Illuminating the Mysteries of the Great Pyramid and the Sphinx

A system of close numerical correlations is revealed between the Martian D&M Pyramid of Cydonia and the Great Pyramid of Giza which, in a system and within the accuracy of the source data, show us new numerically exact bonds between these two objects – in their geographical positions and dimensions – which are expressed in the same mathematical constants and ratios, and even use the same linear measure – the Egyptian cubit.

Key words: Great Pyramid of Giza, Egypt; D&M Pyramid of Cydonia, Mars; Mathematical constants:

, e, and their ratios; Great Pyramid ratio

, tetrahedral constant t.

Since the discovery of this formation in 1976 the D&M Pyramid (named after DiPietro and Molenaar who revealed it) continues to present probably the most attractive artefact among those that were found on other planets. In brief, the situation with this object is described as follows (Emphasize added) [1]

“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" ...

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 …
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! …

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 …

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 Far side 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:

The majority of "active centres" 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. … {SS: to this end, See [2]}”
Geomorphological and geometrical analysis of the D&M Pyramid [3]shows (Emphasize added):
“This investigation of the D&M Pyramid reveals a morphology that is inconsistent with the surrounding geology. The geomorphological processes observed to exist on Mars not only fail to provide a potential mechanism for the D&M Pyramid's formation, but seem to preclude its very existence. Analysis of the object's geometry, and its alignment with other anomalous landforms, reveal intricate relationships that are numerous and logical, and are suggestive of highly sophisticated design. …

The same techniques used for most of this century in air photo interpretation show that the D&M Pyramid may be artificial, or may be a natural landform modified by intelligence. The true nature of this object can be resolved by re-imaging the Cydonia region to obtain high-resolution imagery of the D&M Pyramid and the other enigmatic landforms nearby. The issue of possible ruins at Cydonia is thus the only question involving extraterrestrial intelligence that can be easily resolved with known scientific means.”

All these “non-random” characteristics of the D&M Pyramid provoke us to continue analysis of its geometrical and geographical properties; this automatically presumes advancing a hypothesis that the D&M Pyramid is an artificial structure, or rather a natural landform modified by intelligence, since otherwise a reconstruction the D&M’s design makes no sense. The latter formulation is probably more accurate due to a huge size of the D&M and observed irregularities in its design; the more so since the faces show greater liability to erosion than the edges: this may mean that the edges were somehow reinforced.

In both of these cases there exists no other alternative than to imply that the D&M’s design had followed some plan. If so, it is naturally to suggest that – as in the case with the Great Pyramid of Giza (GP) – the Martian geographical mile and basic side (when taken for the intrinsic units of length) specify – in a system of absolute values and dimensionless ratios – other exterior parameters of both this object and Mars itself, or take the values of primal unit of angular measure, or most important mathematical constants associated with the regular triangles corresponding to the D&M’s floor plan elements.

In case of the Great Pyramid of Giza a series of Earth and Space parameters, as well as the most important mathematical constants are expressed with the use of integers presenting its horizontal and vertical dimensions in inches and cubits. However, only an approximate floor plan of the heavily eroded D&M Pyramid is at our disposal, and without a small enough unit of length. For these reasons in analysis of the exterior of the D&M Pyramid we use the following approach.

1. Firstly, for the primal unit of length being intrinsic to the D&M the length of the shortest side of this pyramid is taken, a = 1. Then, with the use of trigonometry and Torun’s angles [3], the lengths of other quantities: sides, perimeter and area of the D&M are obtained. This approach allows us to parameterize the geometry of the D&M so that:
- with this intrinsic unit of lengthboth the ratios and absolute values of parameters are meaningful;
- for any other unit of length we firstly obtain the length a; then, the value p of any linear (or areal) parameter obtained for the case a=1 takes the value p

a (

, respectively).
In parallel with these calculatedvalues, we consider approximants for the edges presenting the primal D&M elements which are used for evaluating the derived quantities in the same manner. These are the numbers pertaining to the regular triangle (

and

), pentagon (Golden section

), and 45° right triangle (

).
Then, if both approaches give the results which coincide within the source data error we may conclude that the primal and derived approximants are consistent and thus present the intrinsic feature of the D&M.
2. After then, for the unit of length being intrinsic for Mars a Martian geographical mile is taken. In this case, as with the Great Pyramid, both the ratios and absolute values of parameters are also meaningful.

2.1. Linear parameters of the pyramid
Consider the Basic model of the D&M pyramid – the Torun’s floor plan [3, A5.1], which is presented by the pentagon P = ABCDE (Fig. 1). As far as in the below analysis we consider both this pentagon and its fragmentation into the presented triangles, all segments shown in Fig. 1 are called sides.

Fig. 1. Basic Model of the D&M Pyramid [3]
Since the absolute error in the angles specified in [3] makes

, the relative error of the source data does not exceed

(1)
Within this model the pentagon P is symmetric along the axis AO where O is the projection of the pyramid apex. As a consequence, this means the pairwise equality of the following triangles

AOB and

OAE ; (2)

OBC and

OED . (3)

The given angles uniquely define the form of the pentagon P; their properties were studied in [3]. However, its size remains indefinite until we fix up the length of one of its sides. Therefore, with the aim to investigate the linear properties of this object we have to define the length of its elements. For this, evaluate the lengths of its linear elements and areas as the functions of its shortest side AO, the length of which denote by a. Note, that this side also defines the axis of symmetry for the pentagon P and rhombus OBAE; besides, the value a is repeated five times among the considered sides.

Then, in compliance with the law of sines the lengths of the respective sides make

; (4)

; (5)

(6)
The lengths

of altitudes OH, OG, OF to sides AB, BC and CD can be calculated as follows

; (7)

; (8)

(9)
Then, the areas of the triangles

make

; (10)

; (11)

. (12)

The lengths and areas obtained from (4) - (12) call the calculated values.
From (4) - (12) it is clear that the side lengths depend on a linearly, whereas the areas of the triangles –quadratically. This means that for any value of a, viz. for any unit of length, the ratios of linear objects and areas present the dimensionless invariants pertaining to the pentagon P, whereas the absolute values of these sides and areas may present significance only if the accepted unit of length reflects an intrinsic property of the D&M or Mars.

Thus, in the case of the Great Pyramid of Giza an analysis of its sides and height in cubits has led to obtaining of a series of Earth and GP parameters, whereas the ratios of these elements including the ratio of its side and Equator (e.g. See [4]) provide us with a series of other important parameters, whatever unit of length is used.

Similarly we may await that making use of the original Martian unit of length would benefit to disclosure of a series of important relations. And though we do not know it, even if it existed, we can suggest that the Architect of the D&M presumed that we shouldn’t, and designed this pyramid so that a natural unit could be used. The most evident variants for the solutions to this problem are to choose for the unit of length (i) the smallest side of the Pyramid and/or (ii) the integer part of Martian Equator, as it is done on the Earth with the geographical mile. In these cases both the relative and absolute values are meaningful.

For these reasons we firstly consider the basicmodel with the internal measure where the length of the primal side AO is taken for the unit of length, viz. a = 1. Note, that in this case the lengths also present the length ratios of the respective sides to AO.

2.2. Properties of the Basic Model for the case of D&M’s internal unit of length, a = 1
2.2.1. The Planar properties of the Basic Model

Table 1. The Linear properties of the Basic Model for the case of internal measure (a = 1)

Quantity	Internal measure	Approximant		Error	Note
Quantity	Internal measure	symbolic	numerical	Error	Note
1	2	3	4	5	6
a	1	1		*exactly*	Side AB
b	1.4070046		1.414 214	*0.5* %	Side BC
c	1.075103		1.075 048 1.074570	*0.005* % 0.05 %	Side OC
d	1.224 068		1.236 068	0.97 %	Side CD
	0.866 025			exactly
	0.761 538		0.762749	0.16 %
	0.883 890		0.879 637 0.879053	0.48 % 0.55 %
	0.433 013	()		*exactly*	Area of
	0.535 744		0.539 345	0.67 %	Area of
	0.540 971		0.543646 0.543285 0.539 345	0.45 % 0.43 % 0.3 %	Area of
S	2.478 484		2.483 290 2.484 059	*0.19* % 0.22 %	DM base area
	1.612 458		1.618 034	0.35 %
	0.866026		0.865 256	0.09%
P	6.038 077		6.064 495	0.44 %	DM perimeter
	2	2		exactly
	4.038 078		4.064 495	0.6 %
/				0.6 %

- the area of the triangles

presenting the regular pentagon’s central isosceles;

- the area of the triangles

presenting the regular hexagon’s equilateral triangles;

- the length of sides BC, CD, DE of pentagon P presenting those of the regular pentagon;

- the length of sides AB, AE of pentagon P presenting those of the regular hexagon.

Thus from this Table we see that the best approximants for the actual lengths of the sides of the primal pentagon P are expressed in units of those triangles which they represent: the triangle

– in units 1 and

of equilateral triangle, the triangle

– in unit

of right-angled isosceles , and the triangle

- in unit

of the regular pentagon. What is important, an average error in these approximations is less than that for the source data error (1); the only exception in the whole Table is side d.
What is also important the DM-ratio

gives almost exact approximation for the internal side c being one order better than

As far as the angles of the pentagon P differ significantly from 45°, 60°, and 72°, it is not strange that the altitudes and areas of these triangles differ from the regular ones; nevertheless, they are quite accurately approximated by the Golden section and of DM-ratio.
Amazingly, but the base area S and perimeter P are also expressed in simple relations and with the use of the same values –

, g and

, as well as their parts pertaining to the regular hexagon and pentagon (engendering the five-fold symmetry); once again, within the tolerance of the source data error.

Therefore, alongside with their apparent presence in the floor plan of the D&M, the regular square, pentagon and hexagon are essentially incorporated in the mathematical design of the D&M Pyramid (Table 1) through the constants they define when triangulated:

, and

or DM-ratio

. At this, use of the DM-ratio instead of

in the obtained approximants for the side lengths, perimeter and areas results in a significantly lesser error. For distinctness, call them the primal DM constants.

Draw a circle with the centre at O and radius a (Fig. 2) and liken it to a cross-section of a planet. Then, if a diameter G*K* presents the plane of Equator (side view), the central angle

presents (as in Fig. A1) the geographical latitude of the point B, and the central angle

– the geographical latitude of the point E as In a similar way, if a diameter defined by the straight line H*O is taken for the Equator, the central angle

gives the geographical latitude of the point B, but in this new coordinate system. Here the segment OG is the altitude to the base BC, and OH – the altitude to the base AB. As these are altitudes, the respective angles

and

can easily be obtained: , .
Now, if we relate the triangle

to Earth, and triangle

(and the symmetric triangle

) – to Mars, we obtain that the altitude OH, as Equator of Earth, specifies the latitude

= 30° (exactly!), and the altitude OG, as Equator of Mars, specifies the latitude

= 40.4°. Besides, one more important latitude is specified by the vertex E:

= 19.6°. These are apart from the latitudes

= (40.4° + 60°) - 90° = 10.4° (relative to the Mars equator) and

= 90° - 19.6° = 70.4° (relative to the Earth equator; note that in this case E presents the North Pole).

But within the accepted accuracy these are namely the latitudes of the Great Pyramids and the tetrahedral constant
Latitude of the Great Pyramid of Giza:

= 29.9808°

; (13)
Latitude of the D&M Pyramid of Cydonia:

= 40.65°

; (14)
Latitude of the

(or tetrahedral) constant:

= 19.47°

. (15)
To this end it is also interesting to note the following. If b is taken for the unity of length (since the triangle

is likened to Mar’s cross-section), the perimeter makes

(

). For a = 1 the altitude OG specifies the ratio of the Equatorial and Latitudinal lengths of 1 angular unit [A3], or geographical to latitudinal miles:

2.2.3. The Spatial properties of the Basic Model
Though we do not know the actual height of the D&M, we know its estimate (half a mile) and we may suppose that it was somehow associated with the base, as in the case of the Great pyramid of Giza.
To this end consider the following hypothesis:
The height of the D&M was such that the face areas were equal.
Let l be the height of the G&M. Then the areas of the faces corresponding to the triangles

are as follows

; (16)

; (17)

. (18)
The values of Table 1 show that only the areas

and

can be equal; this equality is reached at the height

that satisfies the equation

,
from which we obtain

,
k = 0.216 228. (19)

At this height the areas are as follows
= 0.446 30, = 0.55692, = 0.55692; (20)
= 1.670 768, = 0.892 606, Z = 2.563374; (21)

where

and

are the areas of the faces adjacent to the sides BC, CD, DE and AB, AE which correspond to a regular pentagon and hexagon, respectively, and Z – is the total face area.
Then, though the absolute values of these areas are approximated by the above constants with a relatively large error, the ratio of the areas

and

is also expressed in the primal DM constants

. (22)

As far as the length of the side a is estimated [A5.2] as 1.15 mile, the height l in metresmakes

= 400 (m) (23)
which is close to the existing estimate (half a mile) of the D&M height.

With respect to [A5.2] the estimations of the lengths of the sides that correspond to a make (in meters) 1851, 1851, and 1883, and the lengths of the sides that correspond to c – two values by 2060 (m) each.
Though the values of the radial sides corresponding to a differ slightly, they virtually exactly present the Earth’s geographical mile 1852 m as this value was established by the Int. Geographic Bureau in 1928 to present the average length of 1 arc min of Meridian that changes from 1842.9 at Equator to 1861.6 – at the Pole (

). At this, 1 arc min of Equator makes 1855.33 m.

It is known that the perimeter of the Great Pyramid of Giza (about 921.43 m) makes ½ of the length of the geographical mile (the exact value of its length for the Giza latitude makes 1847.56 m), or ½ of arc min of Equatorial circumference of the Earth [4, 5]. From this we may conclude that the primal side length a of the D&M pyramid makes twice the perimeter of the Great Pyramid of Giza, or 8 times its side.

If so we may presume that applying the Egyptian cubit (with an integer multiple, e.g. 2 or 8) to the D&M may result, as in the case of the GP, in obtaining of some important geographical and/or geometrical information. And it is actually so; thus, consider the number a* = 1.16479 (miles) the rounded-off value of which (1.16) coincides with the average for the existing estimates of primal side length a in miles. Then the perimeter of the D&M in cubits (1 cubit = 0.524 m) makes

Therefore, Within the source data accuracy the perimeter of the D&M pyramid in Egyptian cubits makes the number

= 21600 of arc minutes in a circle (36060 = 21600).

In Martian Equatorial miles [A3], or arc minutes, the primal side length a makes

1.87 (arc min)

(25)
and coincides with the ratio of areas (22) of the pentagon and hexagon related faces.

So, once again it is expressed in the primal DM constants. Moreover, by dividing the length of Equator by this value we obtain

11551. (26)

Within the accepted tolerance this value presents half of the Perihelion cycle of 22 900 Martian years [Table A1]. To this end it is important to note that, in contrast to Earth, this cycle is evidently more pronounced for Mars than the cycle of precession (93 000 Martian years) due to its duration and significant difference in Aphelion/Perihelion distances [6]
Perihelion distance: 1.381 AU

;
Aphelion distance: 1.666 AU

In D&M-latitudinal miles [A3], or D&M-latitudinal arc minutes, the primal side length a coincides with the area S (See Table 1), its approximant and also exactly approximated by

2.4723 (arc min) =

(27)

The above results describing the properties of the D&M Pyramid in terms of mathematical constants give extra evidences to E. Torun’s conclusion [3] that: “the D&M Pyramid displays a complex interplay between five-fold and six-fold symmetry. Both symmetries are present simultaneously, with the front of the pyramid exhibiting six-fold symmetry, and the "ground level" of the pyramid yielding a 36 degree angle that is characteristic of five-fold symmetry”

However, although pentagon-like architectural artefacts are evidently very seldom on the Earth (not counting the Pentagon, USA), we cannot agree with the last statement of another conclusion [3]:
“This object {viz. D&M - SS}has been compared with the elaborate symbolic architecture of antiquity. While much of the geometry is the same utilized by Classical architects, it is important to note that the implementation is totally different. Nowhere in Earth history is this exact type of geometric symbolism to be found.” (Emphasize added).

Indeed, a symbol embodying a regular hexagon and pentagon, or rather the six-pointed and five-pointed stars presents an important artefact of different cultures: the five-pointed star within the Double-Triangle is a well known symbol in esotery and was widely disseminated; you can find it in Lha-khangies (temples with Buddhist statues and images), in every Tzong-pa (Lamaist sanctuary) and frequently – above the relic repository, in Tibet; it was known to the Middle-Age cabbalists as well [8].

The Six-pointed Star (Hexagon) is the symbol, in almost every religion, of the Logos as the first emanation. It is that of Vishnu in India (the Chakra, or wheel), and the glyph of the Tetragrammaton. It presents the "Macrocosm", whereas Pentagon or the five-pointed star represents a man, the Microcosm. So, the five-pointed star within the Double-triangle presents the Microcosm (a man) within the "Macrocosm" (Time-Space). [8, 9]

On the ground of quite accurate Torun’s angular estimates (despite of an excessive erosion of the D&M Pyramid), and with the use of the internal unit of length (a = 1)it is shown that apart from a set of angular ratios [3] the D&M floor plan comprising the triangles which reflect those of the square and regular pentagon and hexagon with an angular error of up to 10 % can be accurately, to within the source data error of up to 0.6%, described in terms of mathematical constants engendered by those regular polygons, viz. by the Golden number

, and

or its approximant – the DM-ratio

. This concerns both the lengths and areas of the D&M elements. At this, the Golden ratio

presents the basic element of this system of correlations. For distinctness, they are called the primal DM constants.

Conversing of the existing estimates of the D&M radii into units of length being intrinsic to Mars – Martian geographical miles – reveals a system of direct correlations between this object and the Great Pyramid of Giza and Earth herself.

However, although this unit of length, when compared with the circumference of the Martian Equator defines the Perihelion cycle for Mars, in the absence of a comparatively small “Martian unit of length” we have no grounds for obtaining the full scope of those great numbers which define the Martian periods (obliquity, precession, etc.) as it was possible with the Great Pyramid of Giza.

Nevertheless, the obtained results show that if the D&M Pyramid is formed with a touch of Intelligence, these intrinsic units of length – the length of the shortest side and the Martian geographical mile have reasonable chances to present the keys for other Martian mysteries.
Besides, the evidences obtained in favour of one more constant – 19.5 – which:

To this end it is important to note, that namely these two numbers,

= 19.5 yr being associated with the duration

= 22.5 yr of the Galactic alignment, define, but in degrees, the most common angle relationship of 22.5° to 19.5° in Cydonia [11]; the more so that this value 19.5 is treated in the analogous way, whether it defines angle, or time: it presents both the point values and intervals of latitudes

19.5° and time

19.5 yr.

In particular, the following results of this study give evidences that the D&M Pyramid design not only integrates a series Mathematical and Martian constants, but a system of relations pertaining to Earth and Great Pyramid of Giza as well.

C1. Alongside with their apparent presence in the floor plan of the D&M, the regular square, pentagon and hexagon are essentially incorporated in the mathematical design of the D&M pyramid (Table 1) through the constants they define:

, and

or DM-ratio

. At this, use of the DM-ratio instead of

in the obtained approximants for the side lengths, perimeter and areas results in a significantly lesser error.

C2. The areas of the D&M faces the bases of which correspond to the regular pentagon are equal just when the Pyramid apex is placed at the specific height, and this height corresponds to the existing estimate of the altitude of the D&M Pyramid. At this, the ratio

of the areas corresponding to the pentagonal and hexagonal faces is expressed through the same primal constants

Hoagland discovered that the geodetic Martian latitude = 40.8681938… almost exactly passes through the D&M Pyramid. Amazingly, another trigonometrical function – Arc cosine of the same ratio – gives thelatitude of the Great Pyramid of Giza = 30.08805…[12]

C3. Within the accepted accuracy the floor plan of the D&M pyramid taken as a cross-section of the Globe in a similar geometrical approach defines the co-ordinated latitudes of the Greatest Pyramids on the Earth (14) and Mars (15), alongside with the latitude (16) equal to

, or Tetrahedral Constant.

C4. The basic dimension of the D&M pyramid – the length a of the sides of the equilateral triangles – virtually coincides with the Terrestrial geographical mile, that is with the length of 1 arc min of Earth’s Equator.

5. The primal linear parameter of the D&M pyramid – the side a – makes twicethe perimeter of the Great Pyramid of Giza, or 8 times the length of its side.
Amazingly, but the Egyptian cubit also plays an important part in these correlations:

C6. Within the source data accuracy the perimeter of the D&M pyramid in Egyptian cubits makes the number of arc minutes in a circle,

= 21600 (viz. 36060’)

C7. In Martian Equatorial miles [A3], or arc minutes, the length of the side a is also expressed in the DM constants,

and coincides with the ratio of areas of the pentagon and hexagon related faces.
C8. Moreover, by dividing the length of Equator (in arc minutes) by the length of side

(in the same units) we obtain

11551. Within the accepted tolerance this value presents half of the Perihelion cycle of 22 900 Martian years [Table A1].
To this end it is important to note that, in contrast to Earth, this cycle is evidently more pronounced for Mars than the cycle of precession (93 000 Martian years) due to its duration and significant difference in Aphelion/Perihelion distances [6]
C9. In D&M-latitudinal miles [A3] (or D&M-latitudinal arc minutes) the length

of the side a coincides with the D&M base area S (See Table 1), its approximant and also exactly approximated by

C10. The formerly obtained correlations for latitudes in degrees and the results of this study relative to length of 1 arc minute of Martian Equator and D&M latitude give us the grounds to resume that the conventional terrestrial grade measure (360°, 60’) is appropriate to D&M Pyramid design and Mars as well.

C11. The incredibly reflective faces is very important feature for both the D&M and the Great Pyramid of Giza; as the Great Pyramid was shining under the Sun, so “the mysterious "D&M" … literally "glow" in the pre-dawn light”.
The concavity of the faces presents one more important feature being common to these two pyramids.
In details these aspects of the D&M are considered in [13] for D&M, and in [4] – for the GP.

C12. Although the pentagonal base of the D&M is not seen among the architectural artefacts, the hexa/pentagonal base of the D&M (Fig. 5.a) provides us with a “unitary” symbol of the universal concept which is reflected by the Terrestrials by the double symbol of the inserted stars (Fig. 5.b).

a) b)
Fig. 5. The Martian and Terrestrial Geometrical presentation of the Concept
“Microcosm” within the “Macrocosm”

At the same time, as far as the new images of the D&M obtained by Mars Odyssey spacecraft have not allowed to increase the accuracy of Torun’s Model (even though they provided us with better images of the most damaged Eastern side), it is still assumed that “more precise angular measurements may also confirm Torun's original model” [7]. (Emphasize added).

November 5, 2007
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