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A DIMENSIONAL ANALYSIS OF MACHU PICCHU AND THE INTIHUATANA STONE COULD REVEAL HOW LONG IT TOOK TO BUILD THEM?
Machu Picchu is considered by many to be one of the 'Navels of the World'. This is a vague term that implies "this is an important place" and we use it in that sense. In the book The Timekeepers of Ancient Earth the importance it has is attributed to its geographical and astronomical function; specifically the importance is derived from the way the monument is exposed to the sun. To us that is not an abstraction but a concrete relationship that can be demonstrated. In this regard the other term that is often used for these places 'Sun Dial' is more specific. Throughout the book we have shown how the obliquity angle of the earth's axis is reflected in the design of many archaeological structures around the world. Machu Picchu, we claim is the quintessential Sun Dial; the entire citadel, we explained how it was built to track the sun. As will be shown here its design tracks the earth's axial tilt over the ages. The Intihuatana stone froze the position of the sun earth relationship in a distant past; assuming the stone was designed and accurately built to geometrically align with the sun at the summer solstice at that location in the southern hemisphere. Any difference with the expected alignment of the sun with the angle of the stone in today's astronomical conditions we believe allows for extrapolation into the distant past to calculate its probable age. To conduct an analysis it would be required to obtain precise measurements of the pyramidal portion of the stone and an accurate reading of its azimuth along the face of the stone supposed to receive the sun at the summer solstice. Few references exist regarding the physical measurements of the Intihuatana Stone. Rolf Müller's measurements taken sometime after his visit to Machu Picchu in the 1920's, are considered to be precise. He published them in the book The Inca Empire, Springer, 1972, quoted by Dieter B. Herman in his article 'On the Intihuatana at Machu Picchu' available on line at www.dbhermann.de. In this study we seek to establish the geographic alignment of the Stone in relation to the geometry of the citadel. In the book we claimed the Citadel is aligned to the winter solstice circle of illumination. That is, the Citadel follows the general direction of the earth axis' tilt at the solstice -23.44⁰(336.56⁰) also, the actual angle of the CI at that latitude is represented in the terraces, i.e. 335.89⁰, see graphic. Also, at the culmination of the summer solstice the sun is perpendicular to the CI which in December 21 has an angle equal to the complement of 335.89⁰ (360.0⁰-335.89⁰ =24.11⁰). This means the sun has a theoretical azimuth of 114.11⁰ at the solstice. Using a solar angle calculator, for December 22, 2015 the calculated azimuth at sunrise at 6:25 will be 114.33⁰. In the graphic overlay above we rotated Müller's diagram to match the current geometry; from his 30.5⁰ to 24.11⁰. The foregoing, confirms the claim the citadel was built with a specific purpose: to reflect the geometry of astronomical events. However, the above data does not associate the Intihuatana stone with the astronomical event. That association exists only through folkloric reference or legend; also, some measurements of the shadows cast have been reported which 'verify' the stone's name "The Sun's hitching Post". We set out to prove whether the geometry of the stone does reflects the solar mechanics as is believed. We were planning to rely on Mr. Müller's measurements since we believe they were made in situ, but we found that constructing the 'parallelogram' shaped diagram of the stone's base using the angles and the lines' lengths given by Müller, it doesn't work out geometrically; the lines cross at the upper left corner, as shown in the schematic above. We realized we needed to confirm his data. Keep in mind that short of going to the site with precise measuring instruments everything else is an approximation. A lot has been said about the stone in relation to the sun's position in the sky, but to our knowledge, there is no definitive proof.

Short of going to Machu Picchu with a Transit and rulers we did the next best thing and cheapest! We Googled for photographs and to our amazement we found several excellent ones taken from various angles. We took the ones having the best alignment with a coordinate grid and proceeded to measure the Stone in absolute units. We used a photograph which appears to have been taken head on at eye level facing the south east face of the stone¹. This is the stone's face which is supposed to receive the sun perpendicularly at the solstice as shown in Müller's diagram. The part of the monument in question is the protruding pyramidal extension on top. Due to the unknown positioning of the camera by the various individuals who took the photographs made the stone's size measurements taken totally random from photo to photo; but the measurements are internally consistent for any one photograph. To eliminate the inconsistency, we took the best photograph, i.e. closest to having a cardinal alignment and normalized its height and width measurements -found their ratio utilizing a computer graphics program. We assigned a length equal to unit (1) to the SE face bottom line and then proceeded to take the other edge measurements which make up the other faces of the stone as they are found in the chosen photographs. Utilizing the 'master reference' we normalized the measurements of the remaining sides. Via this procedure we arrived at a fairly precise set of measures. Since our 'master side' was given a base of unit length, then we were able to multiply it by the corresponding value for the same side and line given by Müller. Following the procedure all of our measurements came substantially close to those given by him. The SW and NE are the wider sides of the stone. The NE face is about 12.9cm shorter due to the gradual increase in height of the base starting on the SE and ending on the NE. Müller doesn't show the difference in one of his diagrams but the length given for the SE corner is equal to the average of the two heights. In the graphic below we show the variance in terms of the ratios of the two heights.

Following these steps we confirmed the shape and size of the stone's base, roughly, but not its cardinal alignment. Müller's diagram shows the SE face of the stone -the reference face- as being aligned at 30.5⁰ vs. North; that is a 6.39⁰ difference vs. the required 24.11⁰ for the sun to impinge perpendicularly on this face at the summer solstice considering the theoretical azimuth the sun at about 114.11⁰. Did Müller make a mistake? Eventually that can be easily confirmed by someone with the proper instrumental measurements. The geometry discussed here tends to prove the stone's lengthwise general alignment with the solstice. Any significant discrepancy found vs. the theoretical value of the stone face receiving the sun would allow one to calculate the age of the stone based on the current drift rate of the earth axis angle and the net change over time, assuming the stone was precisely aligned on the day it was cut and that the rate of change is an average. The earth's axis drift, what causes it and its rate has been a subject of debate since the late 1800's**.

CALCULATIONS

Every degree of earth rotation is equivalent to 69 miles or 111.78 km. or 111,780 m. The earth's axis is reported to oscillate 1.7⁰ within the last ±10,000 years¹ around today's value; currently drifting towards the equator along the 80⁰ meridian at 10cm/yr⁴. To calculate the sun's azimuth at an earlier time by reverse engineering from the stone's azimuth is theoretically possible. From the angle we can calculate an earlier obliquity angle of the earth's axis but a small error in the measurement of the stone's azimuth will result in large time differences as will be shown.

The angle of the CI is found in the SE face of the stone. According to Müller's the angle is 30.5⁰, the equivalent angle for the earth's axis obliquity angle at the equator is 29.61⁰. This value minus the 23.44⁰, the current tilt angle, equals the angle drift that would have occurred since the stone was cut: 6.17⁰*. In meters 6.17X 111,780 = 689,682m. or 6,896,820cm. At the current rate of 10cm/yr the stone would have been cut 689,682 years ago. In this case an error ±0.5⁰ results in an age ranging between 633,792 years and 745,572 years. Incidentally, the oscillation cycle of the earth's axis is 41,000 years. *This difference is highly questionable since the calculated tilt range for the past 5MM years is reported to be only 2.5⁰ ², although larger excursions are claimed to have occurred⁵.

Another common analysis of the stone considers whether the sun casts a shadow at noon time at the solstice; the results reported in the literature are quite variable. The stone is a truncated pyramid with a slanted top and a slanted SW side and not all sides are straight; some of them are slightly bowed. We analyzed the 'shadow' from a different perspective; we measured the angle of the stone's top slant and found it to be 11.44⁰ measured from the SE side and 9.26⁰ from the NW, the opposite side. We hypothesize the stone was given this angle to reflect the sun's maximum altitude at the solstice. The normal to the first angle is equal to 78.56⁰, for the second 80.74⁰; with an average value of 79.56⁰, this would be the maximum solar altitude attained at the solstice. Using a solar angle altitude calculator (susdesign.com), at solar time noon the altitude angle on December 21 will be 79.72⁰. This is a difference of 0.16⁰. Giving this rotation the same treatment as above, the probable age of the stone's top cut is 17,884 years old an age about six thousand years older than Göbekli Tepe.

The angle of the slanted SW side of the stone, which is bowed, was measured; the average slope was found to be about 6.38⁰ from the SE and 7.27⁰ from the NW, the average slope being 6.83⁰. To project no shadow at maximum solar altitude, the sun's altitude would have to be 83.17⁰, resulting in a difference with today's maximum of 79.72⁰ of 3.45⁰, which also far exceeds the maximum calculated range for the axis tilt of 2.5⁰ in the past 5MM years² . Performing the same calculation the stone's age turns out to be 385,641 years old. The slope for the NE side is too steep at about 2.54⁰. This side appears to be not relevant to the analysis. These ages are incompatible with any other dating methods. The premise behind the 'encoding' in the stone and the alignment of the citadel of the earth's motion with respect to the sun needed further support. We could have continued performing this analysis on every angle of the stone and would come up with different ages, it could be argued. However, for us to make any educated assumption about each measurement, we felt it needed to be reconciled with other data points found elsewhere. We started by taking a second look at Müller's data; The SE face angle of 30.5⁰ implies an axis obliquity angle of +29.65⁰ and -29.65⁰ (330.35⁰)at the solstices adjusted for latitude yielding a summer solstice solar azimuth at sunrise of 120.5⁰. Also the winter solstice CI would have an angle of 329.5⁰ at Machu Picchu; the sunrise perpendicular to the CI would have an azimuth of 59.5⁰.

In the book we reported we had measured the angle of the terraces in the plaza. On the SE side of the plaza we find one with a bearing of 330.35⁰; similar to another we had described for the current obliquity angle of -23.44⁰(336.56⁰) and CI of 24.11⁰ at Machu Picchu's latitude. These two terrace angles are found next to each other. This appears not to be coincidental; however, the maximum obliquity angle, as calculated above, is out of range; but, there is something else to consider.

Focusing on the terraces in the plaza we see a pattern similar to the one we had found in the lines on the Nazca plain. In our first publication in 2014 we had speculated the 'star' like arrangement of lines found at various locations represent a historical record of the of the earth's axis angle drift through the ages. The steps in the plaza are not parallel like most of the other terraces in the citadel. They fan out at different angles; within a 10.29⁰span and there are 10 steps. Considering the axis drift has a range of 1.7⁰ it is reasonable (fractional degree changes would have been hard to build) to assume there could be a factor of 10 built into the angular bearing of the steps; which yields a total drift span of 1.02⁰. This translates into 114,015 years recorded in the steps. This would indicate the layout at Machu Picchu was accomplished over that number of years. Many researchers, both at Machu Picchu and Giza have noted the different quality of workmanship and apparent age of the stonework at each one of these locations. It is reasonable to argue that the Intihuatana was the last construction at the site: the stone's slant cut; would correspond with the last layer.

The Machu Picchu steps and the Intihuatana stone appear to serve the same purpose as the 'star' lines in Nazca. We discovered there is a connection between the two sites, as follow: The Müller solar alignment is represented in lines on the Nazca plain. A 'star' line arrangement we labeled Star 2 Node 12 in the above graphic, is located at -14.700⁰, -75.142⁰ it includes one line at 120.5⁰ equal to the solar azimuth shown above. The corresponding perpendicular line at 30.5⁰ corresponds to one of the main lines which form the Monkey Glyph; this is the alleged alignment angle of the SE face of the Intihuatana stone. A line connecting the Monkey glyph with the Star2 node also connects with the Hummingbird glyph at 329.5⁰ the winter CI angle also described earlier. Perpendicular to it is a line which is part of one of the trapezoidal glyphs at Nazca (-14.689⁰, -75.133⁰). This line has an angle of 59.5⁰; the sun's azimuth at the winter solstice. All the reported Intihuatana measurements, we find, are replicated at Nazca. In the book we had claimed "the Nazca Lines are a cartographer's map to different places" this adds a new dimension, in line with that claim: Nazca may have been also the template 'blue print' for the construction of Machu Picchu. Is Müller's angle measurement correct? Finally, if all these line angles are found at Nazca, then it presupposes the equivalent line representing the 'Old North' would be present as well. In fact, a line of a major glyph (-14.685⁰, -75.111⁰) corresponds to the 'old North' with an angle of 351.37⁰ vs. 360⁰, also the corresponding 'Old Celestial Equator' 81.37⁰ vs. 90⁰ is also present (-14.694⁰, -75.178⁰). True Polar Shifts of up to 30⁰ have been reported⁵. The 8.36⁰shift would help explain the fossilized trees found at Mt. Archernar and Beardmore Glacier in Antarctica.

The foregoing analysis of the Intihuatana's top slant and the arrangement of the plaza terraces provide an idea of what the actual age of the last cut on the stone might be. We have shown that Müller's measurement of the stone's alignment is in question. The wider than expected angle of the SE face adds a new enigma: why are the same measurements embedded in lines and glyphs at Nazca and at the citadel? Except for the net difference found vs. the earth's axis obliquity angle -at the Equator- of 6.17⁰, the geometry of the resulting astronomical layout at each location is identical; all lines correspond when superimposed -see first graphic. Until Müller's angle measurement is confirmed, we can only say that today the Intihuatana pyramidal stone is roughly aligned lengthwise in the direction of the sun's azimuth at the summer solstice in Machu Picchu. Regarding the larger than expected axial shift, the recorded shift beyond the calculated maximum could be interpreted as an instantaneous, perhaps, a shift of few thousand years duration which could have produced the forest in Antarctica. The actual 'blip' in time may eventually be read in the trees' age.

¹ Traveler photo by Cvetopipa (July 2014)

²"Obliquity berger -5000000 to 0" by Tfr000 (talk) 15:27, 6 April 2012 (UTC). Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Obliquity_berger_-5000000_to_0.png#/media/File:Obliquity_berger_-5000000_to_0.png

³"Obliquity of the ecliptic laskar" by Tfr000 (talk) 17:58, 21 March 2012 (UTC) - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons - https://commons.wikimedia.org/wiki/File:Obliquity_of_the_ecliptic_laskar.PNG#/media/File:Obliquity_of_the_ecliptic_laskar.PNG

⁴Wikipedia.

⁵ Besse, Jean; Courtillot, Vincent (November 2002). "Apparent and true polar wander and the geometry of the geomagnetic field over the last 200 Myr". Journal of Geophysical Research (Solid Earth) 107 (B11): EPM 6–1.Bibcode:2002JGRB..107.2300B. doi:10.1029/2000JB000050

**Various theories exist as to the degree and type of axial drift and its causes, some involve rather complex mechanisms; the more popular ones are: "Total earth Drift and Polar Drift" for the first, the whole earth shifts with a shift in tilt angle for the second only the mantle shifts.

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