors are led to propose another possibility:
That the "Cydonia equations" may really be attempting to describe, not merely internal energy transport, but internal energy generation -- most evident (because of sheer distance from the Sun) in the overall energy balances of Uranus and Neptune.
Further, the authors believe study of the detailed Voyager infrared Uranus observations support this possibility:
Uranus, because of its extreme obliquity (98 degrees) relative to its orbit, alternates each pole toward and away from the Sun for a quarter of its 84-year revolution. Despite this unique geometric shadowing effect (the Uranian south pole not having "seen" sunlight for over 20 years, at the time of the Voyager Encounter)--
THERE WAS NO GLOBAL TEMPERATURE DIFFERENCE OBSERVED BETWEEN THE DAY AND NIGHTSIDE POLES. Because of the problems cited above with any internal Uranian heat source, and the distinct possibility that (within the error's of Voyager's measurements) Uranus actually possesses zero internal energy, discussion in the literature has attempted to explain this global temperature uniformity as "redistribution of intercepted solar input," via "shallow atmospheric advection" (Friedson and Ingersoll, 1987); if the solar energy is being transported around to the nightside of the planet by a shallow, upper atmospheric mechanism, this would radically decrease (because of the non-necessity for warming the entire nightside atmosphere) the amount of heat (energy) required for transport to the nightside -- otherwise needed to account for Voyager's global-temperature distribution measurements.
A major problem for this model, however, was the Voyager observation that the winds (clocked by observing several discrete clouds) blow in the same direction as the rotation of the planet (Smith, et al., 1986); pre-Encounter theoretical predictions had firmly anticipated a four-day retrograde rotation of the upper atmosphere, driven by external solar radiation. [The planet Venus, where such opposite winds (to the rotation of the planet) are observed, has its atmosphere dynamically-determined by intense external solar radiation.] Further, the fact that the observed Uranian clouds were seen circling the pole in a series of concentric circles (parallel to decreasing latitude) as Voyager approached, leads to difficulties in modeling heat transport to the nightside, pole to pole -- across the latitudinal windflow.
These observations make it at least plausible to the authors that internal energy, not "shallow advection of absorbed solar radiation" constitute the primary driver for the Uranian atmosphere; a final, detailed Voyager infrared observation, would seem to add significant support to this hypothesis.
In scanning both hemispheres -- the dayside South pole and the nightside North -- the Voyager IR instrument detected a small but significant 1-2 K temperature drop in both hemispheres -- at approx. 20 degrees N. and S. latitude (Pearl, et al, op cit). Interpretated as the spacecraft viewing small-scale emissive and reflected temperature profiles of colder, higher clouds (consistent with similar observations made at Jupiter and Saturn -- including measured temperatures of Jupiter's Great Red Spot, which is also colder because it is higher than the surrounding Jovian atmosphere), the Uranus' observations could be interpreted as "some kind of massive 'upwelling' within the Uranian atmosphere," creating condensation products -- clouds -- as the atmosphere rises to higher altitudes--
Narrowly straddling the "plus and minus 19.5-degree latitude" where the "embedded tetrahedral model" of the Cydonia equations would predict -- for an internal, energy-driven "upwelling" on the planet!
The difficulties involved in modeling a process, driven by external solar radiation, which could create such upwellings and then "know" where to create them -- at the "magic 19.5-degree latitude" predicted by Cydonia -- are formidable. In the opinion of the authors, it is much easier to ascribe these symmetrical upwellings to an internal energy source -- released according to the now-familiar Cydonia pattern observed elsewhere in the solar system of "internally-driven energy emission."
The fact that these upwellings appear symmetrically in Voyager IR scans of both hemispheres presents, however, an interesting contradiction to other "planetary upwellings" -- which seem to be restricted to one hemisphere, and to one localized latitude region. Those on Uranus (if the model is applicable) are not.
The apparent enigma is resolved, we think, by the fact that the Uranian magnetic field is radically different from any other planet: aligned at approximately 55 degrees to the inertial spin axis (Ness, et al., 1986). It is at least interesting to propose that somehow this almost right-angles magnetic orientation with respect to the geographic poles "allows" the internal energy processes predicted by the "embedded tetrahedral model" to manifest symmetrically in both hemispheres. If true, this in turn allows some insight into the role of planetary magnetic fields in the "Cydonia phenomenon": in some geometries, that of selective hemispheric suppression of an internal energy-transport mechanism.
Based on all of the above, it is the considered opinion of the authors that at Uranus, the Cydonia "embedded tetrahedral model" reveals itself as not only a mechanism for energy tranport within planets -- but, quite likely, as a process of internal energy generation as well. The implications of verifying this hypothesis -- for all planets where these phenomenon are observed to follow the "Cydonia predictions," including Earth -- the authors think are obvious . . . if not highly significant in terms of other astro-physical environments, where involving a potential "new source" of energy might lead to wholly different fundamental models.
Extension of the Cydonia "Tetrahedral Model" Beyond Planets Heartened by the apparent success of this "embedded tetrahedral model" in empirically predicting surface manifestatons of internal planetary dynamics, the authors decided to extend the model to the Sun. This is the result.
With no solid lithosphere, the solar plasma "photospheric surface" is of course far more like the banded, turbulant atmospheres of the giant outer planets (though much hotter!), than the dense, solid crusts of the "terrestrial" planets. And like the atmospheres of the giant planets, there is a recurring "surface phenomenon" which is measurable -- in terms of a coordinate system referenced with relation to the rotational axis: sunspots.
Though appearing dark against the surrounding photospheric background, sunspots are still measured at approximately 3500 K, and radiate enormous energy per unit area. More significant for our discussion here, though convection within the spot "umbra" (the darkest, central part) is suppressed by intense, local magnetic field strengths (Hale, 1913), there is evidence of enhanced energy emission around the spot itself -- perhaps as much as one or two percent over the normal photospheric background.
When flare activity is considered (which occurs in the intense, tangled magnetic fields between sunspot groups), spots -- as opposed to being "regions of lower solar output" -- are in fact localized areas of "enhanced energy emission" (Brandt, 1966)
The recurring 22-year solar sunspot cycle is made up on average of two back-to-back 11-year components. Sunspots at the beginning of each cycle usually appear in pairs, with opposite magnetic polarity -- N. and S. (though, entire localized "sunspot regions" -- see above -- can also form), at high solar latitudes (-- 40 degrees). The appearance of new additional "spot pairs" (and the dying of "old" ones), as the cycle progresses, subsequently drift North and South (depending on the hemisphere) -- eventually converging late in the cycle in the solar equatorial regions. 11 years later, on average, a new cycle begins, with sunspot pairs of now opposite polarity (compared to the initial cycle) "breaking out" once again at high solar latitudes -- with subsequent spots appearing at decreasing latitudes as the new cycle moves toward its 11-year completion.
To our amazement, when the mean latitude of the majority of sunspot and associated flare activity was examined, the mean in both hemispheres -- from beginning to end of each cycle -- was found to be remarkably close to 20 degrees (Fig. 7)!
The apparent appearance in the Sun of the same phenomenon so successfully predicted by Cydonia for planets -- including the possibility that the effect is somehow related to a new source of energy generation and not "merely" energy transport -- opens up extraordinary possibilities.
These must include consideration that solar luminosity could be a mixture of two energy sources: the "traditional" fusion of hydrogen within the solar core; and another, still inexplicable process, somehow modulated by the general wax and wane of solar magnetic activity! [R. C. Wilson, after many years of careful observations at Mt. Wilson, has demonstrated direct variation of the solar constant in synchronization with the solar sunspot cycle (Wilson, et al., 1980). Newer Solar Max satellite data from above the atmosphere confirm the findings.]
Discussion A detailed discussion of these observations is beyond the scope of this inquiry. However, the authors feel they would be remiss if they did not conclude by at least mentioning two additional areas where future observations could greatly increase our confidence in the reality of this phenomenon -- if not our understanding of its nature.
In view of apparent solar conformance with "circumscribed tetrahedral geometry," one area for further research seems immediately apparent: "exotic stars."
In addition to "flare stars" and other highly-variable stellar objects with surface phenomenon thought to be similar to solar processes, we feel that, if the Cydonia mathematics are attempting to describe not only energy "flow" but somehow "energy generation" -- then the ability to test these ideas via radio astronomy should be the highest in terms of one class of exotic objects in particular: pulsars.
The common link connecting all the objects for which the Cydonia "embedded tetrahedral model" seems to work -- from the planets to the Sun--seems at this stage to be based on one significant association: angular momentum and magnetic fields. Before the adoption of the present, complex "self-excited dynamo theory" (with internal, circulating, conducting "fluids" as the mechanism for general planetary and stellar magnetism), another -- strictly empirical -- hypothesis was proposed: a strikingly simple relationship between the observed total angular momentum of the object, and a resulting dipole.
Termed "Schuster's Hypothesis" (Schuster, 1912), it has been successful in predicting magnetic field strengths (Blackett, 1947; Warwick, 1971) ranging from the earth's, to the sun's, to Jupiter's vast field (20,000 times the terrestrial dipole moment) -- a prediction made over sixty years prior to the 1973-74 Pioneers' 10 and 11 close-up confirmations (Warwick, 1976).
(It should be pointed out, ground-based radio astronomy had successfully deduced Jupiter's magnetic field strength -- thus its conformance with "Schuster's Hypothesis" -- some twenty years before the Pioneer magnetometer observations confirmed the radio data and Schuster -- who's very name and empirical discovery is, inexplicably, completely missing from all NASA literature on planetary magnetism.)
Commented Warwick on the remarkably predictive power of "Schuster's Hypothesis," even in 1971: "Dynamo theory has not yet successfully predicted any cosmical fields. It's use today rests on the ASSUMPTION that no alternative theory corresponds more closely to observations [original emphasis]."
Taking Schuster's 1912 proposal, the authors have plotted contemporary parameters for angular momentum versus observed the magnetic dipole moment (for all planetary objects now visited by spacecraft with magnetometers), and find Schuster's Hypothesis amazingly confirmed (see Fig. 8) -- with the exception of Mars (for which we have extremely poor data -- no American spacecraft since Mariner 4, including Viking, has carried a magnetometer toMars!), and Uranus (discussed below).
It is tempting to propose that what we have observed in terms of the "Cydonia equations" -- a remarkable correlation between external, localized energy emission and a planetary spin-axis, somehow modulated by the orientation (if not the intrinsic value) of the magnetic field -- may be trying to tell us about the physical process underlying "Schuster's Hypothesis": how magnetic fields in spinning bodies form . . . if not how their formation may be associated with internal energy generation.
In terms of an observable, visible connection between these two parameters -- planetary magnetism and planetary surface features -- it is interesting that Warwick in 1976 already had observed:
"It seems likely that, corresponding to atmospheric belts and zones as seen in Jupiter's cloud top level, there is a belt-like structure in the magnetic field not strong enough to alter the dipole structure radically [thus not mappable by spacecraft' instruments -- unless in very close-in orbits] but still sufficient to play a role in magnetospheric diffusion as Neil Brice suggested some years ago. This magnetic fine structure also must play a role in decametric [radio emission] phenomenonology . . .[which], especially [as exhibited in] longitude patterns throughout the Jupiter year, is strictly reproducible over the two decades of radio observations. The conclusion I draw from this fact is that the magnetic fine structure near the [cloud top] surface of Jupiter has remained constant over the same time interval. In support of this conclusion is the more orless constant belt and zone structure of Jupiter since 1950 . . . Such a correlation, which is ultimately [in this theory, associated] with magnetic fine structure, would tend to confirm the existence of magnetohydro-dynamic turbulance near the surface of Jupiter."
Warwick's inferences regarding the role of magnetism and fluid physics, in possibly generating the patterns long-observed in radio astronomy's "decametric observations," would seem to lend support to our own tentative proposals:
That the Cydonia "embedded tetrahedral model" (merely, if we're correct, an equivalent geometrical expression of the far more complex mathematics associated with a "vorticular fluid-flow") is in fact predicting the latitudes and sizes of the Great Red Spots on Jupiter (and now on Neptune!), and the largest volcanic centers of emission on "terrestrial planets," through deep-seated, vorticular magnetohydrodynamic processes -- operating in the highly conducting mantles of these bodies. Asimilar mechanism probably underlies the Sun's conformance to "tetrahedral geometry" as well.
That each "upwelling" (at least in the atmospheres of these two giant planets), seems scaled strictly in terms of the respective size of each respective body, provides further elegant support for an internally-determined "fluid" model.
The fascinating, observed correlation between angular momentum and magnetic dipole moment -- if not field polarity, evidenced by the selective hemispheric appearance of energy emission at 19.5 N. or S. -- has made us wonder, however, about more fundamental correlations . . . . We suspect that the puzzling and periodic "field-reversals" of the terrestrial geomagnetic field are another indication of the "Cydonia embedded tetrahedral model" -- modulated by the constant gravitational "tidal kneading" of the Moon. If this process is involved in periodically "flipping" the entire magnetic field (and the resulting "hot spots"), then the precise physical mechanism should raise provocative questions involvingelectromagnetism, mass, inertia . . . and possibly gravity itself. Sirag, in observing this same remarkable correlation between angular momentum and electromagnetism (1979), raised similar considerations.
If the Cydonia mathematics are attempting to direct us to energy generation and subsequent energy transfer inside astronomical objects, involving a hitherto unknown relationship between two of the four basic forces of the Universe -- gravity and electromagnetism: i.e. a "UnifiedField" -- this process in our opinion cannot help but manifest itself more clearly in astrophysical environments where both parameters have reachedextraordinary values--
Which brings us once again to pulsars. Even the "average" spinning neutron star (the favored "pulsar model") possesses surface gravitational accelerations, angular momentum, and magnetic field strengths billions of times more intense than similar quantities in any solar system object. Moreover, since Schuster's Hypothesis strikingly succeeds in its prediction of even these extraordinary magnetic dipole moments (see Fig. 8), we cannot help but wonder at what rich new confirmations of the "Cydonia mathematics" may lie hidden in existing -- and currently mystifying -- pulsar observations . . .
If in the "Cydonia tetrahedral mathematics" we are truly seeing the deliberate communication of demonstrable astrophysical effects of a long-sought "Unified Field Theory," this in itself would be remarkable confirmation of current efforts to discover such fundamental mathematical connections between Nature's elemental forces. For, most provocative: one leading mathematical approach to successfully modeling such connections is essentially based on a tetrahedral model, and a resulting mathematical expansion into "higher-dimensional, n-space relationships" (recently discovered) between the five Platonic solids (Sirag, 1989). In particlular, these studies relate tetrahedral geometry as being topologically equivalent to three-toruses -- tori extending into "one more dimension than our familiar three." [Many current efforts in pursuit of "unified field models," such as the much-acclaimed "super-string theory," routinely involve up to ten mathematical dimensions. Some more recent theories are exploring twenty-six (Sirag, ibid).]
Phrased in simple terms:
| The routine mathematical representation of vorticular flow in more than three dimensions -- a three-torus -- by means of three-dimensional tetrahedral models, opens up the possibility that the demonstrable geophysical effects of the "Cydonia tetrahedral message" are attempting to communicate the reality of additional dimensions (as opposed to mere mathematical abstractions) -- and the observable reality of vorticular energy flow between adjoining "n-spaces." |
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Such totally unexpected (to non-specialists) and remarkable mathematical correlations -- between as yet unpublished theoretical work into Unified Field Models, and the specific tetrahedral geometry apparently intended at Cydonia -- gives added confidence that such a linkage was in fact intended. If so, there may be an additional confirmation of a such a radical "Cydonia Unified Field Model"--
In the continuing, puzzling departure of some celestial objects from strict "Newtonian mechanics." Careful observation of the outer planets over the last two centuries has revealed that "the motions of Uranus and Neptune cannot be adequately represented within the present gravitational model of the solar system" (Harrington, 1988). While gravitational perturbations originally ascribed to Pluto (Tombaugh and Moore, 1980) have now definitely been eliminated as of adequate magnitude to explain these astrometric deviations (Harrington, op cit), "the suspicion of the existence of a tenth planet" has resulted in renewed efforts (Seidelmann and Harrington, 1988) to search for another unseen object which could gravitationally account for the persistant residuals of Uranus and Neptune.
However, other experts in celestial mechanics are open to the possibility that (as opposed to a new planet) a fundamental modification to gravitational theory itself may in fact be necessary, in order to adequately model the puzzling outer planet motions (Anderson, 1989) -- which, curiously, are most "anomalous" for Uranus.
As a direct consequence of the apparently successful application of the "Cydonia predictions" to Uranus and Neptune, and because of a probable fundamental link this has revealed between angular momentum (rotating mass) and electromagnetism, we suggest a third alternative should be considered:
A derivative "anomalous gravitational effect," somehow created by these interactions. Such an admittedly radical proposal must of course be subject to some stringent observational tests; we believe that Voyager's own 1986 Encounter has fortuitously supplied us with just such an opportunity--
In the form of "anomalous" Voyager 2 X-band range-rate residuals, acquired during the fly-by of Uranus itself. For over a month prior and subsequent to the January, 1986 Encounter, calibrated ranging signals were transmitted to and from the Voyager 2 spacecraft. Examination of five days of ranging data, centered on the timeof Closest Approach, reveals a series of curious, systematic "range errors" -- seen only around the time of periapsis of Voyager to Uranus (see Fig. 9); at this time, the spacecraft exhibited a range error of up to 100 meters (an order of magnitude larger than instrumentation-limited range uncertainties of approx. 9 meters, introduced by the Voyager/DSN radio ranging system itself.) These systematic errors were also inexplicably centered symmetrically on periapsis (Anderson, et al., 1988).
Explanations which would attribute this effect to "group delay of the ranging modulation by free electrons between Earth and Voyager 2" remain unconvincing, when the data are compared to similar residuals (Fig. 9) recorded during Voyager 2's Encounter of Saturn (Campbell, et al, 1989). The latter, by comparison, are "flat" (as opposed to the systematic increase and decrease observed at Uranus): reflecting no similar "symmetrically-centered curve," mirrored around the moment of Closest Approach to Saturn.
We believe therefore that these demonstrably unique, and highly suggestive ranging observations are interpretable as potentially direct evidence of the modification of the Uranian gravitational metric, by some additional "space-time effect" associated with the Cydonia "embedded tetrahedral model." We further suspect that the highly anomalous Uranian magnetic field-geometry Voyager observed with respect to the planetary spin axis (>60 degrees) -- if not the significant departure of the planet's overall magnetic dipole moment from "Schuster's Hypothesis" (see Fig. 8) -- may somehow be involved.
One important reason for raising this possibility now is the imminent Voyager 2 Encounter with Neptune. Uranus and Neptune -- essentially "twins" in terms of mass, angular momentum and (probably) composition -- provide important constraints on several fundamental predictions of the Cydonia mathematics. The great difference in internal energy balance between the two planets (1.14 for Uranus; 2.7 for Neptune), coupled with the unique configuration of the Uranian magnetic field, leads us to propose that the field orientation with respect to the planetary spin axis is an important element in determining in what form the internally-generated energy appears: in Uranus, with an almost 90-degree field orientation, little energy appears as heat. In consonance with conservation of energy, we suggest it may be "appearing" in some other form -- possibly as a change in the local "space-time metric."
The authors, based on this energy discrepancy, and the conformance of Neptune's "Great Red Spot" with the "embedded tetrahedral mathematics," feel a prediction of the Neptune field polarity (opposite the earth's), an estimate of its specific orientation with respect to the planetary spin axis (within 20 degrees), and its overall intensity (approximately one-tenth Saturn's) is now possible--
Before the Voyager August 25, 1989 Encounter. The relevance of these predictions to the question of "anomalous solar system motions" of these respective planets is fundamental: in Uranus, the energy not dissipated as internal heat (in comparison with Neptune) may be appearing as an associated distortion of the "overall metric" connecting the planet with the Sun, resulting in its anomalous motions with respect to current gravitational model for the solar system.
If similar metric distortions are underlying Neptune's (smaller though equally-puzzling) astrometric residuals, we predict that Voyager should experience a smaller (when corrected for the difference in periapsis radii) though significant set of "range residuals" -- when compared to its Encounter at Uranus; in addition, its instruments should verify our predictions of the magnetic field parameters.
In terms of those, one complicating factor is the orbit of Neptune's largest natural satellite: Triton. The essentially circular (though retrograde) orbit, indicates significant internal tidal dissipation of Triton's kinetic energy within Neptune -- and thus a significant source of internal heating, in addition to possible radiogenic and other ("embedded tetrahedral?") sources; Triton's retrograde "drag" on Neptune's interior must also have some (currently unknown) effect on any process that couples overall planetary angular momentum with the creation of a magnetic dipole moment.
Thus, predictions of internal heating, or "exotic gravitational/electromagnetic interactions," indicated by the success of the "Cydonia equations" on the other planets, are at this point somewhat ambiguous when applied to Neptune; the only positive indication so far that this unknown process is somehow effectively operating in the Neptune interior, is the discovery of Neptun
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