by Richard C. Hoagland and Erol O. Torun

from EnterpriseMission Website









  -  Introduction



  -  Background to This Study



  -  The Significance of "e/pi"



  -  The "Message of Cydonia"



  -  Understanding the Meaning of "the Message"



  - "Embedded Tetrahedral Latitude" Discovered At Cydonia


  -  The Significance of the "Cydonia Message"



  - "Cydonia Tetrahedral Model" Extended to the Giant Planets


  -  Extension of the Cydonia "Tetrahedral Model" Beyond Planets


  -  Discussion



  -  In the continuing, puzzling departure of some celestial objects -  from strict "Newtonian mechanics"

  -  Summary

  -  Acknowledgements

  -  References

  -  Table I - Planetary Latitudes of Emergent Energy Phenomena



Return to Martian Structures and Anomalies


Return to Hyperdimensions



Return to Richard Hoagland


























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 controversy -- 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."






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 apparently together (as "e/pi"), redundantly encoded in the fundamental geometry of the layout of the "anomalous features" at Cydonia.

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, trigonometric functions, and radian measure) -- to three significant-figure accuracy (
see Fig. 2).

Hoagland (1988), in re-measuring 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!






Verification of a highly-specific and redundant communication of "circumscribed tetrahedral geometry" -- including its obviously *deliberate* extension to the sitting of the Cydonia Complex on the planet -- would be deemed a phenomenal 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.







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 elegant expression of the ArcTANGENT trigonometric 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. (more info about
Number 19)

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 volcanoes 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 . . .]







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).

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."

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!






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 sitting of active volcanic centers on the surfaces of near-by worlds . . ."






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" (more info about Number 19) 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-by enabled 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 be accounted 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


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" (more info about
Number 19) 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 transport 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.






Heartened by the apparent success of this "embedded tetrahedral model" in empirically predicting surface manifestations 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, turbulent 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.]