The values of observable angles expressed in radian measure.
Examining the ratios formed between the observable angles for equality with mathematically significant numbers.
Examining the sine, cosine, and tangent of measured angles for the presence of mathematically significant numbers.
“These approaches,” explains Torun, “were selected due to their simplicity, their validity in number bases other than decimal, and their independence from our convention of expressing angles as a portion of a 360-degree circle.”
Taking an orthographic projection of the pyramid, Torun measured all visible angles (with a calculated error of ± 0.2 deg).9 A variety of angles offer a variety of ratios. On the premise that an artificial monument would express meaningful measurements and proportions, Torun began to look into these ratios.
To understand his results, it is first necessary to make a brief excursion into the realms of sacred geometry.
SACRED NUMBERS
In the fifth century B.C., initiates of the mathematical and geometric mysteries of the philosopher Pythagoras communicated their fellowship with a secret sign. On meeting a stranger a Pythagorean would offer him an apple. If the stranger was also a Pythagorean he would cut the apple laterally across its core to reveal the pips laid out in the shape of a pentagram.10
The pentagram was a sacred symbol of the Pythagoreans, as it contained within it references to the mathematical measurement known as the “golden section,” or phi ratio:
There seems to be no doubt that Greek architects and sculptors incorporated this ratio in their artifacts. Phidias, a famous Greek sculptor, made use of it. The proportions of the Parthenon illustrate the point.11
Indeed, it was after Phidias that phi was named. Phi has to do with proportion—being the ideal ratio between two lengths that produces the greatest aesthetic effect on the eye when incorporated into the measurement of a work of art or architecture. A rectangle made of sides whose relationship to one another is based on the phi ratio will be more visually pleasing than any other rectangle.
Look at line ABC:
The phi ratio is demonstrated in a figure in which the length AB has the same relationship to the length BC as the length BC has to the entire length AC. For this to be so, the ratio has to be precisely 1:1.61803398.
Why phi produces such an aesthetic effect is a mystery, but the Pythagoreans saw it as reflecting the harmonies of nature—the same figure is found widely throughout the natural world in organic life. The spiraling of a snails shell incorporates phi, as do the distances between leaves on branches.12 The proportions of the human body also relate to phi—which, for example, is the ratio of the length of the body from the head to the navel and from the navel to the feet.
Thus the Pythagoreans claimed “all is number” and used geometry as a metaphor for higher concepts and metaphysical assertions. To them phi expressed beauty—not as a subjective opinion as in “beauty is in the eye of the beholder,” but as a quality intrinsic to the object itself. Beauty is in the beheld.
VESICA PISCIS
Phi is also generated by the most widely used and most sacred of geometric forms—the vesica piscis, “vessel of the fish”—consisting of two overlapping equal circles, the centers of which each stand on the circumference of the other circle.
To the ancient geometers this device represented the union of spirit and matter, heaven and earth.13 In it were generated not only phi, but the constants of the sacred square root series of 2, 3, and 5 and the five regular solids.14 This sacred figure was used as the basis of various ancient monuments including the St. Mary Chapel at Glastonbury Abbey and, according to John Michell, an expert in sacred proportion, the Great Pyramid at Giza.15
The Pythagorean secret sign, the cutting of the apple, was the transmission of a shared wisdom—that of the knowledge of the numerical harmonies of nature revealed through the phi ratios of the pentagram and, by extension, the vesica piscis. This message was nonverbal. All you needed to grasp it was the knowledge of mathematics, the universal language.
But what has this to do with Torun’s model of the D&M Pyramid? He claims it has everything to do with it.
ROSETTA STONE
When DiPietro and Molenaar discovered the pentagonal pyramid they noted its dimensions as 1 mile by 1.6 miles.16 These figures are, of course, extremely close to the golden section ratio.17 In Richard Hoagland’s opinion they may also have a deeper significance. Staring at the “exquisite five-sided bisymmetry” of the D&M Pyramid, he reports:
Another striking aspect of this “magic” ratio suddenly appeared before me: Leonardo da Vinci’s application of these ancient “sacred” proportions … to the human form. And suddenly I comprehended an extraordinary possibility: If I superimposed da Vinci’s famous figure—“a man in a circle”—over the stark geometric outlines of the D&M, the two conformed. The D&M seems to be a striking geometric statement of humanoid proportions arrayed on an alien landscape almost in the shadow of the central “humanoid” resemblance [the Face].18
It was this assertion of Hoagland’s that first caught Torun’s attention. What was a universal constant of aesthetic proportion doing on an inorganic mountain on Mars? Torun’s own findings were to be even more surprising, as the authoritative McDaniel Report confirms:
What Torun discovered was a mathematically rich figure whose geometry contains the mathematical bases for the hexagon, the pentagon, and the classic geometric proportions of the Golden Ratio. Twenty of the model’s internal angles, angle ratios, and trigonometric functions redundantly express three square root values, sqrt 2, sqrt 3, sqrt 5, and two mathematical constants, pi (the ratio of the circumference of a circle to its diameter) and e (the base of the natural logarithms)…. Except for sqrt 2 and sqrt 3, the constants do not appear alone, but in seven different mathematical combinations. The most redundant values discovered were e/pi, e/sqrt 5, and sqrt 3. These values were repeated four times each in at least two different modes of measurement.19
The D&M Pyramid, in other words, seems to be a veritable textbook of the same numerical forms that were deemed sacred by the Pythagoreans because of their universal harmonic qualities.
VERIFICATION
We must admit that we are impressed by Torun’s model, with its amazing ability to yield geometric constants. But wouldn’t any pentagonal figure produce the same results?
Keith Morgan, an electronics technician, devised a FORTRAN computer program at Howard University, Washington, D.C., to answer this question. Keeping the two front 60-degree angles, he adjusted the “ridge-lines” of the opposite face throughout a range of different angles, generating 680 variations on the pyramidal form. His conclusions confirmed the uniqueness of Torun’s model showing it to be the only pentagonal form with front angles of 60 degrees that could generate the vesica piscis and, simultaneously, the values of phi, pi, e, sqrt 2, sqrt 3, and sqrt 5, and the only one which could represent them all (save phi) across the three measurements of angle ratio, radian measure, and trigonometric functions!20
Clearly Torun has uncovered not only a rich geometric minefield, but a unique one in the bargain, a giant rock containing the Pythagorean constants—a true philosophers’ stone.
ALCHEMY
In the ancient art of alchemy, it was the task of the alchemist to find the lapis exillis—the philosophers’ stone—that turned base metals into gold. This stone was said to have “fallen from heaven,” like the meteoric Benben stone of Heliopolis that is spoken of in ancient Egyptian tradition, a pyramidal stone associated with rebirth.
The Benben stone bore arcane knowledge about the nature of the universe—“On the stone is encoded the cipher of life’s mysteries”21—and it was supposed to redeem spirituality from base matter, the pecuniary aspects of the process being metaphors for spiritual transformation.22
Now this pyramidal lapis, “the cipher of life’s mysteries,” is depicted as a stone—and yet it encompasses all matter, being composed of “de re animali, vegeta
bili et minerali.”23 It was also said to grow from “flesh and blood” and to possess a body, soul, and spirit.24 The lapis is thus intrinsically connected with rebirth, new life, and growth.
Strangely, Torun finds similar qualities referred to in the measurement e/sqrt 5 found in the Martian pyramidal stone:
The relationships between e and sqrt 5 may also be suggestive of biology. Five-sided symmetry is not characteristic of non-living systems. Life-forms on Earth, however, often exhibit five-sided symmetry, especially in the plant kingdom. The constant e, the base of the natural logarithms, is also known as the law of organic growth. It is a way of describing growth where the increment of growth is always proportional to the size of the growing quantity, as is often the case in biological systems. Most formulae devised for the study of organic growth, whether for population studies, or predictions of microbial and plant growth, incorporate the number e as a factor. The relationship between e and sqrt 5 might therefore be interpreted as being symbolic of “the exponential growth of life.”25
Torun supports his interpretation of these numbers as a biological metaphor by pointing to the fact that the D&M Pyramid possesses another characteristic of living things—bilateral symmetry—and “by the alignment of the D&M Pyramid’s axis of bilateral symmetry with the one object in Cydonia Mensae that most clearly resembles a living thing: the Face.”26
MESSAGE
The Pythagorean philosophers saw the vesica piscis (whose organic constants and geometric numbers are mirrored in the D&M Pyramid) as a powerful symbol of the joining of heaven and earth, spirit and matter. The pyramidal philosophers’ stone served exactly the same function, and yet, in the rhyme of the fourteenth-century alchemist Arnaldus de Villanova quoted at the beginning of this chapter: “The fools rejected it.”
Like the philosophers’ stone, it is Torun’s claim that the D&M Pyramid is some sort of cipher—a latter-day Rosetta Stone—for the whole Cydonia region, revealing a message of intelligent design. As we shall see, the same essential design features recur repeatedly among all the monuments of Cydonia. The structures seem to work together, like the instruments in an orchestra, to create an infinite mathematical symphony.
13
Coincidences
Gentlemen, you do not have a science, unless you can express it in numbers.
ARTHUR EDDINGTON, BRITISH ASTRONOMER WHO
VERIFIED EINSTEIN’S GENERAL
THEORY OF RELATIVITY
LET US remind ourselves of the mathematical characteristics of the D&M Pyramid. Among other features, its angles and dimensions yield a total of 10 pi ratios, 10 e values, and 4 e/pi values. It also redundantly “prints out” the values of sqrt 2, sqrt 3, and sqrt 5.
Such insistent repetition of geometrically significant data is not a normal characteristic of naturally formed structures. Moreover extremely accurate measurements from the Viking photographs indicate another curious indicator of intelligent design: the apex of the D&M Pyramid stands at 40.86 degrees north latitude. The tangent of 40.86 is 0.865—the precise value of the ratio e/pi that is repeated four times in the internal structure of the pyramid.1
As the Artificial Origins at Cydonia researchers point out, it is almost as though the great pentagonal monument is telling us that “it knows where it is” on Mars.2
TIME FOR T
Another notable point about latitude 40.86 degrees north as it runs through the apex of the D&M Pyramid is that it is subtended from the monument’s nearest corner-diagonal by an angle of precisely 19.5 degrees. This is an angle that crops up several times elsewhere within the structure. It is also a highly significant angle within a field of mathematics known as “energetic-synergetic geometry” that was pioneered by the American engineering genius R. Buckminster Fuller (1895–1983). The system takes as its basic unit the tetrahedron (a pyramid shape with four sides including the base—each side being an equilateral triangle) and builds from it a number of astonishing structures, most famously the geodesic dome.
A curious “rule” or constant has been revealed by this geometry and commented on by Richard Hoagland, Stanley McDaniel, Erol Torun, and other AOC researchers. The rule is that when you place a tetrahedron inside an exactly circumscribing rotating sphere so that one of its four vertices touches either the north or the south pole of that sphere, then the other three vertices, each separated by 120 degrees of longitude, will be found at latitude 19.5 degrees south (when the first vertex is at the north pole) or at latitude 19.5 degrees north (when the first vertex is at the south pole).3 The figure of 19.5 is therefore known as t, the tetrahedral constant.4
MOUNDS
Torun and Hoagland have always claimed that the tetrahedral numbers yielded by the D&M Pyramid must be significant. This claim, in our view, gains in credibility from recent discoveries by Horace W. Crater, a professor of physics at the Tennessee Space Institute. Working with Stan McDaniel, Crater has found the same specific measurements cropping up in other structures in Cydonia—particularly in the City, with its enigmatic complex of sixteen oval mounds (four of which are directly aligned with the D&M Pyramid).
Hitherto we have only commented in passing on the existence of these bright, uniformly shaped mounds, each 300 to 700 feet in diameter and 100 feet high, dotted around the foothills of the City and stretching out toward the south. Four of them form the regular cross shape of the City Square lining up not only with the D&M Pyramid but also, remarkably, with the mouth of the Face.
MISSED TARGET
When NASA re-imaged sections of the Cydonia landscape in April 1998 (see under heading “Unexpected News,” chapter 15), the four mounds forming the cross-hairs of the City Square were selected, on the advice of pro-artificiality scientists, as a rather apt target to follow the controversial re-imaging of the Face.
Unfortunately Mars Global Surveyor missed the Square and caught a swathe of land about a kilometer to its left (as seen from above), which included just a single mound and a couple of the least impressive outcrops of the City. Though other intriguing objects dot the surface of this image, unseen by the earlier Viking orbiters (such as a strange ring of small pyramidal structures and a larger pyramidal structure on the edge of a rocky outcrop for which we will have to await further analysis) little information was obtained on the enigmatic mounds themselves that could aid classification of these features and their alignments.
The only mound captured by Mars Global Surveyor is seen to be a regular, oval-shaped ridged knoll—and, unfortunately, as we have no other high-resolution images to compare it with, it is impossible to tell if it is a natural formation or whether it is similarly structured to the other mounds photographed by Viking and thus suggestive of artificiality.
The one thing that the mounds do tell us clearly about themselves, however, is their own precise locations on the surface of Mars. These locations were studied from the original Viking frames by Horace Crater and were reported on by Crater and McDaniel in their joint paper “Mound Configurations on the Martian Cydonian Plain: A Geometric and Probabilistic Analysis.”
“THEIR ARRANGEMENT WAS NOT NATURAL …”
Probably no one is better qualified to evaluate the patterns formed by the mounds than Horace Crater. A specialist in theoretical particle physics, he is a world expert on the transformation of experimental data patterns into mathematical forms, from which further patterns can then be predicted.
“Like many,” says Professor Crater, “I was interested in the controversy surrounding the Cydonian Face, but at a distance. It was not until late 1993 that my involvement with the Mars anomaly research began.”
Crater started out skeptical, saying of Torun’s reconstruction of the D&M Pyramid:
It was my suspicion that proportions with such redundancy could occur with reasonable odds in any semisymmetrical five-sided figure. Of the various five-sided figures I examined, many showed proportions like those of Torun’s measurements. As I increased the precision of my calculations, however, I came up with a surprising result.
At greater levels of precision only the Torun model appeared with significant redundancy.
This unexpected result stimulated my interest in the Cydonia region. I began to investigate a number of small mound-like features found there. These “mounds” are small enough to make measurements of their geometric relationships relatively precise, within a determinable margin of error. What I found astounded me. Their arrangement was not random.5
ANALYSIS
In his paper Crater relates how he began his investigation by labeling the sixteen mounds A through P, not in any strict order due to their positioning on the planet, but in the order he studied them. His first target was the E-A-D group of mounds—those closest to the D&M Pyramid, some miles south of the City. As Hoagland had shown as early as 1992, these three mounds form a perfect isosceles triangle.6
Crater based his measurements of E-A-D on orthographic prints, which corrected camera tilt to establish a workable Mercator projection, and found that the angles of this triangle were as follows: 70.9 (± 2.9) degrees; 54.3 (± 2.2) degrees; and 53.5 (± 2.2) degrees. These results were strikingly similar, he realized, to the angles of the plane formed inside a tetrahedron when you take its cross section from one axis so that it bisects the opposite face. These angles are, respectively, 70.5 degrees, 54.75 degrees, and 54.75 degrees. Furthermore, when the angles of the ideal tetrahedral cross section are expressed in radians, “We see that all of them are simple linear functions of [the] tetrahedral constant, t, equivalent to 19.5 degrees.”7
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