Introduction

In this article I focus on archaeogeodesy study results from the perspective of Machupicchu. Both herein and in new KML files and KML file updates I've adopted the current orthography used by Peru's Ministerio de Cultura; Machupicchu instead of Machu Picchu, Waynapicchu rather than Huayna Picchu, and Intiwatana replacing Intihuatana. Previous archaeoastronomy and archaeogeodesy writings provide context for this article, as recently reiterated in Chavin de Huantar Archaeogeodesy where Machupicchu (Machu Picchu) is prominently mentioned (Figure 1). This article repeats some of the material in previous articles, quotes some of them, and links back to relevant material.

Figure 1. Chavin de Huantar, Machupicchu, and Phi.

Machupicchu has been one focus of my archaeogeodesy research from the beginning. A lack of confidence in accuracy of coordinates delayed releasing most results until recently. Crowd sourcing and data mining global positioning system (GPS) data online and site maps with UTM grids provided sufficient information to refine coordinate accuracy and proceed. Coordinates for several monuments at other sites have also been recently updated. Given precision of results such as seen in Figure 1, the most accurate possible determination of coordinates is quite relevant and necessary.

Some coordinates still require refinement and I am mostly withholding those preliminary results. All results need to be considered in the context of site coordinate margins of error, thus discussion of methods and related factors precedes presentation of the results.

Methods, Coordinates, Precision, and Margins of Error

Precise geodesy is a recent historical achievement. Geodesy as a science measuring the earth has attained the mathematical precision needed to assess if past civilizations measured the earth with precision. New technology makes the ability to determine the latitude and longitude of a monument, be that a petroglyph or a modern property marker, accessible with known accuracy to anyone with a hand held GPS device or even a smart phone. Personal computers with access to satellite imagery displayed by a geograhic information system (GIS) allow determinations of latitude and longitude of any visible feature on the earth. While vastly improved precision in recent decades capacitates this study, margins of error need to be understood in quantifiable terms and must be accounted for in determining confidence in interpretations of data and results.

Accurate determination of ancient monument coordinates can be problematic. Over recent decades it has become much easier with the capacity for precise GPS readings, the world geodetic system (WGS84), online availability of maps and shared photography, and, when precise data is not elsewhere found, GIS tools such as Google Earth (GE). Over time the precision of GE has improved as updates with higher resolution imagery and refined imagery placement increase the area useful for archaeological research.

To the extent possible I use GPS data for site coordinates and when recording my own data I annote the GPS margin of error. GPS data varies in accuracy and margin of error is not always noted. Comparison of my own GPS data of Eastern Woodlands and Southwest monuments with GE high resolution imagery positioning confims accuracy sufficient for many purposes of my studies. The GPS data also allowed observing improved accuracy over time with GE high resolution imagery updates. The GE historical imagery tool enables viewing older satellite images. I database the access date when recording GE-based coordinates to time reference the data capture. The Ministerio de Cultura's Machupicchu detailed site map with a UTM grid verified the accuracy of GPS-determined coordinates, revealed the GE image displacements, and improved the accuracy of some older GPS data and several GE determined coordinates.

What most impacts GE precision in relation to Machupicchu is the terrain, vertical mountains. Obliqueness of camera position is a significant accuracy issue with GIS tools employing satellite imagery. Almost all satellite imagery has an oblique camera angle, and with steeper terrain the margin of error is in all probability proportionally greater. This aspect of employing photography is correlated with departures from a flat landscape in the image capture area in combination with the degree of obliqueness. Needless to say, in and around Machupicchu the vertical landscape exacerbates this problem, necessitating quantification of the error variable. To get a good idea of how oblique camera angles impact point positions in the imagery, use the Historical Imagery tool in GE to compare the various captures of the pyramids at Giza. While the pyramid bases remain in almost exactly the same positions, the apparent area of the pyramid sides changes immensely and the apex coordinates shift considerably. With monuments of any prominence, I always use the monument base to determine centerpoints, rather than the apparent summit in a satellite photo.

Now consider the relative height difference between the Great Pyramid and the mountains around Machupicchu or even just the difference between the buildings at Machupicchu and the summits of Montaña Machupicchu and Waynapicchu and the problems presented by overlaying 2-D satellite images on 3-D terrain are ready discernible. The following image (Figure 3) illustrating the margin of error at Waynapicchu is included in the KML file along with individual placemarks for the tessellated red lines. You can interact with the Historical Imagery tool in GE and observe the placement of the mountain summit shift to the endpoints of the dated lines.

With steep terrain, GPS data for monuments and summits overcomes the significant problem of resolving accurate coordinates from satellite imagery. I continue to research and request GPS data and any contributions of data are appreciated. The number of useful online data resources has increased recently. GPS is becoming ubiquitous with cellular phone usage. Phone apps record user positions and hikers post track files and photography on websites dedicated to sharing data. For both Machupicchu and Waynapicchu peaks, I crowdsourced the currently available data on wikiloc.com to derive mean coordinate and thereby reduce margins of error in the coordinates. Steep terrain presents problems for GPS accuracy also by limiting the number of satellites having reception, but at least atop the peaks that problem is negated and hikers also tend to remain still there for an interval. Cellular phones are not the most precise GPS devices, so I determined mean coordinates from numerous waypoints and favored tracks produced by dedicated GPS devices. Tracks with higher margins of error are discernable and I avoided these.

The margin of error in imagery placement for the Machupicchu ruins is far less than for Waynapicchu summit, but nonetheless more than seen in flat terrain. Crowd sourcing the wikiloc.com GPS tracks was useful in determining which historical imagery placement of the architectural features and paths was most accurate in relation to the GPS data, thus providing more confidence in the accuracy of updated and added coordinates and improved quantification of margins of error. Visitors to the site are delimited to walking specific paths and this fact increased the utility of their GPS tracks. Detailed archaeology site maps, information in The Machu Picchu Guidebook, and the multitude of shared photographs online, all in combination, now provides high confidence in the monument coordinates.

In GE Layers, one option is to display terrain elevation. In the Preferences 3D View tab several terrain options are available, an elevation exaggeration setting and an option for high quality terrain modeling. A digital elevation model (DEM) of satellite data employs vector points and terrain imagery is folded accordingly (Figure 4). This variable from precise is best illustrated with the problem of determination of mountain summits as illustrated in Figure 4, also a placemark in the KML file. Unlike the Waynapicchu summit, photography of mountain summit snow fields often provides far less visual data for summit determination.

Topography, Digital Elevation Models, and Margins of Error I created this Huascaran summit image to illustrate GIS elevation modeling inaccuracy in comparison to precise GPS data. Viewing a grid pattern overlay from various angles and tilts and outlining topographic contour patterns in GE readily illustrates the inherent pattern in digital elevation modeling of satellite data. The limitations of DEMs are made both readily visible and quantifiable. The added blue lines illustrate the distance between folds in the topographic model, about 75 meters. The shape of the 6720m contour pattern, the red outline, reflects the digital model fold pattern's highest topographic elevation rather than the actual topography. My "huasc" placemark represents the recent Instituto Nacional de Investigación en Glaciares y Ecosistemas de Montaña (INAIGEM) +/- 3.5mm GPS summit determination. The highest elevation in the DEM does not even include the summit because the modeling is based on a pattern of vectors spaced about 75 meters apart. Figure 4. Huascaran summit and DEM comparison. INAIGEM determined summit: UTM: 18L 213778.255, 8990724.614, 6756.697 +/- 3.5mm The highest resolution Shuttle Radar Topography Mission (SRTM) DEM released to the public has a spatial resolution of about 30 meters. The Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) GDEM-2 is considered a more accurate representation than the SRTM elevation model in rugged mountainous terrain, improving on the original GDEM with global resolution of 90 meters and 30 meters in the United States.

Geodetic Scale

While analysis of the geodetic relationships of monuments requires coordinate precision to provide confidence in results, the context of the relationship considered impacts confidence immensely. The globe is about 40 million meters in circumference and the arc distance from Machupicchu to Newark Earthworks is nearly six million meters. Thus when considering the location of a monument at Newark in relation to Machupicchu, a GPS datum with plus or minus five meter accuracy has a margin of error of five parts in near six million, about one part in one million. Monuments spaced at great distances from each other offer very high confidence in study results when the margin of error is only one part per million. Likewise, to monumentalize an astronomical constant with mathematical precision, the monuments need to be spaced across vast distances. The methodolgy of determining site coordinates herein is certainly sufficient for analysis at this scale and for more proximate considerations too.

For intra-site anaylsis, precise GPS or direct survey is necessary to attain reasonable confidence in the results. Comparing a monument at Machupicchu to the nearby Waynapicchu and Machupicchu peaks, given near precise datums, the margin of error would be nearer five parts in two thousand, or one part in four hundred. The intra-site results in the KML file are in a separate folder due to the lessor geodetic scale and correlated lower confidence in results. Given the correlation between larger scale and much greater confidence in the results, I prefer researching regional and global monument relationships and geodetic properties such as latitude. As previously discussed, I almost entirely eschew determinations on the scale of single monuments, such as alignments of megaliths in a stone circle or openings in earthworks. Of course, global scale relationships and site latitudes provide context and possibly greater confidence in intra-site analyses.

The longitude difference of Cerro Salcantay's DEM coordinate and the Intiwatana is just over fifty meters. The Cerro Salcantay coordinates (sallq) are based on digital elevation modeling, unlike the GPS coordinates for Apu Salkantay (salka). When two sites on a meridian are compared with a third site, the margin of error proportion can be even more significant. Expressing this difference in relation to another longitude difference in an east-west ratio, the ratio's margin of error can result in zero confidence. These ratios can only be considered approximate and while many such preliminary results may be interesting and are recorded, they are not presented. The few DEM coordinates and results in the KML file and herein discussed represent larger scale relationships. Interesting relationships between Machupicchu and surrounding peaks with only DEM coordinate data are all considered preliminary and few are presented herein or in results placemarks.

Tectonic Motion

Another important factor impacts ancient monument coordinate accuracy, the older the construct the greater the monument displacement due to tectonic motion. I included a tectonics folder in the Chavin KML file to model Chavin's motion since the Neolithic, about the time the Giza pyramids were built (east of Chavin 108.31 degrees, cosine = -0.314159). While greater scale of monuument spacing enables greater accuracy in relationship results and higher precision in correlated numerical expressions, with global distances the tectonic motion variable is a factor, and proportionately moreso the more ancient the monuments. Individual site tectonic motion modeling is required to correct coordinates for ancient positions. Some coordinates in the archaeogeodesy.xls applet have recalculated positions for the Neolithic epoch (-4739, when obliquity was 24.0°).

An advantage to study of Inca monuments in a global context is their recent antiquity, thus tectonic motion is less a factor. With the tectonic motion variable, which individual monuments or mountains and which relationships are considered is also a significant factor. At the Newark Earthworks complex, tectonic motion is near exactly east-west so site latitude is consistent over long time periods. Thus, tectonic motion is less a factor in accuracy of Newark's north-south arc distances and ratios. This is just one example of the complex considerations inherent due to tectonic motion. Margins of error are variated by each site's tectonic motion, antiquity, and the particular relationship, a complex combination unique to each consideration.

Machupicchu and Phi

Figure 5. Machupicchu, Newark, and Chichen Itza phi ratio.

Some of the earliest archaeogeodesy results I posted focused on Newark Earthworks, including a phi relationship to Machupicchu (Figure 5). The result illustrated in Figure 1 was posted with the focus on Chavin de Huantar. With impressive precision, the spatial interrelationship of Machupicchu and Chavin presented the phi ratio, 1.0 : 0.618034 = 1.0 : 1.618034, as the ratio of arc distance to latitude difference (arc : n-s). Figure 6 illustrates the relationship between the updated Intiwatana data at Machupicchu and the centerpoint of the circular plaza at Chavin (in Spanish, Chavin Plaza Circular), the discovery location of a prismatic granite monolith, the Tello Obelisk. Chavin's 21 meter in diameter sunken circular court is centered on the axis of the U-shaped Old Temple platform mound, the initial monumental construction phase at Chavin.

Using coordinates for the quadrangular granite sculptures at the two sites, the bedrock Intiwatana at Machupicchu (GPS coordinates by Gullberg, margin of error unstated) and the sunken circular courtyard (GE centerpoint coordinate, 2016 imagery), the location where Julio C. Tello discovered the obelisk during excavation work, the relationship is precise (Figure 1.) With the recently updated Intiwatana coordinate based on the UTM grid of the Programa de Investigaciones Arqueológicas e Interdisciplinarias en el Santuario Histórico de Machupicchu (PIAISHM) "Llacta de Machupicchu" map, the arc to n-s ratio is 1.0 : 0.618027 (Figure 6). This five meter margin of error is still within the margin of error expectation for the site coordinates. Accuracy of South American GPS has improved since Dr. Steven Gullberg, et al., took the Intiwatana readings supplied by his co-author, Dr. Kim Malville.

Figure 6. Machupicchu, Chavin de Huantar, and Phi.

The Chavin-Machupicchu phi relationship has phi context. In 2015, I first posted a Golden Ratio Longitudes KML file illustrating the golden mean longitude ratios of Newark Earthworks with other major American monument complexes, reporting "six major American monument complexes have three golden mean longitude ratios with Newark Earthworks" (Figure 7). Machupicchu and Chichen Itza have phi longitude relationships with Newark Earthworks while their arc distance ratios also present lunar astronomy constants (Figure 5).

 
Figure 7. Monument Longitudes and Phi Ratios.

The recently updated Ancient Monument Phi Ratios KML includes additional Machupicchu phi monument relationships with Tikal, Piramide del Sol, Cahokia, Huascaran, Huaca de la Luna, Chan Chan, Sillustani, Tiwanaku, Acre Triple Square, Tumi Chucua Zanja, and Loma X. Results related to Machupicchu are placemarked in the Machupicchu Arcs KML, some are listed in Table 1. The current KML versions have updated Chichen Itza's Castillo and Caracol coordinates, results, and graphics (Figure 8) based on recent high resolution updates in Google Earth. The KML placemarks display more detailed information about site relationships than the tables and discussion herein, and more relationships.

Figure 8. Monument Longitudes and Phi Ratios
 

Latitude Geometry and Astronomy

Machupicchu is framed by two lunar orbit baselines, to the north lunar motion per rotation (0.036501 circumference, 13.140376°) and mean daily lunar motion (0.036601 circumference, 13.176352°) to the south near the summit of Montaña Machupicchu and transecting Llactapata (Figure 9). In other words, Machupicchu is situated between latitudes equating to the circumference of the earth divided by the number of rotations per lunar orbit and the number of days per lunar orbit. This latitude property suggests symbolic positioning, unlike other major monuments situated in relation to a more practical geodetic attribute, the trigonometry of their latitudes previously discussed in Ancient Monument Latitudes Evidence Accurate Astronomy.

Huaca del Sol on the Pacific coast, the largest ancient adobe structure in the Americas, is transected by the latitude with tangent equaling one-seventh (sol.kmz). Both Chavin and Sechin Alto are transected by latitudes formed by one-sixth right triangles; at Chavin the latitude sine equals one-sixth, at Sechin the tangent is one-sixth. At Monks Mound, the largest prehistoric monument north of Mexico and the largest-in-volume prehistoric earthen construction in the Americas, the latitude with tangent equaling four-fifths transects the mound.

Figure 9. The Machupicchu and Llactapata latitude equates to a lunar orbit constant.

The latitude of Machupicchu between 0.036501 and 0.036601 circumference south of the equator is, in and of itself, noteworthy. These two numbers also repeat frequently in the study results, as discussed below. The four baselines in Table 2, determined with two additional lunar numbers, are placemarked in the KML file along with the listed sites.

Machupicchu, Chavin, and Illimani

Mountain peaks figure prominently in Andean culture to this day as well as in Machupicchu's archaeogeodesy study results. Prominences in the landsape aid in orienting and way finding and have a role in surveying the earth, so it's completely unsurprising that mountains might be important in ancient geodesy. One method of measuring the scale of the earth is to measure the dip of the horizon, best accomplished from the highest mountains providing the greatest area to triangulate.

Machupicchu is surrounded by prominent mountain ranges. The Cordillera Urubamba due east of Machupicchu includes two prominent high peaks, Waynawillca (Nevado Veronica) and Sahuasiray (Nevado Ccolque Cruz). Nevado Salcantay, with the two highest summit points in the Vilcabamba range, is due south of the ancient monuments. Pumasillo to the southwest of Machupicchu and Llactapata, in the direction of the December solstice setting sun, is the high point of the Sacsarayoc massif and the second highest peak in the Vilcabamba range.

Nevado Illimani at 6,438 meters is the second highest peak in the Cordillera Real to the east and south of the Lake Titicaca basin in Bolivia and Peru. After writing the Chavin article, I updated the Illimani summit coordinate using online GPS tracks of seven ascents (Apu Illimani, ailli). Unlike the highest peaks around Machupicchu, Illimani has been climbed many times by mountaineers sharing their data. The arc distances from Machupicchu to Chavin and to Illimani are equal (Figure 10). The two arcs also form a great circle which extends to the Moche Huacas, Huaca del Sol and Huaca de la Luna, on the Pacific Coast (Figure 11). A direct line from Chavin to Illimani passes midway between Montaña Machupicchu and the Intiwatana. In Figure 10, the two proximate lines in the lower left are great circles from Chavin to Illimani.

Figure 10. Machupicchu, Chavin, and Illimani arcs and great circles. View larger scale.

The great circle from Apu Illimani to the Moche huacas extends to one of the oldest occupation sites and monuments on the Pacific coast, Huaca Prieta. The latitude differences between Apu Illimani, Huaca Prieta, and the Machupicchu Intiwatana express the ratios 5.0 : 3.0 at Huaca Prieta and 3.0 : 2.0 at the Intihuatana.

The great circle passes near an isolated and prominent pyramidal summit, Nevado Tarata, the highest peak in the Huaguruncho mountain range. The provisional DEM coordinates for Tarata indicated an arc distances ratio to Machupicchu and to Chavin of 1.0 : 0.365, so I sought to obtain precise data. I found no GPS data and very little climbing history. To refine summit coordinates I used online photography in combination with the DEM, Google Earth, and satellite photography. While I cannot state the margin of error with precision, assuming improvement of the DEM coordinate I decided to present some preliminary results nonetheless (Figure 11). These and other interesting DEM based results not presented herein await reliable GPS data from mountain climbers. In particular, I still lack GPS for one of the Salcantay peaks due south of Machupicchu, plus Nevado Veronica and Waynawillca east of the ruins.

Figure 11. A great circle of major monuments aligns with Apu Illimani.

Salkantay directly south of Machupicchu, is the second highest prominence in Peru (after Huascaran) rising 2,540 m above the surrounding topography. Only two Peruvian peaks are higher than Illimani, 6,634 meter Yerupaja and 6,768 meter Huascaran near Chavin and Casma-Sechin. Recently Peruvian officials determined the summit of Huascaran with high precision (INAIGEM Expedición Científica al Huascarán 2017 GPS survey). The highest peak in the Andes, Aconcagua in Chile, the highest point above Earth’s center, Chimborazo in Ecuador, and the highest peaks in North America have published GPS readings and, not surpringingly at least to me, figure prominently in the study results.

Lunar Standstills and Lunar Orbit Inclination

I previously discussed spatial relationships between Chavin, Pachacamac, and Newark Earthworks expressing lunar astronomy constants. In a 2009 entry in Newark Archaeogeodesy, Assessing Evidence of Geospatial Intelligence in the Americas, I cited the relationships of Newark Earthworks to Tikal, Chichen Itza, and Pachacamac. These major sites and Machupicchu have interrelationships expressing pi, phi, astronomy constants, and astrogeodetic modules. The mathematical expressions are often the spatial relationship of three monuments and the same proportions are expressed redundantly by various combinations of and different geometric relationships of the same major monuments. I find these to be strikingly impressive achievements of not only nesting the monumental arrangement on a spherical earth but also of both understanding fundamental cosmic motions and accurately determining the corresponding astronomical constant values.

Western cultural astronomy is focused on the apparent 18.613 year lunar standstill period of the varying angular span of moonrises and moonsets on the horizon. Lunar orbit inclination turning is the sidereal cycle producing this apparent lunar motion period and the predominant constant of these two as expressed by the monuments. This precessing motion, lunar orbit inclination turning, slowly gyrates the intersection of lunar orbit and the ecliptic and consequently where eclipses occur in relation to fixed space. Lunar node retrograde rotation around the ecliptic equals 18.5990 solar orbits. Ancient monuments precisely evidencing both, the 18.5990 orbits sidereal cycle and the 18.613 year apparent motion period (Table 3), imply advanced understanding of lunar motions and another precession motion, the also gyrating orientation of eath's axis of inclination caused by lunar motion. Precession of the rotation axis inclination produces the difference between the sidereal cycle (ot) and the lunar standsill period (os).

Figure 12. Newark, Chichen Itza, Pachacamac, and Chavin de Huantar.

In 2018 in Chavin de Huantar Archaeogeodesy I also discussed the longitude difference between Chavin's Circular Plaza, Chichen Itza Castillo, and the centerpoints of the Newark Octagon and Observatory Circle expressing both these astronomy values with precision, albeit with a shift in decimal position (Figure 12). The arc distances ratio from Chichen Itza to Machupicchu and Newark Earthworks (three sites also expressing phi in their longitude difference ratio) redundantly express the same two lunar standstill astronomy constants (Figure 13).

Figure 13. Newark Earthworks, Chichen Itza, and Machupicchu.

The same two lunar motion constants are again expressed by ratios of the difference in degrees of latitude and longitude between Machupicchu and Newark (Figure 14). The astronomy constant values are expressed accurately from the Intiwatana to two features of the Octagon and also from Montaña Machupicchu to Observatory Circle. From Waynapicchu the arcs ratio to Pachacamac's Templo Pintado and Huaca de la Luna equals 1.0 : 1.861246. From Machupicchu Torreon the arc distances to Pachacamac and Huaca de la Luna have a ratio of 1.0 : 1.861258.

Figure 14. Machupicchu and Newark Octagon, latitude and longitude differences.

More Machupicchu site relationships expressing the lunar major constants are listed in Table 4. The numbers expressed in relation to the astronomy constants match the relationships of the monument centerpoints with a few meters margins of error. In the KML placemark balloons I state the relationship margins of error relative to epoch specific constant values. Lunar standstill related constants have greater temporal variance than the fundamental astronomy motions, resulting in greater relevance for epoch equated determination.

Machupicchu and Fundamental Astronomy

Earth's rotation on its axis is a precise clock, a steadfast motion easily and accurately subdivided and counted using fixed space. Lunar orbits per rotation is the observable fundamental motion ratio. Machupicchu's latitude between 0.03650 and 0.03660 of earth circumference south of the equator equates with these astronomy ratios. Lunar motion observed from the earth is not as steadfast as earth's rotation rate, hence long periods of observation are a requisite for accurate determination of lunar astronomy constants.

In 1997 when I launched this domain I published The Àryabhatiya of Àryabhata, the oldest precise astronomical constant? with my prior observation of the earliest accurate astronomy constant known in literature to me at the time. Àryabhata's 500 A.D. ratio had gone relatively unnoticed to astronomers, and apparently entirely unnoticed as a precise sidereal ratio. Astronomers readily recognize the number of days per lunar orbit but not necessarily the number of rotations per lunar orbit, known and accurately stated by Àryabhata's ratio in 500 A.D. In 2009 I again discussed Àryabhata and integral ratios of astronomy cycles and periods in Ancient Astronomy, Integers, Great Ratios, and Aristarchus.

Also in 2009 I posted Eclipses, Cosmic Clockwork of the Ancients, presenting fundamental astronomy from a heliocentric perspective, a task which compelled modifications to prevailing astronomy terminology. Geocentric astronomy lacked two of the three fundamental variables in Table 5, requisite factors to quantify astronomy constants, due to ignorance of solar orbits and earth rotations. Albeit, a count of rotations was apparent as a tally of fixed space seemingly revolving around the earth. Language is slow to accommodate new knowledge, we still say the sun rises even though we know the earth rotates and the periodic words day and year have been adapted erroneously to name sidereal cycles. My terminology replaces those errors. This task should have fallen to astronomers centuries ago, not to an archaeologist.

Several times now recognition of a previous culture possessing knowledge I could not explain has forced contemplating how they knew what they knew. The focus on eclipses and formulating helioocentric astronomy fundamentals led to understanding the simple method of determination of astronomy constants I presented in Ancient Astronomy, Integers, Great Ratios, and Aristarchus. The knowledge derives from the observable fundamental motions, lunar orbit and rotation, a ratio redundantly expressed in the Machupicchu results. Western geocentric culture's ignorance of the existence of the variables precluded using the method to determine the constants as well as knowledge of the ratios of those motions. I've discussed this in greater detail in previous articles linked above and in the Chavin Archaeogeodesy page. The method should have been obvious to astronomers centuries ago, rather than discovered by an archaeologist in this century. The important point, the fundamental 1.0 : 0.036501 ratio of lunar orbits and rotations, the two observable fundamental cosmic motions and the key to quantifying astronomical constants, is a culturally distinct astronomy and the monument relationships indicate Indigenous American astronomers understood heliocentric astronomy and geodesy long ago.

Machupicchu and Lunar Astronomy

The Chavin page discussed various monument relationships expressing the lunar constants. In 2015 I noted the arc distances from Monks Mound to Tikal and to Tiwanaku express the ratio.

Figure 15. Machupicchu and lunar orbit constants.

Machupicchu"s latitude between 0.036501 and 0.036601 circumferences contextualizes relationships with other monuments expressing these astronomical ratios (Figure 15). The arc distance from the Intiwatana to Teotihuacan's Pyramid of the Sun equals 0.3650130 diameters, and from Waynapicchu to Piramide of the Moon 0.3650105 diameters. The north-south ratio from Monks Mound to Machupicchu Courtyard and Pyramid of the Sun is 1.0 : 0.366011. North-south from Aconcagua to Llactapata II and to the Caracol the ratio is 1.0 : 0.365010, to Llactapata A and Caracol the ratio is 1.0 : 0.365243. At Llactapata B the north-south ratio to Observatory Circle and Aconcagua is 1.0 :0.366008. The Chavin article presented other 1.0 : 0.36501 ratios expressed by major monuments.

Decades ago I noticed the arc distances from Karnak, Egypt to the North American great pyramids at Teotihuacan and Cholula closely approximate the diameter of the earth, circumference divided by pi. Arcs from the Second Pylon entrance of Karnak's Great Hypostyle Hall to Teotihuacan's Piramide de la Luna and to the Cholula Pyramid average 114.59147° or circumference divided by 3.1415951, within 10 meters of expressing pi precisely. Given spanning the Atlantic and the antiquity, this minimal margin of error is less than plate motion. With the arc distance from Machupicchu to Teotihuacan equaling 0.3650130 diameters, of course at Teotihuacan the ratio of arc distances presents the same number. From the Sun Pyramid at Teotihuacan the arcs ratio to Waynapicchu and to Mut Temple at Karnak today equals 1.0 : 0.3650107. Table 6 presents selected results equated to Machupicchu's latitue.

Additionally, from Teotihuacan's Quetzalcoatl Pyramid the east-west ratio (longitude differences) to the Intiwatana and to Karnak's Mut Temple is 1.0 : 0.200001 and the longitude difference between Teotihuacan and Karnak is 0.36528 circumference. With the Teotihuacan, Machupicchu, and Karnak triad of monumental sites another utility of shifting decimals is noted, expressing several very different constant values. From the Moon Pyramid the arcs ratio to Cerro Machupicchi and to Karnak's Hypostyle Hall equals 1.0 : 0.3652432, within eight meters of precisely expressing the numerical string for days per year. Even though 365.24 and 0.003650 are very different numbers, with the decimals shifted both have accurate representation in the ratio of arcs presented by the same three monument complexes. Not only is decimal shift considered advantageous for accurate expression of the relevant number strings, decimal shift may also function to improve accuracy of calculations and/or modeling the geometry.