Joseph C. Keller, M. D. –
(B. A., cum laude, Mathematics, Harvard) –
The Great Pyramid, the Cholula Pyramid, and the Pyramids of the Sun and Moon: working together to prove that “2012” isn’t nonsense
Abstract. Following Petrie, the position of the pole at the time of the pyramid builders is determined two ways. At their oppositions in early 2013, Arcturus and Algieba reach within an arcminute of the zenith, at the pyramids of Cholula, and of the Sun and Moon, resp., using those poles. The distance between the pyramids of Cholula and the Moon, tells the correct precession rate. The distance and angle between the pyramids of Moon and Sun, and the sizes and shapes of the pyramids themselves, tell us that the pyramid builders assumed an inexact second derivative of the ecliptic pole position, apparently for the sake of using round numbers, and correcting this improves the accuracy to a few arcseconds for Arcturus. Similar plans explain the positions of the three main Giza pyramids, Monk’s mound at Cahokia, and the Bosnian pyramids.
I. Introduction to the stars.
Arcturus (Greek, “the bearkeeper”) is the brightest star in the northern hemisphere, though some sources say Vega, depending on photometric details. Algieba (Arabic, perhaps originally “the mane” of Leo, according to Burnham) is a famous second magnitude double star, one of the fifty brightest stars in the sky. These stars are in the spring sky, about a radian apart. Both stars are nearby orange giants, with large proper motions.
Based on their negative radial velocities and on trigonometry, increases in their proper motions during the last 5000 yr, would cause 45″ & 5″ overestimates of declination for Arcturus & Algieba resp., if only the instantaneous proper motion, arcsec/yr, and not its time derivative, arcsec/yr^2 (and in the case of Arcturus, the significant time second derivative arcsec/yr^3) were known 5000 yrs ago. The accuracy of placement of the Cholula pyramid with respect to Arcturus, explained in Sec. VI, implies that not only accurate proper motions, but also first and second derivatives of proper motions, or the mathematical equivalent (e.g. a presumption of oblique rectilinear motion) were known to the builders.
The orbit of the double star, Algieba A & B, is highly eccentric and highly inclined to our line of sight, so it would have been difficult for the builders to average the center-of-light motion. This would have been easier with Castor AB; see the Addendum re Giza. Algieba AB’s orbit is vague and ambiguous, but by measuring and quadratically extrapolating positions on Burnham’s graph, “Celestial Handbook”, vol. 2, p. 1063, and using mass and magnitude estimates from Kaler’s stars.astro.illinois.edu webpage, I find that Algieba AB’s epoch 2013.0 difference between center-of-light and center-of-mass motion is, in declination, +2.02 mas/yr.
II. Positions of the stars in early 2013.
From their 2000.0 AD positions (Bright Star Catalog) in 2000.0 AD celestial coordinates, and the “rigorous formulae” on p. B18 of the 1990 Astronomical Almanac, I found their declinations in 2013.0 AD celestial coordinates, including proper motions. These declinations are:
Arcturus +19deg06’55”
Algieba +19deg46’32” (primary component)
There is no atmospheric refraction correction at the zenith, but the nutation and starlight aberration corrections will affect the accuracy considerably. At opposition, the change in declination due to aberration and to nutation in longitude (chart at www.pietro.org) can be estimated as
(-20.5″-15″) * sin(23.44) * cos(34, resp. 25)
= -12″ & -13″ for Arcturus & Algieba, resp.
For nutation in obliquity, the effects are
-6″ * sin( -34, resp. +25)
= +3″ & -3″ for Arcturus & Algieba, resp.
The oppositions occur near 2013.32 & 2013.15, resp., giving by precession from 2013.0 another
-5″ & -3″ in declination, resp. (omitting -0.6″ Proper Motion for Arcturus)
This gives declinations at their oppositions in early 2013, as observed when at the zenith:
Arcturus +19deg06’41”
Algieba +19deg46’13”
III. Introduction to the pyramids.
The Great Pyramid of Giza, the pyramid of Cholula, and the pyramid of the Moon are among the world’s largest known pyramids both in height and volume. The Great Pyramid of Giza is a close #2 (to Giza’s pyramid of Khafre) in the world for height. The pyramid of Cholula, thought to have been begun in the 3rd century BC, is #1 in the world for volume and #2 in Mexico for height. The pyramid of the Moon is somewhat smaller than its more famous, nearby, larger teammate in Teotihuacan, the pyramid of the Sun; the pyramid of the Sun is #3 in the world for volume and #1 in Mexico for height. Although most findings of this paper apply almost as well to the pyramid of the Sun as to the nearby pyramid of the Moon, I choose the pyramid of the Moon because according to Wikipedia, the pyramid of the Moon “covers a structure older than the pyramid of the Sun”.
According to the geodesy teaching website plone.itc.nl/geometrics (“Geometric Aspects of Mapping, 3. Reference Surfaces for Mapping”) the plumb line (i.e. perpendicular to the, “geodetic” surface i.e. “geoid”) commonly deviates from the perpendicular to Earth’s reference ellipsoid by up to 50″ near mountains, but in flat regions uncommonly more than 10″. A low-resolution world map of the geoid (Uotila, 1962; cited in Heiskanen & Moritz, “Physical Geodesy”, 1967, Fig. 21, p. 157) shows that both Giza and Teotihuacan/Cholula are in regions where the geoid is extraordinarily close to the reference ellipsoid, about -6m and +2m, resp. At Giza, the gradient of the geoid height is relatively large but lies EW so would affect observed declinations little. At Teotihuacan/Cholula, there is a moderate NS geoid gradient equivalent to about 2″ northward tilt of the plumb line.
IV. The latitudes of the pyramids.
The geographic latitudes are:
19deg03’27” (pyramid of Cholula)
19deg41’59” (pyramid of the Moon)
A large error, is the motion of Earth’s pole over thousands of years. Flinders Petrie (p. 125; Ch. 13, Sec. 93 in Birdsall’s online edition) says that the most reliable structures in the Giza pyramids indicate that the pole at the time of their foundation, lay 5’40” +/- 10″ west of the present true north. Petrie remarks that this indicates a rate of geographic pole migration only a few times greater than measured in recent centuries, and that physically, quantitatively, it is consistent with major changes in ocean currents. Likely, a pole shift would change Giza’s latitude a comparable amount, so I assume that the Great Pyramid was originally at exactly 30N. If the Great Pyramid serves as a monitor of the pole change since the foundation of the greatest Mexican pyramids, then Cholula, and the pyramids of Sun and Moon, would have been 180″ and 177″ nearer the pole, resp., when founded. Their geographic latitudes then would have been:
19deg06’27” (pyramid of Cholula)
19deg44’56” (pyramid of the Moon)
With this ancient pole, Arcturus would have been observed only 14″ too far north to match this original Cholula perfectly, and Algieba 77″ too far north to match this original pyramid of the Moon perfectly, at their respective oppositions in early 2013.
If no latitude change happened at Giza (maybe that site was chosen for this reason, determining its longitude; then the site away from 30N was chosen because of the architectural convenience of the Giza plateau) then these pyramids would have been 227″ and 226″ nearer the pole. Their geographic latitudes would have been:
19deg07’14” (pyramid of Cholula)
19deg45’45” (pyramid of the Moon)
Arcturus & Algieba would have been 33″ too far south & 28″ too far north, resp. This is a better fit: 14^2+77^2 = 6125 > 1873 = 33^2+28^2. The declinations of these stars change about -50″ * sin(23.44)*cos(34 or 25) = -16″ or -18″/yr, resp. Using the oppositions in early 2012 or early 2014, the sum of squared errors would be 17^2+46^2=2405 or 49^2+10^2=2501, resp.
V. The pyramids tell the precession rate.
The foregoing, determines the pyramid latitudes; but the longitudes remain free choices. Suitability of sites, would remove one degree of freedom, but another degree of freedom, the small difference between the longitudes, remains available.
Apparently the builders wished to tell us what precession rate they used in their calculation, by setting the geocentric arc between the two pyramids, equal to that precession rate times an interval of time that we would know or guess. Wikipedia gives geographic coordinates for the pyramids to about 0.0001deg = 10m. Geographic latitude, g, is converted to geocentric latitude, f, by tan(f) = tan(g) * (296.0/297.0)^2. The results are
of Cholula: long 98.3019W geog lat 19.0575N geocen lat 18.93854N
of the Moon: long 98.8440 geog lat 19.6996 geocen lat 19.57727
The geocentric angular separation of these pyramids, is 2946.5″, precise to 4 figures. The time period used seems to have been based on Jupiter’s tropical period, 360deg / (dL/dt), where L = the mean longitude of Jupiter referred to Earth’s ecliptic and mean equinox of date. Clemence, Astronomical Journal 52:89+ (1946), p. 90, gives L = 10930690″/cyr plus insignificant secular terms and a sinusoidal 900-yr “Great Inequality” term which is 1/10^4 as large and therefore barely significant (and perhaps averaged out by the pyramid builders). Because of the 5::2 resonance of Jupiter and Saturn, 5x Jupiter’s tropical period is a reasonable time interval to use. This implies a precession rate of
2946.5″ / (5 * 360*3600/10930690*100)
= 2946.5″ / (5 * 11.856525 Julian yr)
= 49.702″/yr, where the last digit is not significant
The Newcomb precession formula chosen by Clemence (op. cit., p. 90) amounts to 50.238″ + 0.02216″ * T per yr, where T is centuries from 1850.0 and I have omitted a barely significant quadratic term. The precession rate implied by the pyramid spacing, corresponds to 1850.0AD – 2420 = 571BC, which is halfway between 2013.0AD and 3154BC; 2013AD – 3154BC = 5166yr (last two digits not significant)(Mayan Long Count = 5125.26yr)(I’m keeping more digits than I show here, so your last digit might not agree). Thus the inter-pyramid spacing refers to either the start of the Mayan Long count, 3114BC, or else the reference year of Hindu astronomy according to Bailly and Playfair, 3102BC, and that spacing was designed to suggest the mean precession rate between that time and the present.
We thus confirm that the builders’ precession error was probably no more than, and perhaps much less than, about 0.005″ * 5000 = 25″. This corresponds to a declination error for either star, of at most about 25″ * sin(23.44) = 10″, and perhaps much less.
VI. The pyramids tell the ecliptic motion.
Proceeding as in (V), I find that the geocentric angular separation of the pyramids of Moon and Sun, suggests that the builders estimated the arclength rate of ecliptic pole motion, as 25.46″/(5 Jupiter tropical periods), if I use the Wikipedia coordinates. Instead I’ll use the separation I measured on the paper map of Rene Millon and Armando Cerda, Univ. of Rochester, 1970 (as Teotihuacan appeared in 1962); I prefer this figure, 26.108″, because I was able to make sure that the measurement was from the centers of the topmost marked rectangles on the pyramids.
Precession moves in a circular cone around the perpendicular to the ecliptic, but because both Venus and Jupiter affect the ecliptic considerably, specifying ecliptic motion is more complicated. The second difference of the ecliptic pole position using 100 yr intervals, often is inadequate to predict the change of pole position over 5000 yr.
Let’s guess the thoughts of the builders. Their other calculations were so competent, that surely they knew their contemporary arclength rate, 31.72″/ 5 Jupiter periods (at 3113BC, according to data from NASA’s online Lambda utility, by either first or third order numerical differentiation with central differences on 100 year intervals) and direction of ecliptic motion. The Lambda utility data indicate that the rate slowed roughly linearly to the present 27.87″ (average rate over a modern 800 yr interval) with a mean rate since 5125 yr ago, of 29.38″. However, the builders, according to the distance between the pyramids of Moon and Sun, apparently assumed a more drastic linear slowdown.
The actual mean radius of curvature of the ecliptic pole path over the last 5125 yr, has been about 1.4606deg (circumcircle of endpoints & time midpoint). Likely the builders used a constant radius of curvature for the pole path, to get a second order approximation method which we moderns, the intended audience of their message, could guess from the layout of their monuments. The slope of the interpyramid line Moon-Sun, is W of N, 1.53 +/- 0.22deg including 1 sigma rounding error, from the Wikipedia coordinates, or else 2.106deg from my measurements on Millon’s map. Maybe this slope was intended to indicate the assumed radius of curvature of the motion of the ecliptic pole.
There is reason to believe that the real interpyramid line slope is 1.594deg W of N: originally, before Petrie’s presumed changes of the latitude and true north at Giza, this would have been 1.667deg, which I argue, is the true original figure. Millon’s map and other sources (primarily James W. Dow, American Antiquity 32:326-334, 1967) give 15deg25′ E of N, as the slope of streets and buildings at Teotihuacan and, approximately, of some other Central American pyramids. Giulio Magli of Milan (paper on ArXiv.org) calls this slope the “Nord” slope and the other prominent slope found at Teotihuacan (following Dow), 16deg30′ E of N, the “Est” slope. The deviation of the interpyramid line W of N, probably a little less than 2 deg, could be added to either of these slopes. So one could speak of 15, 16, 17 or 18 deg slopes; 15 and 17 are heard oftenest.
On the messageboard of the late Dr. Tom Van Flandern, I have explained quantitatively how decreasing oblateness would provide energy for Earth to undergo torque-free precession at 18.90deg with period one year. This would put the pole at the same place every winter solstice, providing, climatologically, a pole-like location with latitude consistent with Charles Hapgood’s estimates of the latitudes of the last three Ice Age poles (60, 72, 63). If 1.669deg is added to both the 15deg25′ and 16deg30′ slopes of Teotihuacan, and I use, converting the Wikipedia coordinates, 19.5737N as the mean geocentric latitude of the pyramids of Moon & Sun, then 16deg30’+1.669deg becomes the angle between the line of longitude, and the great circle drawn tangent to a latitude circle at 15deg25’+1.669deg from the pole. This colatitude is 17.1deg, not far from my estimate, 18.9. Note that 360/1.669 = 215.7, and 360/1.66667 = 216 = 6^3, a round number in base 6.
Alternatively, I could use the mean geographic, not geocentric, Teotihuacan pyramid latitude. Then the solution is 1.460deg, exactly equal to the actual circumcircle radius of curvature, of the ecliptic pole path, mentioned above. So apparently the builders knew the exact value of the curvature of the ecliptic pole path 5000 yrs ahead, but exploited the two equations resulting from use of geocentric and geographic latitudes, to encode both curvatures in the chosen 15deg25′ & 16deg30′ angles: the actual 1.460deg circumcircle radius of curvature of the ecliptic pole path since 3114BC, and a cautious “round number” underestimate of that curvature, a 360/6^3 = 1.6667deg radius. If the former radius had been used but not correct, future scientists might never have deciphered what radius was used.
Millon warned that he had not had time to correct some cartographic details of his map; one of these details might have been exact equality of NS and EW scale. I can cause the 2.106deg interpyramid slope to become equal to 1.594deg (that is, the 2.106+15.417 deg between interpyramid and “Nord” lines, to become 1.594+15.417) by stretching NS distances and shrinking EW distances, by a factor q = 1.01464. This slightly increases the interpyramid distance, to 26.49″. Corroborating this, I find that the angle between the map’s NS arrow, and the bottom of the map (which closely parallels the nearest horizontal grid line) is 90 – 15.800deg which would be corrected to 90 – 15deg25′, if I use q = 1.01299; with this q, the interpyramid slope becomes 1.650deg.
The height and base of the pyramid of the Moon (George L. Cowgill, “An Update on Teotihuacan”, online, written for “Antiquity”) are 46 m & 149NS x 168EW m; of the pyramid of the Sun (when “completed” c. 200AD, according to the Metropolitan Museum of Art, “Heilbrunn Timeline of Art History”, online) 63 m & 215 m square. Knowing the 15.417+1.594(?) = 17.011deg offset of the interpyramid line from their faces, let’s use the Pythagorean theorem to find four distances:
1) It would be practical to lay chains straight down the north & south sides of the pyramids of the Sun & Moon, resp., then along the ground in a straight line from there. This distance, slightly longer than the great circle distance from peak to peak, corresponds to a total arc (using as denominator the Earth radius 6378.3km, corrected for latitude and altitude, and assuming perfect pyramidal shape) of 27.79″. We recognize this as a round number in base 6, 360deg/216^2=27.778″, analogous to the other round number, 360deg/216 = 1.667deg, which the builders chose for the original interpyramid line rotation W of N. It soon will become apparent that this arc, 27.778″, is what the builders assumed for the time-average arclength speed, per 5 Jupiter tropical periods, of the ecliptic pole path.
2) Suppose instead, the chain is laid straight down the north & east sides of the pyramids of Sun & Moon, resp., then from there along the ground. This distance corresponds to a total arc of 29.55″, near the 29.38″ which we think has been the actual time-average arclength ecliptic pole speed during the last 5125 yr.
3) Suppose the chain is laid down the west & east sides of the pyramids of Sun & Moon, and from there along the ground. This distance corresponds to a total arc of 32.51″, near the 31.72″ which we think was the actual arclength ecliptic pole speed at the start of the prediction, 5125 yrs ago. This arc also arises from the circumference of the circumcircle of the base of the pyramid of the Sun; using a square 215m base, this corresponds to 30.89″. Except for a factor of 2, the arc arises more accurately from the circumcircle circumference at Cholula; at the latitude and altitude of Cholula, its usually given square base side of 450m corresponds to 32.33″ times 2. The square perimeter at Cholula likewise corresponds to 29.10″ * 2, and the square perimeter of the Sun pyramid to 27.81″; these arcs approximate the actual average rate of ecliptic pole motion 29.38″, and the assumed round number rate 27.778″.
4) Finally suppose the chain is laid down the west & south sides. This distance corresponds to a total arc of 30.18″, again somewhat near 31.72″, the actual ecliptic pole speed at the start of the prediction.
Summarizing:
west+east –> original ecliptic pole speed 5125 yr ago
north+east –> actual mean ecliptic pole speed since 5125 yr ago
west+south –> original ecliptic pole speed 5125 yr ago
north+south –> assumed mean ecliptic pole speed since 5125 yr ago
Geometrically, the west+east and west+south distances must differ by more than the other two differ; this precludes their simultaneously being the original ecliptic pole speed. But their average is (32.51+30.18)/2 = 31.34″ ~ 31.72″.
The slopes of the corner edges of the idealized Moon & Sun pyramids are 22.3 & 22.5deg, resp.; these slopes seem intended to equal pi/8 radians. The Sun pyramid seems intended to be square. Above, I showed that the interpyramid line slope (offset from NS) seems intended to show the assumed radius of curvature of the ecliptic pole path. If the pyramids must be orthogonal to the famous 15.5 degree orientation, then there remain only four adjustable variables: the base side of the (square) Sun pyramid, the two different base sides of the (rectangular) Moon pyramid, and the interpyramid distance. Above, I show that these four variables have been chosen to satisfy four constraints:
i) the circumcircle circumference arclength, of the Sun pyramid, measured in geocentric angle, equals the original (~5125 yr ago), actual and assumed, arclength rate of ecliptic pole motion per 5 tropical Jupiter periods;
ii) the via north face Sun pyramid & south face Moon pyramid, chain distance described above, measured as in (i), equals the assumed average rate of ecliptic pole motion during the last 5125 yr;
iii) the via north face Sun pyramid & east face Moon pyramid, chain distance equals the actual average rate of ecliptic pole motion during the last 5125 yr
iv) the via west face Sun pyramid & south or east face Moon pyramid, chain distances, averaged, equal the original rate of ecliptic pole motion.
By measuring on both Millon’s 1:2000 maps (for the Sun pyramid, the large map, and for diagonals also field map N3E1, p. 45 in his map book; for the Moon pyramid, only the large map, because the field map is divided, and I did not measure Moon pyramid diagonals) and averaging these similar results from the two maps, I find dimensions significantly different from the consensus values above. Millon helped by drawing rectangles to outline the main levels of the pyramids. I assumed the center of the top level to be the peak of the pyramid. The Sun pyramid’s peak is decentered eastward and the Sun pyramid is significantly non-equilateral and non-rectangular. Retaining the consensus values of the pyramid heights, and again assuming perfect pyramid shape (others know more than I about what formerly was on top) I repeat (i)-(iv) above:
i) the Sun pyramid’s NW-SE diagonal is 0.5% longer than the other, corresponding to 0.005rad = 17′ hingelike northward motion of the west side. Its longest peak – corner line, the NW line, is thus
sqrt((102.8+208.2*0.005/2)^2+109.5^2) = 150.55m;
which times 2*pi for this largest circumcircle, corresponds to 30.59″ geocentric, vs. 31.72″ predicted.
ii) the via north face Sun pyramid & south face Moon pyramid, chain distance is 27.84″ vs. 27.778″ predicted.
iii) the via north face Sun pyramid & east face Moon pyramid, chain distance is 29.35″ vs. 29.38″ predicted.
iv) the via west face Sun pyramid & south or east face Moon pyramid, chain distances (necessarily much different) averaged, equal (30.28+32.28)/2 = 31.28″ vs. 31.72″ predicted. By adding 5.0% (5.5m) to the horizontal distance from the Sun pyramid’s peak to its west edge, the values of (i) and (iv) become equal at 31.41″, vs. 31.72″ predicted, without affecting (ii) or (iii).
The length/width ratio of the pyramid of the Moon is 168m EW /149m NS = 1.128. The ratio, of the original ecliptic pole speed which seems to be indicated by my map measurements of the Sun pyramid, 31.42″, to the assumed round number speed 360deg/216^2, is 1.1311.
From Millon’s two largest maps, I find differences in the diagonals of the Sun pyramid’s ten level rectangles as outlined by Millon. The lowest three levels are nearly rectangular, but above these, the NW-SE diagonal is always much longer. The 4th & 5th levels’ diagonals always differ a little more than 1%, and the 6th through 10th differ on average more than 3%. This corresponds to a slant, for the top five level parallelograms, averaging 1.77 +/- SEM 0.11deg, and for the top seven, averaging 1.49 +/- SEM 0.15. Again this might indicate the assumed and actual radii of curvature of the ecliptic pole path, 1.6667 & 1.4606deg, resp.
In a plane approximation, the differential equation of Earth’s pole precession about the ecliptic becomes linear. If the ecliptic pole moves slightly, the perturbation terms obey a nonhomogeneous linear differential equation which I solve by one Runge-Kutta step. The perturbation, is the actual Lambda utility ecliptic pole position minus the builders’ estimate (the builders’ estimate appears to be mean arclength motion of 27.778″ per 5 Jupiter tropical periods, linearly decreasing from the actual NASA Lambda rate, 31.72″, 5125 yr ago, along a circle of radius 1.6667deg, originally tangent to the actual Lambda direction 5125 yr ago). My result is that the Earth’s actual rotation pole is at 102″ greater longitude, and 41″ greater obliquity relative to the invariant plane, than the builders’ parameters predicted.
Solving the spherical case numerically with a computer (approximating the actual ecliptic pole sufficiently precisely with 4th order central difference interpolation of NASA Lambda values) I find that if I assume that the torque on Earth is proportional to sin(obliquity)*cos(obliquity), and adjust the precession rate accordingly based on the present rate and obliquity, the solution using the actual ecliptic pole positions, predicts the longitude of Earth’s present rotation axis much less well than if I assume the torque is proportional simply to sin(obliquity). Be that as it may, the results are 85.6″ longitude & 41.2″ obliquity, and 99.1″ longitude & 41.1″ obliquity, for the first and second assumptions, resp. So, the difference between the second, apparently valid, assumption, when solved with a computer, and the approximate back-of-envelope calculation of the preceding paragraph, is negligible.
The slower longitude regression, relative to the invariant plane, is due only to Earth’s rotation pole being nearer to the ecliptic pole; it does not indicate a different angular precession rate. The actual 99″ greater pole longitude, relative to the invariant plane, would cause 99″ * sin(23.44) * cos(34 or 25) = +33″ or +36″ change in declination for Arcturus or Algieba, resp. The 41″ greater obliquity, relative to the invariant plane, would cause 41″ * sin(-34 or +25) = -23″ or +17″ change, resp.
Now, my first geographic pole correction, based on the additional assumption of original 30.000N latitude for the Great Pyramid, seems best. Had the ecliptic change been what the builders predicted, then Arcturus & Algieba, at their oppositions in early 2013, would have missed by only +14-33+23 = +4″ and +77-36-17 = +24″, resp. The variance, 4^2 + 24^2 = 592, is far smaller than for any alternative considered.
For Arcturus the error is as small as can be expected in view of Petrie’s 10″ uncertainty of the old pole direction at Giza, and the 10″ uncertainty of reference spheroid vs. geodetic vertical. Correction for an approx. 2″ northward tilt of the plumb line at Teotihuacan/Cholula (as for southern Mexico generally, according to Uotila; see Sec. III) lessens the errors to +2″ & +22″, resp.
At its opposition in 2014AD, correcting for a year’s change of precession and nutation, Algieba would be misplaced by only 22-18+1 = +5″. Maybe the delay of Algieba’s alignment, by one year, was intended to tell us that some phenomenon, accompanying the end of the Mayan Long Count, persists for only a year.
In Sec. I, we learned that the epoch 2013.0 difference between center-of-light and center-of-mass proper motion for Algieba AB, corresponds to +10.4″ J2000 declination in 5125 yr. By Kepler’s law of equal areas (which applies even to inclined orbits) Algieba B moves 14x slower relative to Algieba A, at its apparent farthest (near the present location) than at its apparent nearest. Such grossly nonsinusoidal motion might have been only roughly averaged by the builders.
VII. Conclusion.
John Major Jenkins, as quoted in Geoff Stray’s popular book on 2012, has expounded the idea that the Central American pyramids might be “precessional alarm clocks”, but Jenkins has looked for relationships mainly to special points on the sky path of Venus, or of the Pleiades, heeding the known Mayan interest in these objects. Such relationships exist but are approximate to the order of a degree, not arcseconds, like the relationship I discover here.
Someone invested huge resources of calculation and construction, to align the zeniths of two of the world’s largest and most famous pyramids, within an arcminute (and within a few arcseconds, if what appear to be deliberate second order ecliptic motion underestimates by the builders, had been corrected) with two of the world’s most prominent stars, at their oppositions in early 2013. Alignment would happen then (rather than more than a decade later) if the geographic pole shift determined by Petrie from Giza, had not occurred.
By encoding the average precession rate, in the distance between the Cholula and Moon pyramids, and the ecliptic motion, in the separation and shapes of the Moon and Sun pyramids, the builders told us their plan was deliberate, accounted for precession and ecliptic motion, and was very ancient. They enabled us to correct their calculations, if only we could otherwise determine the foundation date, e.g. by knowing that the Mayan Long Count started in 3114BC, or knowing the 3102BC Kali Yuga reference year of Hindu astronomy, or guessing from the chronology of Manetho which starts c. 3110BC (or by assuming the encoded precession rate to be accurate and using the date implied by it, to evaluate the encoded ecliptic motion). No significant correction to precession rate is indicated, and only a plausible second order correction to the ecliptic. This correction improves the accuracy, especially for Arcturus, to no worse than a few arcseconds at the stars’ oppositions in early 2013.
Addendum June 1, 2012 re Giza:
A theory has been popularized, that the layout of the Giza pyramids is related to the layout of the stars in Orion’s belt, after some amount of precession. Years ago I tried, with several mapping schemes, to verify this quantitatively, but was unable.
I discovered today that instead, the layout of the Giza pyramids is related to Castor and Pollux, those other near together bright stars visible from north temperate latitude. Pollux is a nearby, large proper motion orange giant like Arcturus and Algieba. Castor is a nearby, non-giant blue-white star. Using online Bright Star catalog epoch 2000.0, J2000.0 positions (given to the nearest arcsec), the NASA Lambda utility for conversion to mean 2013.0 coordinates, Bright Star catalog Proper Motions for 13 years, and finally my own approximate corrections for nutation in longitude and obliquity, aberration of starlight, and the extra fractional year’s precession, I find that the observed (near the zenith, refraction ~ tan(phi) ~ phi, so is only a tiny minification) declinations when their midpoint reaches opposition early in 2013 are:
Castor 27deg59’25.5″
Pollux 31deg51’19.9″
The midpoint of these lies at declination 29deg55′ at 2013.0AD. The builders of Giza apparently seized this random, rough coincidence and used it to mark the end of the Mayan Long Count in a way that still allowed them to put the Great Pyramid at (then) exactly 30N.
Let x1 (resp. x2) denote the absolute declination difference Pollux (resp. Castor) minus 30.0deg.
x1/(x1+x2) = 0.519929
The Great Pyramid of Cheops (Khufu) corresponds to Pollux, Mycerinus (Menkaure) to Castor, and Chephren (Khafre) to the observer. Let y1 (resp. y2) denote the distances Cheops-Chephren (resp. Chephren-Mycerinus) measured by Petrie, and theta be Petrie’s break angle between the lines Cheops-Chephren and Chephren-Mycerinus.
y1/(y1+y2*cos(theta)) = 0.520723
To get this agreement, we projected y2 onto the line of y1, hence the cosine. The difference, +0.000794, corresponds to a mere +11″ declination shift for the Castor-Pollux midpoint, actual vs. Giza-based prediction.
The foregoing, based on declinations, is most natural to us, but requires measurement at two points in time. Another way to measure discrepancies from the zenith, would be at one point in time, when the stars’ EW discrepancies from the zenith are equal, sighting upward through a rectangular grid. Suppose first that it is the RA discrepancies, not the EW discrepancies, that are equal. Curvature of the parallels, would displace Castor or Pollux northward, in radians,
(1-cos(RA difference/2))*cos(31d51′ or 27d59′)*sin(same)
which divided by the stars’ declination difference, is 0.001804 or 0.001668. For the EW discrepancies to be equal, the RA discrepancy is proportional to q = cos(mean declination)/cos(declination), so the above RA discrepancies must be multiplied by q, and the northward displacements multiplied by q^2, giving 0.001804*1.0412 = 0.001878 and 0.001668*0.9634 = 0.001607. The ratio of NS displacements becomes x1/(x1+x2) = (0.519929-0.001607)/(1-0.001607+0.001878) = 0.518182. Without any projection, y1/(y1+y2) = 0.517483. The difference, -0.000699, corresponds to a -10″ declination shift of the Castor-Pollux midpoint.
Precession changes the declination about 50.28″*sin(23.44)*sin(115.2-90) = 8.5″/yr. The accuracy of the two methods above, implies a best fit of +0.5″ +/- SEM 7.4″. So, January 2013AD is indicated at a significance exceeding 1 sigma per yr.
Petrie’s distance from Chephren to Mycerinus, if doubled, corresponds to 29.3859″ geocentric arc, adjusted for latitude, the altitude of the Giza plateau near these pyramids, and the barely significant nonperpendicularity, of the reference ellipsoid, to Earth’s radius. Fourth order central difference interpolation of the NASA Lambda positions of the mean ecliptic pole through the last 5125 yr, gives an integrated arclength travel of 29.3819″ per 5 tropical Jupiter periods (this time unit is suggested by the spacing of the Mexican pyramids discussed in the other sections of this paper). The difference corresponds to only 2.4 inch error in Petrie’s one-way distance. The foregoing findings already constrain the Cheops-Chephren distance, but if doubled it happens to correspond to 31.52″, near the actual 31.72″ initial ecliptic pole rate.
Petrie’s angle between NS and the Cheops-Chephren line, is 43deg22’52”; though Petrie determined the old NS line of the pyramids, as 5’40” +/- 10″ W of modern N, he used for a round number, 5′ in this determination. Adding 40″ gives 43.392deg. From my interpolation based on the entire 5125 yr interval, the initial rate of change of the ecliptic pole, makes a 42.53deg angle with its equinox of date. If instead I find the ecliptic pole rate of change from values +/- 100 yr, I find 42.63deg. So the two distances and two angles of the Giza plan, manage approximately to encode five data: two different definitions of the Pollux-Castor straddle of the 30th parallel, at their opposition in 2013; the initial and mean rates of ecliptic pole motion over the last 5125 yr; and the initial direction of ecliptic pole motion.
At Newcomb’s precession rate of 5125 yr ago, 49.15″/yr, the equinox 5166 yr ago (see Sec. V) would have been at 0.560deg greater longitude than 5125 yr ago. This would improve the agreement of the smoothed initial ecliptic pole direction, to 42.53+0.56=43.09, vs. 43.392 at Giza.
Castor A & B differ from Algieba A & B: Castor AB’s orbit is only moderately inclined to our line of sight, and its true orbit only moderately eccentric. So, the difference between Castor AB’s center-of-light position and center-of-mass position, is roughly sinusoidal and could more easily have been averaged out by the builders.
Summary of addendum re Giza. The builders chose the distance ratio and break angle at Giza, to quantify as simply as possible in two natural ways, the exact deviation of the Castor-Pollux (a.k.a. “Dioscuri”) midpoint from the zenith over 30.0N at those stars’ opposition in 2013. The absolute distances and the Cheops-Chephren direction, quantify the initial direction, and the initial and mean rates, of ecliptic pole motion from ~5125 yr ago.
Addendum June 3, 2012 re Cahokia:
From information in the Bright Star Catalog, I find Vega’s observed declination, employing all corrections as elsewhere in this paper, observed at the zenith at its opposition in mid-2013, to be +38deg47’57”. Wikipedia gives the geographic latitude of the Cahokia mounds near St. Louis, USA (the largest, about 30m tall, is named “Monk’s Mound” because a Trappist monastery was nearby) as N38deg39’14”.
The same correction for the post-Giza pole shift that I use elsewhere in this paper, gives the presumed original Cahokia mounds’ latitude as 38deg42’48”. Now I consider the effect on Vega, of correcting the ecliptic pole motion underestimate used in Mexico (see Sec VI):
99″ greater actual longitude of Earth’s rotation pole –>
99″*sin(23.44)*sin(-9.2) = -6.3″ change in Vega’s declination
41″ greater actual obliquity –>
-41″*cos(9.2) = -40.5″ change.
Thus if the Teotihuacan (and Cahokia) builders’ ecliptic pole estimate had been exact, the declination of Vega would have become 38deg48’44”. This 5’56” discrepancy between Vega and Monk’s mound, amounts to 24% of Vega’s proper motion in declination over 5125 yr. Maybe the error is due to an unknown physical or optical effect of Vega’s rapid pole-on rotation.
Ptolemy’s star positions indicate that Arcturus’ proper motion (2.25″/yr) has been near its present value for 2000 yr, but that Vega’s (0.33″/yr) very much has not. Aided by a computer, I consider four triangles:
1. Arcturus-zeta-epsilon Bootis
2. Vega-beta-eta Lyrae
3. Vega-beta-theta Lyrae
4. Vega-gamma-eta Lyrae
5. Vega-gamma-theta Lyrae
Ptolemy’s distances within a constellation should be his most accurate data. The two dimmer stars in each triangle were chosen to make all the angles of the triangle large (giving a well-conditioned system of equations) and for their small (modern) proper motions (about 50 mas/yr for the dimmer stars in Bootes and not more than 4 mas/yr for the dimmer stars in Lyra).
Because Ptolemy’s positions for these stars are all rounded to sixths or quarters of a degree, the sides of implied uncertainty boxes for the stars are 5′ or 10′. Defining error as sum of squared side length differences between Ptolemy, and modern proper motion corrected data, I find the least error among the 4^3=64 choices of corners of these uncertainty boxes.
For Ptolemy’s Almagest, I use Peters & Knobel’s translation, “Catalogue II”, 1915. For Arcturus’ triangle I find minimum root-mean-square side length error of 13′ (with best choice of uncertainty box corners) at 0 AD (near experts’ estimated true epoch of Ptolemy’s catalog) with twice that rms error, at 1300AD & 1100BC. By contrast, for Vega’s four triangles I find minimum rms errors ranging from 3′ to 5′, at times ranging from 4450AD to 5700AD, mean 5225AD.
Flamsteed’s (adjusted for precession to 1690.0AD by the editor, Baily) “British Catalog” (microfilm version at the Iowa State Univ. library) has uncertainty boxes 1′ on a side in right ascension (the relevant stars in Lyra all are given only to 1′ RA) and 5″ in declination. Flamsteed somewhat corroborates my finding about Ptolemy. For the four Vega triangles, the minimum rms errors range from 0.07′ to 0.24′, at times from 1715AD to 1745AD, mean 1730AD. I gather from Baily’s explanations, that Flamsteed’s observations began in 1689 and most were made during the early 1690s. If Flamsteed’s actual mean epoch were 1695AD and Ptolemy’s c. 150BC (i.e. Ptolemy used Hipparchus’ observations corrected for precession, as many believe) then a change in Vega’s proper motion, mas/yr/yr, sufficient to move the date of best fit by +35yr in 305yr, would move it 35*(2150/305)^2 = +1740yr in 2150 yr. With a higher order change, mas/yr^3, the indicated +5375yr shift seems possible.
Measuring subjectively the closest rectangular approximation to Monk’s mound, from Young & Fowler, “Cahokia” (2000) Fig. 11, p. 125, I find NS::EW = 1.12. This is the same as for Teotihuacan’s Pyramid of the Moon, but with NS & EW interchanged.
Summary of addendum re Cahokia. Cahokia is to Vega, as Cholula is to Arcturus. Vega’s rapid pole-on rotation might have caused a ~20% error in predicted proper motion, causing Cahokia to be misplaced by 0.1degree latitude. Ptolemy’s and Flamsteed’s catalogs support drastic long-term changes in Vega’s proper motion, which the builders might only roughly have estimated.
Addendum June 10, 2012 re Bosnia:
In the foregoing, I’ve discussed pyramids of stone, brick, and earth. Whether the Bosnian pyramid is an artificially shaped hill or built from the ground up, is less important now, than whether it is another “precessional alarm clock” for December 2012, and to what catastrophe, these precessional alarm clocks are trying to alert us.
The epoch 2000.0 position of Capella in J2012.94 coordinates (Capella’s next opposition) is RA 5:17:38.75, Decl +46:00:41.0. Proper motion from 2000.0 to 2012.94, nutation and aberration decrease this figure for Capella’s declination by about 9″, to +46d00’32”.
My position for the main Bosnian pyramid is interpolated by measuring to the end of the marked gradient, which is at least near the peak, and interpolating the corner coordinates of the online ( www.bosnianpyramid.com ) topographic map of Amer Smailbegovic, Ph.D. My result is E 18d10’37.0″, N 43d58’37.0″ (Wikipedia gives ~1′ farther north than my result, but theirs might not refer to the peak of the main pyramid; my coordinates differ only 1″ in latitude and 2″ in longitude from those of Enver Buza). The pole shift which Petrie and I infer from Giza, again gives an original latitude of 44d00’56”. This is 88″ farther north than needed to make Capella graze the horizon, neglecting atmospheric refraction. That is, 44d’00’56” + 46d00’32” = 90d01’28”. The true (refractionless) grazing could have been determined, from near-zenith observations, with spherical trigonometry.
Assuming the round-number ecliptic pole motion underestimate (which seems to have been used in Mexico but not at Giza) as explained in Sec. VI, Capella’s actual position now is about 33″ north of what the builders predicted. So for Capella to graze the horizon, the main Bosnian pyramid is really only 88-33=55″ farther north than it should be, according to what the builders expected. Enver Buza gives the third Bosnian pyramid, Buc^ki Gaj, as the equivalent of 69″ S of the main Bosnian pyramid, Visoc^ica. That would be only 55-69 = -14″ latitude from perfect theoretical position re Capella.
On Dr. Smailbegovic’s topographic map of the Visoc^ica pyramid, I marked by eye, the points of greatest curvature of all the level curves at the NE & NW edges, then drew least squares lines by eye to approximate these edges. Including a small correction for Earth’s flattening, I find that the orientations of these edges are 45.505deg E of N & 35.7475deg W of N, midpoint 4.879deg E of N. In my Sec. VI discussion of Teotihuacan, I noted the relationships:
cos(geographic lat.)*sin(x+d1) = sin(y+d1)
cos(geocentric lat.)*sin(x+d1+d2) = sin(y+d1+d2)
At Teotihuacan, y=15deg25′, x=16deg30′; d1=1.460deg, d1+d2=(5/3)deg, which seem to signify the actual and assumed radii of curvature of the ecliptic pole path during the last 5125 yr. Instead of the mean latitude of the pyramids of Moon & Sun in Mexico, let’s now use the theoretical ideal mean latitude of the pyramids of Dautovci & Visoc^ica in Bosnia, (geographic) 44deg00’01”. Instead of y = the famous 15 deg alignment of Teotihuacan, let’s use y = the 4.879deg alignment which I measure on the Visoc^ica pyramid, and d1 = (5/3)deg. Then d2 = 272″. Before the pole shift indicated by Giza’s layout, true north at the Bosnian pyramids would have been 374.5″ W of N, but because of their latitude, this amounts to only 270.3″ on the globe. So the small alignment of Visoc^ica away from true north, seems as at Teotihuacan to encode important angles.
Using the 2.5m contour intervals, I find that the average slope of the NW edge is 20.25deg and of the NE edge about the same, 20.92deg. Similarity to the edge slopes (22.3deg & 22.5deg) at Teotihuacan, also suggests artificiality. Because my drawn NW & NE edges do not meet exactly at the peak of Visoc^ica, I can only give a range of possible NS::EW ratios for that pyramid: 1.1049 to 1.1656, mean 1.135; this is similar to the pyramids of Cahokia in Illinois, and the Moon (exchanging directions) in Teotihuacan.
Summary of addendum re Bosnia. Regardless of how the Bosnian pyramid might have been produced or arisen from the natural material, its latitude, assuming only the pole shift deduced from Giza, is only 1.5′ too far north to be perfectly consistent with Capella’s circle of travel grazing the horizon at Capella’s opposition in late 2012.
