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THE PYRAMIDS OF GIZA: A GEOMETRIC ANALYSIS OF ITS LOCATION & DESIGN
The pyramids of Egypt, in particular the ones at Giza, are perhaps one of the most studied archaeological sites. It can be safely stated that most methods of human inquiry have been used in their study. All those studies and analyses, physical instrumental and otherwise, have spawned a myriad of theories about how and why they were built and by whom and what their ages might be. None of these studies, however, have answered to a reasonable certainty the questions: Why are they located where they are found? Was their locale casually selected; what determined it? Is there a reason for their alignment bearing north and for their relative sizes, heights and shapes? Are these monument sites geographically related to others around the world in any measurable way? Earlier research had shown that many archaeological structures are aligned with the cardinal points, the pyramids of Giza are the most prominent but this alignment is found in temples and monuments as far south as the Temple of Kalasasaya in Tihuanaku, Bolivia and as far east as the Temple of Angkor Wat in Cambodia. This is well known. In the book Nasca & Easter Island, we published the discovery that monuments which do not align with the cardinal pints, almost invariably, their azimuths set the direction for a great circle which when followed around the earth coincide or align with other archaeological sites. Also, sometimes, when two such circles intersect, one or both points of intersection are marked with a monument as well. The pyramid of Cheops is one such intersection point. Giza's Location and Layout Reflect the Earth's Plate Tectonics In the book N & EI we reported two alignments for the pyramids of Giza, specifically, for the pyramid of Cheops. The apex of this pyramid is the point where two great circles (ellipses) intersect: A great circle that can be drawn following the direction of the Abbas Giant's mace, in southern UK, and another that follows the direction of the Ahu Akahanga, in Easter Island (EI). This geometry we argued, fixes the relationship between the locations of these three archaeological sites; which lead us to conclude, these sites are related to each other in a planned design. The Ahu Akahanga great circle -see 3D Graphic below-we reported, could be attributed with having had the purpose of setting the latitude for the location of the pyramid of Cheops. In that study we showed that the arc segment of the Akahanga circle which connects the Cheops pyramid with Basrah, reaches the northern most latitude of 30.4⁰N at a point nearly half the distance between the two locations at longitude 41⁰E. The Cheops pyramid has latitude 30⁰N and longitude 31⁰E. The circle is made up of seventeen archaeological sites, whose coordinates were drawn in the Google earth photos shown and trigonometrically plotted in 3D, thus confirming the precision of the alignment. The Abbas Giant's mace circle centers the pyramid ten degrees west of the maximum latitude. The Abbas circle shown in green is comprised of thirteen sites. In the 3D plot the sites -the red dots- for each circle are shown. The Ahu Akahanga circle has additional points not corresponding to archaeological sites; some show the crossing of the Meridian, the Equator and the Tropics of Cancer and Capricorn, plus a few additional points to enhance the illustration. The Abbas circle does not exhibit these. The immediate question this prompts is; what determined the geometrical position of these two great circles? One can argue that Easter Island being relatively small and at a distance of about 16,000km. from Giza, it could be considered to be a point and any great circle passing through it could be jogged to match the pyramid's apex. However, the Ahu marks not only the specific point in the island the arc should pass through, but also its direction. The Ahu is not aligned with the cardinal points but has an azimuth of 75.77⁰; a great circle in that direction reaches the pyramid of Cheops therefore we concluded this anchor point was designed and built for this purpose. In the book we reviewed several other alignments of similar importance and purpose; with other Ahus in the Island and several structures elsewhere. The case for the Abbas Giant's circle is different. It could have been drawn anywhere pointing his mace in any direction, except, we found the direction of his mace traces a great circle which follows almost 3,000km. of the South Indian Ocean fault line, in the book N&EI we referred to it as the SIO Glyph. We named it so due to its unique convoluted shape and continued using the term glyph for the other earth's ocean fault lines that were encountered and have unique shapes. One such glyph is shown below, which we will discuss later. The two facts just described lead us to conclude that the pyramid's location was very likely selected to be there as prescribed by the fault line and EI's Ahu Akahanga, and ventured to say that the pyramids' arrangement and north alignment were probably not just due to stellar alignments, as has been claimed for many years by a number of researchers^3. The two circles are not only determined by the azimuths of the Ahu and the Mace, but, each circle encompasses other alignments around the globe; the first with seventeen archaeological sites and the second with thirteen sites. These circles and the sites are identified in the Google^©maps shown. However, the foregoing explanation of the geometry involved in fixing the site for the pyramids said nothing regarding the physical layout of the site or the number of structures present or their shape. The stellar alignment with the constellations of either Orion or Cygnus, the various researchers claim they have, remained unresolved and in doubt nonetheless. We sought to find a rationale that would explain the diagonal alignment of the pyramids that makes the alignment with Baalbek and Mount Ararat possible, as has been claimed^4. The continuation of the study resulted in further discoveries and data that reaffirmed the original argument and explained the physical layout of the complex: the pyramids’ location, heights, bearing and distance between them. These variables are dependent on a different set of earth's fault lines or ridges in the ocean, located in the South Pacific Ocean at -25.250714° -154.263026°. The fault lines can be seen with Google^© earth by viewing the ocean closely without historical mapping or by downloading a file^5 which overlays the ocean lines. The first question we sought to answer was the reason for the north-south alignment for the three pyramids. That alignment as well as their claimed alignment with constellations exhibited, in our estimation, the same problem; the North Star Polaris is as much a moving target as the constellations are; which we had argued diminishes the possibility of an ancient civilization having designed a layout of this magnitude based on alignments with stars -moving targets, particularly with the presumed limited technology available to them, despite the myriad theories that exist of how it was done. One such theory quoted in Building the Great Pyramid^7 is Otto Neugebauer's "On the Orientation of Pyramids" In his theory he proposes a wooden or stone model called a pyramidion; "The pyramidion is precisely aligned north to south" and the shadows cast by it by the sun at various times." The method says nothing about the time element involved over the twenty to thirty years^9 it is estimated it took to build them with crews of up to 4,000 men^10. Again, the wooden model relies on Polaris and as mentioned there the method was only useful in winter. Creating a geo-centric referencing system is the subject of study by the International Terrestrial Reference System, for which various datum have been developed to create global standards for global positioning, aeronautics, space, defense, geographic and point of interest POI applications-. "The International Terrestrial Reference System (ITRS) describes procedures for creating reference frames suitable for use with measurements on or near the Earth's surface. This is done in much the same way that a physical standard might be described as a set of procedures for creating a realization of that standard." ^6 From the data collected in our earlier study, it became apparent to us that the tectonic plates positioning were used for this purpose in ancient times. The current work appears to confirm that belief and strongly suggests the fault lines could become a useful standard like other geodetic standards the ITRS uses, such as triangulation pillars or bronze disc markers placed in various areas of the earth indicating coordinate points accurate to ten decimal places. In the first part of the published research, we established a procedure to initiate the geographic evaluation of the positioning for any structure within a site; first we determined if the structure is aligned with the Cardinal axis and which feature(s) of the structure are involved in the alignment. The Giza pyramids’ alignment with the Cardinal points is common knowledge, well documented by numerous studies found in the literature^7, the reference given is a fairly comprehensive such references are available review of pyramid facts, many of which reflect its preferred theories. In the current study we used the Chephren pyramid to trace global ellipses following the Cardinal points, each ellipse starting at the southeast corner of the pyramid, at 90⁰ from each other; thus their intersection point, half way around the world, determines the pyramid's antipode point on earth. In N & EI book chapter: Ocean Lines, we described a fault line found south of the French Polynesian islands of Moorea and Tahiti in the South Pacific Ocean (dubbed SPO Glyph) –a tracing of it is shown left. In the Chapter we discussed how its shape had been copied and engraved as a glyph at Nazca by the ancient authors of the Nazca Lines: it is the perch on which the famous Monkey Glyph stands -see graphic in the Image File Page-. The graphic shows that the fault line is composed of a vertical line which is located at the 150° longitude meridian, starting in Antarctica and ending south of Tahiti. There it folds back going south-west, then returns northward in a zig-zag manner. The pyramids’ antipode point is located at the second fold of the fault line at about 30.0°S 149.0°W. In the graphic we also show the location of this point with respect to the fault line glyph. In the graphic we enlarged (~650 times) a layout of the pyramids drawn to scale to depict their relationship. The ocean fault line has two important characteristics: The vertical line runs north-south, and is located exactly on the 150° longitude meridian. The pyramid of Cheops location is at about 31°E, so, at its antipode point its longitude is about 149° and the pyramids' NS alignment runs parallel to it. This line of the SPO Glyph is what we believe determined the North alignment of the pyramids and every other structure thus aligned; such as the Temple of Kalsasaya in Tihuanaco, instead of their alignments being based on the mobile Polar Star. The SPO Glyph has a natural earth line that happens to precisely run north south; as such it is a perfect 'immutable' anchor, which at this point of our research appears to be unique amongst all fault lines; if anything it is perhaps the longest and most obvious line. We believe this line, which runs north all the way from Antarctica for 6,000km, is the Ancient's original geo-centric based source for the "North" concept. We were able to safely come to the conclusion that the Pyramids' geometry is related to this Glyph by surveying the other components of the fault line. The other features of the fault clearly show the connection the pyramids’ layout has with the entire fault line's layout, i.e. there is a nearly one to one (1:1) relationship between all their geometric measurements: geographic alignment and positioning on the earth, the heights of the pyramids with the glyph's peaks' heights, and the distance between them. Also, the overall layout's pattern; it is shown as a white 'step' line in the graphic below. Graphic 1 Now the comparison in numbers: The line that follows the 150° longitude meridian ends just south of Tahiti, there as mentioned, it folds back SW at an angle of 195° and continues for about 2,000km. From there it returns north-west following a symmetrical zig zag path, eventually reaching past the island of Moorea. The zig zag line has three major saw tooth-like folds at almost right angles with each other; the peaks that point east, not their corresponding troughs. The peaks of the three teeth are distanced from each other as follow: 752km. between the largest in the south and the middle one to the north and 716km. (.95) between the middle one and the third to the north of it. The ratio of the distances is shown in parenthesis. The first two peaks are near equally sized while the third one is the smallest. Their heights are: 479(1), 459.4(.96), 285(.59) km. The ratio of their heights is shown in parenthesis. Similar measurements were taken for the Giza pyramids, the distance between their apices are: Cheops-Chephren-490mts., Chephren-Mycerinus-440mts. and their ratio (.90). Likewise the pyramids’ heights (original) and their ratios are: 147(1), 143(.97), 109(.74)mts; comparing the ratios; heights and distance show near correlation with the fault line’s geometry. Recall that the pyramids' geographic location is antipode to the SPO Glyph. These numerical relationships can be better appreciated visually in Graphic 1 shown above. In the graphic, superimposed on a map to scale of the pyramids, is a tracing of the earth’s SPO glyph also drawn to scale and then reduced close to the pyramids’ size. The graphic of the SPO glyph was moved north along its NS axis, over the North Pole until it reached Giza, then, it was moved about a degree east to align with the Chephren pyramid. It is important to emphasize that at this location the SPO glyph's north becomes its south; as such it makes patent the geometrical relationship it has with the pyramids' layout; the glyph was geometrically translated. In this position the angle of the peaks is nearly the same as that of the pyramids' arrangement of ~39⁰; also each peak coincides with each pyramid in the same order; from largest to smallest. If the pyramid of Cheops' base were to be enlarged a thousand times it would fit snuggly in the square section of the glyph on the upper left. In the graphic this relationship is shown in dotted lines. This fact addresses the question; why are they square not any other shape? Also, the size of the pyramid's base may have been set to be a multiple of the size of the square section; once the shape and size of the base and the height are given all the other measures are derived from these^7. After reducing the SPO glyph's size a thousand times, curiously, the difference in areas divided by a thousand and then the cube root taken the result is equal to the number of pyramids: (283^2 - 230^2)/1,000 = 3^3 In Graphic 2, the arc segments for both the Abbas Giant's Mace and Ahu Akahanga's are shown as they cross over the pyramid of Cheops' apex. The Abbas Giant ellipse line is about 15⁰ off the pyramid's diagonal. Following is a second geometric analysis. We took the SPO Glyph tracing and rotated it along its NS axis to obtain a mirror image, followed by a rotation of about 75⁰ clockwise about its center. This transformation yielded extraordinary results. While maintaining the overall alignment of the SPO glyph's peaks with the pyramids, some of its lines became aligned with other features of their layout, this time pointing out their relationship with the Sphinx and the causeway that connects it with the Chephren pyramid. Simultaneously it aligned the southern line of the glyph's peak which corresponds to the pyramid of Cheops, with the diagonal line across it which is closely followed by the ellipse that lines up with the Abbas Giant's Mace and the SIO Glyph. This exercise showed the closed reference matrix that exists between all these elements. The geometric layout of the Giza complex reflects the exact location and layout of two of earth's most salient geological features: The South Pacific and South Indian Ocean's fault lines. Summary; the SPO Gyph set the 150⁰ meridian great ellipse that determined north, the alignments the pyramids were given, but, not their location along its circumference. The SIO Glyph alignment ellipse would have placed the pyramids 8km southeast of Alexandria the point of its intersection with the 30⁰(150⁰) meridian. The actual location is approximately one degree south and one degree east from the ellipsis' intersection. This location at the antipode point corresponds to approximately the midpoint of the line that connects the two large peaks on the SPO glyph; which may have determined the one degree displacement in both directions. The Abbas Giant and the Ahu Akahanga appear to have been designed as geodetic markers to help unravel the logic in the design and layout as was exposed here. Analyzing the pyramids by themselves would have not revealed the underlying (literally) rationale for their location, layout and sizes. The triangulation offered by the two geodetic markers provided the path. In our analysis the SPO Glyph was aligned NS inverted from its natural alignment in the South Pacific Ocean, also, it was positioned over the eastern side of the pyramid of Chephren; the alignment line we used to find the pyramids antipode point- shown in the graphic as a star near the southwest corner of the pyramid of Cheops. The geometric translation of the SPO fault lines tracing -to scale- results in a near perfect superimposition of its layout over the pyramids' site; this indicates that the variable layout of the pyramids was fashioned after the immutable fault lines' layout. Graphic 2 The relationships described here are only a small part of a greater globe spanning arrangement. Graphic 4 below is a Google^© earth map of the SOP and SIO fault lines, it shows their locations relative to Australia, Antarctica, Easter Island and South America, superimposed with the alignment arc lines-ellipsis- previously mentioned. In addition it shows another great ellipse (tan) that connects the pyramid of Cheops with the SPO fault line going through the Sajama desert. This latter connection will be discussed below. In the graphic on the lower left one can see the 3,000km. segment of the SIO fault line that is closely followed by the Abbas Giant's ellipse. The Ahu Akahanga line appears in blue color and Easter Island is shown (not to scale) in the upper right quadrant. The SIO glyph has a spur line headed southward at 202.79⁰ starting at -42.434°S 89.837°W and running for 634km. Following this line over Antarctica it crosses the Sajama desert and the Isla del Sol in Lake Titicaca; the line is shown at the bottom of Graphic 4. The Sajama desert region has been claimed to have been the seat of an earlier civilization^8, specifically it is claimed to have been a settlement of the people of Atlantis. The Sajama region encompasses the pre-Inca civilization region of Lake Titicaca where the citadels of Tihuanaco and Puma Punku are found. This spur line intersects the tan ellipse that connects the pyramid of Cheops and the SPO glyph going through the Sajama desert. The results of our research of the Sajama Desert show features in the desert determine the position of the Cheops-Sajama-SPO ellipse. That triangulation makes the Sajama desert a Point of Interest. It seems that the ancient structures in the region of Titicaca, Tihuanaco and Puma Punku, were also designed following the same geographical scheme, as we have discussed, was used for the Giza pyramids, although, to a perhaps lower level of geometric sophistication. Graphic 3 In graphic 3 the green line is an outline of the base of the Akapana Pyramid in Tihuanaco, enlarged to scale, then laid over the Giza layout. Although the peak ratios for this outline are not as close as for the SPO Glyph and the Giza pyramids, it still reflects similar geometry. These relationships may be emphasized by the fact that the antipode point for the pyramids is equidistant from Sajama and the point where the Abbas Giant alignment ellipse begins to run parallel to the SIO fault line, thus forming an 'isosceles spherical triangle' with the sides ratio of 1:.66:.66, which coincidentally is approximately the same as Easter Island's shape and has the same side ratios. The graphic below illustrates these relationships. Graphic 4 In the book N & EI we observed the near absence of archaeological alignments the Giza pyramids have with other archaeological structures, unlike most of the others we presented there do; save for the diagonal alignments mentioned and a few in the Cardinal directions: immediately north across the Mediterranean in Turkey we find Aspendos and Altinkaya. To the east are the Rajajeel and Junapani circles, which appear to be directional markers. The importance these have were discussed in the book along with other circles. The geometric alignments discussed here appear to have as their main purpose to set the location of the pyramids on earth; the alignments found for the pyramids answer, for this site, the basic questions that set the objectives for this research: " Why are archaeological structures located where they are and what determined it?" From the evidence discussed here and other previously reported, we come to a conclusion: The pyramids had as their purpose to rise the geological 'markers' inherent in the ocean plate fault lines to the surface; in full view for a future (our current) civilization, to learn about the astronomical facts the earth's plates encode. The pyramids also serve as gigantic sextants that appear to complete the astronomical connection with the stars, thus placing earth in its universal context. An extraordinary legacy from a technologically advanced ancient civilization. Bibliography 1. HARDOY JORGE E. 1973 Pre-Columbian Cities-. Walker and Company NY 2. http://www.archaeological.org/Archaeology%20of%20America%20and%20Canada.kml. 3. http://www.math.nus.edu.sg/aslaksen/gem-projects/hm/0102-1-pyramids/page1002.htm 4. Z. SITCHIN 2008 The End of Days. Harper Collins 5. http://mw2.google.com/mw-earth-vectordb/blog/Google_Earth_Seafloor Updates_06_2011.kmz. 6. Wikipeadia.com 7. http://www.cheops-pyramide.ch/khufu-pyramid/pyramid-alignment.html 8. DONNELLY IGNATIUS Atlantis, the Antediluvian World, at sacred-texts.com 9. curiosity.discovery.com, www.ancient.eu.com 10. http://www.bbc.co.uk/pressoffice/pressreleases/stories/2002/09_september/13/pyramid_pack.pdf 11. http://archaeology.about.com/od/assyria/fl/The-Aqueduct-at-Jerwan-Iraq.htm 12. http://archaeology.about.com/od/dathroughdeterms/qt/Desert-Kites.htm 13. http://www.antiquity.ac.uk/projgall/beck308/ 14. http://www.tandfonline.com/doi/abs/10.1080/00438243.2010.498640#.VACAH6NCJuA 15. Villamarin, Arturo. April 2012. Nasca & Easter Island. An ancient Global Plan Revealed. Amazon
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