Requiem for Relativity

The Mother Lode of Archaeoastronomy

The library is closing so I don't have time for details.

The angle of Teotihuacan is usually said to be 15deg25'. This is between North and the Avenue, east of North, but there is another obvious angle: the angle between North and the inter-Pyramid line (centers of Pyramids of Moon & Sun), west of North, which I measured on Millon's hardcopy map at the Iowa State Univ. library, as 2.11deg +/- 0.02deg.

These angles, and their sum, correspond to three astronomical angles at the present epoch. These angles are at their intersection at the autumnal equinox, between three circles on the celestial sphere (generally not great circles). All the circles contain Regulus and the autumnal equinox of date. Also:

Circle #1 contains Spica.

Circle #2 contains the vernal equinox.

Circle #3 contains the North Pole (Polaris gives an even better fit).

For the equinox, ecliptic and positions of 2000.0 AD, #3 & #1 intersect at 17.76748deg, and for 2020.0 AD, 17.55209deg.

For the intersection of #2 & #1 (#2 lies between #1 & #3) the angles are resp. 2.32217 & 2.26664.

The agreement is perfect, by linear extrapolation, at 2077.7 AD for the former and 2022.7 AD for the latter. A small pole-shift adjustment to true north, based on the alignment at Giza determined by Petrie, does not affect the latter but improves the former to 2050.1 AD.

Teotihuacan - Mother Lode of Archaeoastronomy (update of June 22 post)

Teotihuacan is a "precessional alarm clock" (phrase due to John Major Jenkins). At the time for which the "alarm clock" is set,

1) the mean declination of Algieba equals the geographic latitude of the Pyramid of the Sun - Pyramid of the Moon midpoint;

2) the angles between four circles (not great circles) through the autumnal equinox on the celestial sphere, are the same as the angles between the four lines, through the centers of the tops of the Pyramids of Sun and Moon; geographic north-south; orientation of the Avenue of the Dead, etc. (often called the 15.5deg orientation); and the orientation of some other structures including the early lower south face of the Feathered Serpent Pyramid, a.k.a. Temple of Quetzalcoatl (this is often called the 16.5deg orientation).

The three circles all contain Regulus and the autumnal equinox. One circle contains Polaris, one contains Spica and one contains Dschubba.

The circle containing Polaris makes, at the equator, an angle 9.10669deg south of east at the equinox and epoch 2000.0 and 9.23943 at 2020.0.

The circle containing Spica makes 26.68628 at 2000.0 and 26.63725 at 2020.0.

The circle containing Dschubba makes 24.76731 at 2000.0 and 24.76606 at 2020.0.

My measurement on Millon's map (hardcopy at Iowa State Univ. library) finds that the Pyramid of Sun - Pyramid of Moon line is 2.1064deg W of N (I'll call this the "interpyramid angle"); Wikipedia gives coordinates that imply the mean expected and maximum-likelihood interpyramid angle is 1.5231 but because of large rounding error, possibly as large as 2.2840. So, tentatively I'll estimate the angle as the average (2.1064 + 1.5231)/2 = 1.81475 +/- 0.29165.

Unlike the interpyramid angle, the Avenue of the Dead angle has been studied extensively (see Sprajc, Latin American Antiquity 11:402-415, 2000). The two most prominent estimates are 15deg28' (favored by Sprajc) and 15deg25' (favored by Millon) E of N. So, I'll use 15deg26.5' = 15.44167.

Yet another angle found in some structures seems to have been most carefully estimated as 16deg26' or 16deg29' (the latter is S of E along the early lower south face of the Feathered Serpent Pyramid a.k.a. Temple of Quetzalcoatl). So, I'll use 16deg27.5' = 16.45833. This angle will be discussed in the section "Feathered Serpent and Barbarossa", below.

Behold, the Spica circle is 1.91897deg, for 2000.0, and 1.87119deg, for 2020.0, S of the Dschubba circle. By linear extrapolation this difference equals, at 2043.6 +/- 122.1 AD, the angle between the interpyramid line and the NS line, 1.81475 +/- 0.29165deg.

Much smaller error bars occur for the angle between the Avenue of the Dead and the NS line. The Dschubba circle is 15.57959, for 2000.0, and 15.52663, for 2020.0, S of the Polaris circle. By linear extrapolation this equals 15.44167 at 2052.1 +/- 9.4 AD.


Returning to the discussion of (1), the mean declination of Algieba, for the equinox of date and including proper motion, is 19.84167 for 2000.0 and 19.73980 for 2020.0. The geographic latitude of the midpoint of the Pyramids of Sun and Moon is (19.6925 + 19.6996)/2 = 19.69605. Extrapolation gives equality at 2028.6 AD.

From my measurement on Millon's map, the difference in geographic latitude, between the Pyramids of Sun and Moon, is 26.0725". Wikipedia gives 25.5600" +/- 0.1800" rounding error. So, I'll use the mean, 25.81625" +/- 0.25625". Algieba's aberration in declination at conjunction is +7.4073", and at opposition -7.5794". Its parallax in declination is 0.0077" more positive at opposition than at conjunction. Due to eccentricity, the time from conjunction forward to opposition is considerably less than 0.5 sidereal year; the mean declination change due to precession and proper motion is -9.0217". (These are first order calculations in eccentricity and the last digit or two might be inaccurate.) So we see that ignoring nutation, the declination change from conjunction forward to opposition is currently 24.0007", only 1.81555" +/- 0.25625" less than the interpyramid change in geographic latitude.

Now let us examine the 1980 IAU theory of nutation as published in the 1984 Astronomical Almanac. The largest, IAU #1, 18.6 yr terms give, for the declination of Algieba, -6.2003" in amplitude for the sine term and 3.8901" for the cosine term. The most negative rate of change occurs for theta = arctan(3.8901/6.2003) = 32.104deg phase, and this rate of change is -1.2354" per 0.5 Julian year. At this same phase, the rate of change due to the IAU #2, 9.3 yr terms is +0.0448" per 0.5 Julian yr. The second largest term is IAU #9, the 0.5 yr term, but this term hardly contributes because it is almost exactly one period from Algieba's conjunction to opposition. The amplitudes, for change in Algieba's declination, of the remaining 37-9 = 28 sine terms, with period one year or less, on the first page of the list (the sine terms are all bigger than the cosine terms) sum to 0.2331"; twice this is an estimate of their largest possible contribution. So, we have estimated the most negative possible contribution of nutation, to the change in Algieba's declination from conjunction to opposition, as -1.2354+0.0448-2*0.2331 = -1.6568", remarkably close to the -1.81555 +/- 0.25625 implied by the interpyramid spacing. In brief, the latitude difference of the Pyramids of Sun and Moon tells us the largest possible change in Algieba's apparent declination, from conjunction forward to opposition, at the present epoch.


"Feathered Serpent and Barbarossa"

In 2007 I discovered on online sky surveys, that our Sun has a satellite solar system, most likely chiefly a pair of very cold brown dwarfs, orbiting the Sun at about 300AU near the positive dipole of the so-called Cosmic Microwave Background. By 2009 I had calculated the center of mass orbit, amassed many lines of supporting evidence, and given a lecture on the subject to the regional amateur astronomy society. I invested much effort, including many personal interviews, in seeking to convince professional astronomers to investigate the matter, without success. There has been a small, ambiguously successful effort by amateur astronomers but with only marginally adequate equipment and minimal resources of observing time. It would not be too great a digression to mention here one of the simplest and most compelling bits of evidence: the exceptionally great, unexplained "interstellar" absorption of light from two stars in that direction: 61 Leonis and Theta Crateris; this is evidence of a nebula associated with this small satellite solar system.

Rechecking my old notes, I recalled that the main object, which I named "Barbarossa", was fitted by me with an orbit of eccentricity 0.61 and period 6340 yr. I found the heliocentric position for the winter solstice, 2012, and estimated that the orbit lay within a degree of its outbound latus rectum at that time. The object was crossing lines of declination at an angle arctan(0.50) S of E.

With the above information I now can calculate the circle through Regulus, the autumnal equinox, and Barbarossa. This circle corresponds to the "16.5 degree" line at Teotihuacan, because (interpolating between 2020.0AD & 2040.0AD positions) at 2025.1 AD it makes the same angle with the Regulus - autumnal equinox - Dschubba circle, that the "16.5 degree" S of E line makes with geographic north. This circle on the celestial sphere moves rapidly, its slope at the autumnal equinox moving more than 0.5 degrees per year, because Barbarossa's period is only about a fourth the precession period, and the circle is small because of Barbarossa's position. So if my orbit is accurate, the error bar on 2025.1 AD is small. Note that this agrees well with the estimate based on the declination of Algieba, 2028.6 AD. Just as the Hindu reference year 3102BC equals the Mayan reference year 3114BC plus 12 yr., likewise 2025AD equals the 2012AD winter solstice, i.e. approx. 2013AD, plus 12 yr.

Finally there is the line through the top centers of the Pyramids of Moon and Sun. Let these points be M and S, resp. The south face of the Feathered Serpent Pyramid (Barbarossa lies near the constellation Hydra = Feathered Serpent?) extends to a line segment which intersects the line MS, at F, a point east of the Feathered Serpent Pyramid. Let the southeast corner of the Feathered Serpent Pyramid, be C. According to my rough, hasty measurements on Millon's map, we have

CF / (CF + FS + SM) = 45.77/360 [45.72/360]

FS / (CF + FS + SM) = 192.64/360 [195.33/360]

The numbers in brackets are from my remeasurement July 9 on Millon's small map, Fig. 13A, in the hardback "Urbanization" vol. 1 accompanying the maps; this is because I didn't have time to reshelve the map myself so it is unfindable during a prolonged library "reshelving" process, a common problem at ISU.

The former number, using the average 45.745, exactly equals the fraction of the circular arc southward from the autumnal equinox to Barbarossa, in the equinox-Barbarossa-Regulus circle, at 2027.0AD. The latter number, using the average 193.985, exactly equals the fraction of the circular arc from the equinox to Polaris (not via Regulus) of the equinox-Polaris-Regulus circle, at 2052.3AD though with a large error bar.

Thus three dates with small error bars,

2028.6, 2025.1, 2027.0AD,

the first date corresponding to the declination of Algieba and the latter two corresponding to the direction and distance to my discovery, Barbarossa, from the equinox along the circle including Regulus, cluster with mean 2026.9 +/- SEM 1.0 AD.

TEOTIHUACAN AND GIZA HAVE SAME ARCHAEOASTRONOMICAL PLAN AND INDICATE SIMILAR DATES

Joseph C. Keller

Joseph C. Keller, 16201 620th Ave., Roland, Iowa U.S.A. 50236
(josephckeller@gmail.com)


Abstract. The main angles of the urban plan of Teotihuacan, all correspond to sky angles involving Venus, Jupiter, Regulus, Spica, and the autumnal equinox point, at 01:43 GMT Sept. 8, 2016. Equality between the latitudes of Teotihuacan and Cholula, and the declinations of Algieba and Arcturus, corroborates the correspondence. Analogous angles at Giza, correspond to 20:59 GMT Sept. 16, 2015. The plan at Giza is analogous to that at Teotihuacan, but Venus, Regulus, and Spica are changed to Jupiter, Arcturus and Regulus, and clockwise changed to counterclockwise. The apparent small pole shift, discovered by Petrie, since the construction of Giza, also is indicated at Teotihuacan. The interpyramid angles at Giza, however, give evidence of a larger pole shift, whose ratio to the smaller one, is the ratio of 11,400 yrs ago (Plato's date for the fall of Atlantis) and the well-established historical date of Giza construction, 4500 yrs ago.



Prerequisites. This paper must assume knowledge of the elements of spherical astronomy (i.e. celestial coordinates and equinox precession) available in introductory college astronomy texts and in popular field guides to astronomy. The calculations of this article, require facility in U. S. high school geometry, trigonometry and analytic geometry. Readers lacking such facility can take the calculations on faith, without impairing their understanding of the rest of this article. As on many pocket calculators and in the BASIC computer language, a star (*) denotes multiplication.

Giza vs. Teotihuacan. Many authors have noted the high precision of Teotihuacan's urban plan, which contrasts with the variable and approximate use of similar angles in later Mesoamerican urban plans which seem to mimic Teotihuacan. Just as Egyptian temples often were rebuilt again and again over millennia, on the same foundation with the same location and orientation, the Teotihuacan structures might reflect a design much older, than the oldest dates yet found by archaeological dating of the structures now extant. This paper makes the case, via an implied small shift in Earth's geographic pole relative to the land mass, that Teotihuacan's design is of the same vintage as the design of Giza.

Cholula and the Mayan Long Count. As an instructive preliminary example, let us consider the correspondence between the Cholula pyramid, and the declination of the star Arcturus. The geographic latitude of the Cholula pyramid is 19.0575deg N. The declination of Arcturus was +19.182408 at 2000.0AD, using 2000.0AD celestial coordinates. The change in declination due to precession, may be calculated from the handy "rigorous" formula found in editions of the Astronomical Almanac c. 1990, or obtained directly online through the "NASA Lambda" utility (these are "mean" precession formulas, i.e. they ignore the changes of a few arcseconds, due to aberration and nutation); using the former here, I find that the change is -0.004614deg per year. Also, the star itself is moving; this "proper motion" change, which is extraordinarily large in the case of Arcturus, in declination is -0.000555deg/yr. Lastly there is the small alteration in the geographic latitude of Cholula, which seems to have occurred since its foundation. Most likely this is practically the same as for Teotihuacan, which has been estimated below as -1.231*177.1 arcseconds = -0.06056deg. Equating the original geographic latitude of the Cholula pyramid, to the modern ("mean", ignoring aberration and nutation) declination of Arcturus, is by solving the equation

19.0575 + 0.06056 = 19.182408 + T * ( -0.004614 - 0.000555)

which gives T = 12.45. That is, if Earth's pole had not shifted at all, Arcturus would have reached the zenith over the Cholula pyramid in June 2012, ignoring aberration and nutation. Aberration affects Arcturus' declination by as much as 0.0020deg and nutation's effect is comparably large, so these effects are equivalent to as much as about half a year error. This looks suspiciously like the end of the Mayan Long Count, December 21, 2012.

The pole shift has been estimated below, as a multiple of the Giza pole shift, from linear interpolation and extrapolation involving the mean declination of Algieba, which is accurate enough to be used with the ground direction measurements with their relatively large error bars. So, this pole shift estimate cannot be precise enough to be tested against the effects of aberration and nutation. Another approach must be taken: see the section, "Arcturus and Algieba..." below.


Analysis of Teotihuacan Ground Directions. At Teotihuacan, we have six important directions:

#1. The pyramid of the Moon - pyramid of the Sun line. On Millon's paper map at the Iowa State University library, I have measured this as 2.1064 +/- 0.0243 deg W of N. The error bar is my crude estimate of the error due to my ability to read fractions of a millimeter on a wooden ruler: it ignores error due to stretching and slight tearing of this 40 year old paper map. So, the true error bar is probably larger and likely considerably larger; a nominal error of 3 sigma should not be dismissed, as it is likely overestimated.

#2. The line to geographic north. As Petrie noted, the most reliable orientations at Giza suggest true north at Giza has changed by a few arcminutes, and Petrie thought this could be due to climate change causing major change in ocean currents. Earth mantle changes also are possible. Many have remarked that likely, the Great Pyramid originally was exactly 30deg N. Petrie estimated that at the time of construction of the Great Pyramid, true north there was 5'40" +/- 10" W of true N today. Also, the Great Pyramid today is 1'15.6" S of 30.0deg N. As viewed from Teotihuacan, this small pole shift, relative to Earth's crust, would have given a true north 262.3" E of true N today, and a latitude 177.1" greater than today.

#3. The Avenue of the Dead orientation (the "15.5deg angle"). Millon (1973) favors the value 15deg25' E of N; Sprajc (2000) favors the competing value 15deg28'. These values were found by competing investigators decades ago. As Sprajc's publication is the more recent, I'll favor him in my adopted round-number compromise value, 15deg27', E of present-day N.

#4. The "16.5deg angle". There is another cluster of orientations of structures near 16.5deg S of E. According to Sprajc's citations, it seems that the most carefully determined values are 16.3deg, 16deg26', and 16deg29', so I'll adopt the compromise 16deg27.5' S of present-day E.

#5. The pyramid of the Moon - pyramid of the Feathered Serpent (a.k.a. Temple of Quetzalcoatl) line. By measuring the two acute interpyramid angles (always to the centers, i.e. intersection of diagonals, of the top marked levels of the pyramids) on Millon's map with a steel meter stick, and a sheet of paper as a right angle (weighting the two ways of doing this, according to the distance involved) I find that this is 8.91deg E of the Moon-Sun line (see #1 above). The two methods afford an error estimate of 0.03deg.

#6. The pyramid of the Sun - pyramid of the Feathered Serpent line. As in #5, I find that this is 15.04deg E of the Moon-Sun line. An error estimate as in #5, would give 0.06deg.

It turns out that #1 through #3, correspond to sky angles characteristic of 2013AD, as does the declination of Algieba.
Let's consider the problem of signifying a time, using only the universal language of astronomy and mathematics. A time a few thousand years in the future, could be specified by the angles of spherical triangles which include the equinox as one of their corners and stars as the other corners. This scheme would be easiest to decipher, if the angle specified always is the angle at the equinox. Then, this collection of triangles amounts to a collection of great circles, each through one star and the equinox. The concept of great circle may be eliminated, by using instead, a circle (generally not a great circle) which is uniquely specified by two stars plus the equinox. The most obvious such circle is the circle containing Regulus, Spica, and the autumnal equinox. These stars are both bright, both blue (color coding) and both near the ecliptic and the equinox.

(1) At the equinox, this Regulus-autumnal equinox-Spica circle makes the angle 26.6863deg S of E, at 2000.0AD, and 26.6373deg S of E, at 2020.0AD, using the mean position of the equinox and of the stars (including proper motion) involved. Let's denote this angle by (1).
Now two additional obvious circles are:

(2) Through Regulus and the two equinoxes; this is a great circle which, as explained in (1), makes the angle 24.36411 at 2000.0 and 24.37061 at 2020.0.

(3) Through Regulus, the autumnal equinox, and the north pole; this makes the angle 8.91880 at 2000.0 and 9.08516 at 2020.0.
From now on, all these angles will be interpolated linearly in the domain 2000-2020AD. The Teotihuacan ground angle #1 minus #3, does not involve the geographic pole. If the real ground angle #2 minus #1, is 2*0.0243deg, i.e. 2 sigma, greater than my measured value, and #3 minus #2 is 15deg27' as adopted, then (1) minus (3) equals #1 minus #3, at 2015.09AD. If 3 sigma, then at 2012.83AD. Each of these dates implies a certain angle (1) minus (2), and thus a certain pole shift E of N, for the time of pyramid construction, so that (1) minus (2) shall equal #1 minus (the ancient) #2. The 2 sigma error implies a pole shift 1.720 times the Giza shift (if only pole shifts proportional to the Giza shift are considered) and the 3 sigma error implies a pole shift 1.473 times. (Recall that my value of sigma is an underestimate, so these "2 sigma" and "3 sigma" errors are really somewhat smaller sigma values.) The former implies that the mean declination of Algieba equals the geographic latitude of the Moon pyramid - Sun pyramid midpoint, at 2011.98AD, and the latter implies 2014.36AD. Interpolating, a 2.670 sigma error implies 1.5544 times the Giza shift, and that both the (1) minus (2), and the Algieba declination, criteria are met at 2013.58AD, in remarkable agreement with the Mayan calendar.

If the true, or intended, direction of the Avenue of the Dead (perhaps more accurately translated "Memorial Drive") is 15deg24', then the solution is mid-2016AD rather than mid-2013AD. Putting the finest point on it, 15deg27.6' corresponds to the Mayan end date in Dec. 2012; 15deg24.1' corresponds to the Venus date described below, in Sept. 2016. The true direction can vary this much, depending on how various structures are weighted in determining the direction. A few years disagreement might not be surprising with the Mayan Calendar starting at 3114BC, Manetho's chronology at about 3110BC, and the Kali Yuga of India at 3102BC. If 15deg24.1' is adopted, then a repeat of the foregoing analysis is consistent with a pole shift 1.231 times that implied at Giza - almost equal, considering the 3%, maybe more, error that Petrie assumed.

With the 15deg24.1' value, the meaning of directions #5 and #6 becomes clear. These directions correspond to the directions, at the autumnal equinox, of circles on the celestial sphere, through Regulus-Venus-autumnal equinox, and Spica-Venus-autumnal equinox, respectively, on Sept. 8, 2016 at 01h43m GMT. Assuming the mean positions of Regulus and Spica at 2016.68AD (disregarding aberration and nutation) and my measured directions #5 and #6, the point of intersection of the two circles, is at RA = 190.43414, Declination = -3.59348. At the abovementioned time, Venus is 52 arcseconds south of this intersection point. Thus 01:43 is the time that Venus' actual RA equals that of the intersection point; Venus' actual declination equals that of the point, at 01:02.

The JPL Horizons ephemeris includes aberration of light, and nutation, effects in its Venus position; these effects are not included in the mean stellar positions. However, the aberration correction to the stellar positions, only changes the above 52" declination error, to 54", and only makes the time of RA agreement, three minutes later. Rough estimates indicate that correction of stellar positions for nutation, and correction of planetary positions for possible non-geocentric observatory locations, also are an order of magnitude smaller than the roughly one arcminute error observed.

Thus the three sides of the triangle made by the three largest pyramids at Teotihuacan, correspond to the three subsets of two, possible within the set containing Regulus, Spica, and Venus. To say it another way, The pyramid of the Moon corresponds to Regulus, of the Sun to Spica, and of the Feathered Serpent to Venus.

Finally the significance of direction #4 becomes obvious. It pertains to Jupiter. At the time Venus fits the angles, namely 01h39m GMT Sept. 8, 2016, the apparent right ascension of Jupiter is 180.16124 and declination +1.12245. That Jupiter is so near the equinox at this time, is significant in itself, but direction #4 puts an even finer point on it. For this time, direction #1 (see first page) corresponds to the sky angle (1), 26.6453deg S of E. From this I subtract

my measured angle 2.1064deg

and my implied measurement error +0.0243deg*3.285

to get the sky angle corresponding to present-day geographic N. I then subtract

(90deg + 16deg17.1')

to get the sky angle, E of N, arctan(0.16124/1.12245), which is Jupiter's direction from the equinox (spherical trigonometry is not needed at this precision). Jupiter's distance from the equinox is so tiny, that the difference between 16deg17.1' and 16deg27.5' corresponds to only 0.00347deg = 0.83 seconds of right ascension = 12.5 arcseconds. A stricter analogy between the 15.5 and 16.5 degree directions, would be a circle through the fall equinox, Jupiter, and the N pole. This gives an additional 4.8' E of N, implying 16deg12.3' for the 16.5 degree angle at Teotihuacan.

The small difference in latitude between the pyramids of the Sun and Moon, corresponds, at least approximately, to the largest shift in the declination of the star Algieba, that can occur during the course of a year, due to aberration, precession and nutation (based on inspection of the 1980 IAU nutation amplitudes).


Arcturus and Algieba, Cholula and Teotihuacan, and Giza

As noted above, Regulus and Spica are a color-coded blue pair of stars; likewise Arcturus and Algieba are a color-coded orange pair, with Arcturus being to Cholula what Algieba is to Teotihuacan. Let's assume that if there had been no pole shift, then Arcturus would have reached the zenith over the Cholula pyramid at 2012.97AD (end of Mayan Long Count) and Algieba the zenith over the pyramid of the Moon at Teotihuacan, at 2016.68AD, the date found above, when Venus and Jupiter correspond to some of the angles at Teotihuacan. Let's also assume, as above, that the pole shift is some multiple of that implied by Petrie's survey at Giza.

Both stars must satisfy the equation:

present pyramid geographic latitude
- N*(latitude effect of Giza's implied pole shift)
= star's J2000.0/epoch2000.0 declination + T*(rate of change due to precession and proper motion)
+ nutation-caused difference from mean position
+ aberration-caused difference from mean position

The nutation terms are -1.60" for Arcturus at 2012.97 and -1.84" for Algieba at 2016.68, using the principal (18.6 year) nutation period only, whose amplitude is about 10 times bigger than all the other, shorter, nutation periods' amplitudes combined. Stellar parallax, a fraction of an arcsecond, is ignored. The aberration terms are approximately -5.60" and +6.26" for Arcturus and Algieba, resp. The effect of the Giza pole shift at Teotihuacan is -177.1" geographic latitude and at Cholula slightly more, -179.4".

Solving gives N = 1.121 for Cholula and N = 1.182 for Teotihuacan (using the pyramid of the Moon; its unique design and privileged location at the terminus of the Avenue of the Dead, recommend it). These are the multiples of the Giza pole shift, that would have had to occur, for Arcturus to have been at the zenith over the Cholula pyramid, if that pyramid had remained at its original latitude, at the end of the Mayan Long Count, 2012.97AD; and for Algieba likewise to be at the zenith over Teotihuacan at 2016.68AD, the time implied by the correspondence, between Teotihuacan's ground directions and the planets Venus and Jupiter. This suggests two rival predictions, 2012.97AD and 2016.68AD, and that the latter prediction was the earlier. On the other hand, if nutation and aberration effects are ignored, solving gives N = 1.161 for Cholula and N = 1.157 for Teotihuacan, so, since these are practically equal, maybe the plans of the structures were contemporaneous.

The implied pole shifts are almost the same as inferred by Petrie: for example, 1.16 is 16 percent too large, 4.3 sigma, using Petrie's apparently rather arbitrary 3.75 percent error bar (the shift of the Great Pyramid to exactly 30N, partly cancels the effect, at Mexico, of the 5'40" E of N shift, and thus somewhat magnifies the effect of Petrie's estimated 10" error bar for the E of N shift). With nearby mountains, the geodetic quantity "xi", i.e. the north-south deflection of an actual plumb bob from the geographic vertical of the reference ellipsoid, should amount to several arcseconds at these locations, and might explain the discrepancy from Petrie. The author met with three faculty members at Iowa State Univ.; neither he nor any of them were able to obtain the needed geodetic information, even approximately, online or anywhere. One of the faculty members, an active full professor and geophysicist, told the author that the condition of the U. S. Geodetic Survey website is so poor that it is impractical use it. No response, other than the automated confirmation, was obtained from the author's online request to the Bureau Gravimetrique International. Perhaps a reader can find the geodetic "xi" data.


The Analogy with Giza

One might object, that there are many stars, planets and abstract points in the sky. Yet, among the abstract points (poles, solstices, equinoxes) the autumnal equinox is the natural choice because of the proximity of bright stars. Venus and Jupiter are the brightest planets, the brightest astronomical objects in the sky aside from the Sun and Luna.

Regarding the stars, let us note that Sirius is the brightest, but it is far from the autumnal equinox. The brightest star which is closer to the autumnal equinox than Sirius, is Arcturus (brightest star in the northern hemisphere according to the usual photometric definitions). The brightest closer than Arcturus, is Spica. Arcturus is a natural alternative to Spica, Jupiter an alternative to Venus, and, for a clockwise angle in the sky, counterclockwise on the ground (as if the ground figure were to be viewed from the center of Earth) rather than clockwise as at Teotihuacan. Also, Arcturus at Giza plays the dominant role, like Regulus at Teotihuacan.

The angles between the three large pyramids, are known more accurately at Giza than at Teotihuacan, because of Petrie's survey. The centers of the Giza pyramids are more precisely defined, because of their simple square pyramidal shape; the pyramid of the Sun at Teotihuacan, for example, has, in contrast to the simplicity of the Giza pyramids, a well-known torsion to its successive levels, visible on Millon's map. Two of the corners of the pyramid of Menkaure at Giza, deviate four arcminutes apiece from 90 degrees; but the other ten corners of the main Giza pyramids, all deviate no more than about an arcminute, often much less. The accuracy of the corners implies the potential for an even greater accuracy of the interpyramid angles, because these are determined by averaging four corners on each pyramid; and the interpyramid distance is greater than the pyramid side length, so any position error becomes proportionally smaller.

At Teotihuacan, the direction, at the autumnal equinox, of the circle through Regulus and the two equinoxes, corresponds to geographic North on the ground. At Giza, it is the circle through Arcturus and the two equinoxes.

At Teotihuacan, the direction of the circle through Regulus, Spica, and the autumnal equinox, corresponds to the line through the pyramids of Moon and Sun. At Giza, it is the circle through Arcturus, Regulus, and the autumnal equinox, which corresponds to the line through the pyramids of Menkaure and Khafre.

At Teotihuacan, the direction (these circle directions always are taken where they pass through the autumnal equinox) of the circle through Regulus, Venus, and the autumnal equinox, corresponds to the line through the pyramids of Moon and Feathered Serpent. At Giza, we substitute Arcturus for Regulus and Jupiter for Venus; the corresponding line is through Menkaure and Khufu.

Likewise at Teotihuacan, the direction of the circle through Spica, Venus, and the equinox, corresponds to the line through the pyramids of Feathered Serpent and Sun. At Giza, we substitute Regulus for Spica (and again Jupiter for Venus); the corresponding line is through Khafre and Khufu.

The intersection of the sky circles implied at Giza, by the two preceding paragraphs (using mean star positions for 2015.72AD) is at the point RA = 159.88794, Declination = +9.46645. Jupiter's apparent position is 73 arcseconds south of this point, at 20:59 GMT Sept. 16, 2015; at 14:46, Jupiter's apparent declination equals that of the point. As for Venus, if the star positions are corrected for aberration and nutation, it should make only a few arcseconds difference in the error, and a few minutes difference in the time.

So, for Teotihuacan and Giza, Venus and Jupiter, respectively, pass about an arcminute south of the predicted position. They do so, less than nine days apart on consecutive Septembers, 2016 and 2015AD.

At Teotihuacan, the orientation of the pyramid of Moon - pyramid of Sun line, is roughly consistent with the presumed pole shift since the construction of Giza (this is discussed above). Using mean stellar positions (i.e. excluding nutation, aberration and parallax) the latitudes of the pyramids of the Moon and Cholula are consistent with 1.159 +/- 0.002 times the Giza pole shift, as estimated by Petrie and from the discrepancy between the pyramid of Khufu and the exact 30th parallel. Taking 2550BC as the well-documented approximate historical date of the plan of the Giza pyramids, this suggests that the date of the plan of the Cholula and Teotihuacan pyramids is (2014AD - 2550BC) * 1.159 = 5289 yrs ago = 3276BC; this is not far from the start year of the Mayan Long Count, 3114BC.

The main elements of the Giza plan might be much older yet. The interpyramid lines at Giza are determined accurately enough to afford an independent estimate of the pole shift. If the angle between the Khafre-Menkaure line, and ancient North, is to equal the angle at 2015.72AD between the sky circles described above, there must have been a pole shift not 1 or 1.159, but rather 2.513 times the (implied, according to Petrie, by the NSEW pyramid orientations; and by the deviation of Khufu's geographic latitude from exactly 30N) shift since Giza. If the pole shift, on the average, is approximately linear in time, this corresponds to 9455BC. Socrates was killed in 399BC; Critias' lecture on Atlantis, recorded by Plato, would have been no more than a few years prior to that, so the pole shift implied by the interpyramid lines at Giza corresponds to the date of the fall of Atlantis (9000 years prior to Critias' lecture) given by Plato.

The north-south extent of the Teotihuacan, and Giza, pyramid triads, was chosen to provide confirmation of the declinations of Venus and Jupiter at the specified times, and perhaps a more accurate, certainly a more simply obtained, value for those declinations. The formula is:

A * B = C * C

where A = planet declination (absolute value)

B = north-south extent of pyramid triad, in geographic latitude

C = a standard small angle, specifically 0.2500 deg. This (or more precisely 0.250014deg, accurate for tens of thousands of years according to Simon, 1994) is Luna's mean angular radius using the 1982 IAU value for Luna's mean radius (0.2725076 times Earth's equatorial radius) viewed at Luna's mean apogee (geocentric distance = mean semimajor axis * (1 + mean eccentricity) ) from a point on Earth's equator at which Luna happens to be at the zenith. Of historical interest is Hansen's wholly ground-based, painstaking but low-tech 19th century value (cited in Newcomb, 1912) 15'34.08" = 0.259467deg at "mean distance" (conventionally this signifies the length of the semimajor axis) which when corrected to mean apogee (using Simon's modern orbital elements) as seen at the zenith from Earth's equator, becomes 0.249749deg.

Alternatively one may note that 360 = 2*2*2*3*3*5 is, like 12 = 2*2*3, and 2, a natural numerical base, and 1/360 quadrant = 90/360 = 0.2500deg; or, that to the nearest whole number, there are 30 days in a synodic month and 12 synodic months in a year, and 30*12 = 360, a number of days prominent in both the Mesoamerican and the Egyptian yearly calendars.

For Giza, I use the distance between the centers of the first and third pyramid, according to Petrie. For Teotihuacan, I use my steel ruler distance between the pyramids of Moon and Feathered Serpent on Millon's map.

At Giza, an accurate north direction is important for finding NS distance; at Teotihuacan much less so, because the line is nearly NS, and the cosine of a small angle, is insensitive to that angle. At Giza, the distance must be multiplied by

sin(56.95479 - 5.931556 + 12deg12'22" - 340" * 2.513)

where the first two of the four inner terms are the Menkaure-Khafre slope and the Khafre-Menkaure-Khufu angle according to Petrie's coordinates; the third inner term is Petrie's conversion of his temporary survey coordinates to his true N; and the last term is my calculated pole shift needed to get a match to the angle between the sky circles through Arcturus (see above). At Teotihuacan, the distance must be multiplied by

cos(2.1064 + 3.285*0.0243 - 8.9068 - 262.3" * 1.159)

where the first two terms, are my measurement of the Sun-Moon line on Millon's map, and the correction implied by the sky angles; the third term is my measurement of the Feathered Serpent - Moon - Sun angle, and the fourth is the pole shift implied by Arcturus and Algieba over the pyramids of Cholula and Moon (see that section).

To convert NS distance to geographic latitude difference, I use the first derivative formula (calculated via parametric latitude) according to the triaxial standard Earth ellipsoid (Zhongolovich, 1952). I use Millon's altitude for Teotihuacan, 2276m; and the approximate median altitude for the Giza plateau, 70m, seen on various online maps. In the derivative formula I use the mean of the geographic latitudes of the pyramids of Moon and Feathered Serpent, found by a small correction to the geographic latitude of the pyramid of the Sun as given by Millon; I assume exactly 30.0N for the median geographic latitude of Giza.

The result is, that to make the A*B = C*C, where C = 0.25deg, formula hold exactly, the declinations of Jupiter and Venus must be +9.424345 and -3.54843, resp. The time of geocentric observation of Jupiter must be later, 03:42 GMT Sept. 17, 2015 (RA = 159.94429), instead of 20:59 GMT Sept. 16; and the time of observation of Venus must be earlier, 22:56 GMT Sept. 7, 2016 (RA = 190.30489), instead of 01:43 GMT Sept. 8. For Jupiter as observed from Giza, the time is 03:39; and for Venus observed from Teotihuacan, 22:54 (these times still omit the corrections, also of only a few minutes of time, for the effects of aberration and nutation on stellar positions).

It might be, that the date discrepancy, is because Venus lies far from the autumnal equinox, at the intermediate time, March 2016, which really was desired to be indicated. So, they bracketed the date, indicating T - dT and T + dT at Giza and Teotihuacan, resp.


Appendix. Outline of mathematical methods; astronomical data sources.

Geometry and Analytic Geometry.

Problem A. Find the angle at the equinox point O, of a circle on the celestial sphere, through the equinox and two other points P and Q.

Method: Form the vectors P-O and Q-O. Find the y and z components of their Gibbs-Heaviside cross product (a.k.a. vector cross product) then arctan(y/z).

Problem B. Given circles C1 and C2 on the celestial sphere, each through the equinox O and points P1 or P2 respectively, which make angles A1 or A2 at O. Find the intersection of C1 and C2.

Method: For each circle, write the equation of the plane through O, cutting the celestial sphere perpendicularly at angle A. Also write the equation of the plane through O which is tangent to the sphere. Find the linear combination of these planes, which contains P. Then find the intersection of the two planes corresponding to circles C1 and C2, then the point where this line pierces the celestial sphere.

The angle-preserving maps at Giza and Teotihuacan, mapping circles through the autumnal equinox on the celestial sphere, to lines on the plane, are known mathematically as "stereographic projection". Teotihuacan uses a standard stereographic projection with the autumnal equinox as the source point known as the "pole" at the top of the sphere; Giza performs a reflection of the standard projection, by turning the plane upside down so clockwise becomes counterclockwise and vice versa.


Astronomical Data and Calculations.

For precise planetary coordinates at future times, the author uses the online "Horizons" ephemeris of the Jet Propulsion Laboratory of California, which gives "apparent" positions in the coordinates of the equinox and ecliptic of date, including nutation and aberration effects. Many sources, online and hardcopy, give stellar coordinates to the nearest arcsecond, with an estimate of the proper motion; any of these is suitable for the work of this paper. However, with a view toward possible greater precision requirements in the future, the author has used the most recent and precise catalog information he could find on the online VizieR service: the proper motions and International Coordinate Reference System (ICRS) epoch 2000.0 coordinates of the Hipparcos "main" catalog, for Regulus and Spica; and, the proper motions and J2000 epoch 2000.0 coordinates of the Catalog of Stars with High Proper Motions (Ivanov, 2008) for Arcturus and Algieba. The difference between the ICRS and J2000 coordinate systems is apparently somewhat vague, theoretically no more than about 0.02 arcsecond, but maybe several times that in practice.

The stellar coordinates must be converted to the "mean" equinox and ecliptic of date, using the online NASA Lambda service or else the straightforward yet rigorous formula in, e.g., the years 1990 and 2000 Astronomical Almanac (p. B18); then the nutation and aberration effects can be added to that, using well-known approximate formulas, e.g. the year 2000 Astronomical Almanac pp. B20 and B17, resp. In the aberration formula, the Earth velocity for 2000AD may be used as a substitute for the same date in 2012AD or 2016AD because these years all are at the same place in the leap year cycle. The most thorough list of coefficients for calculating nutation, is found in the 1984 Astronomical Almanac.


References.

United States Naval Observatory. Astronomical Almanac, 1965, 1984, 1990, 2000.

Jet Propulsion Laboratory. "JPL Horizons Ephemeris" online.

Millon, Rene. The Teotihuacan Map. (University of Texas Press, 1973).

Newcomb, Researches on the Motion of the Moon, Part II, 1912.

Plato, "Critias" (c. 400 BC).

Petrie, W. M. F. The Pyramids and Temples of Gizeh (1883). Online, Oaten and Birdsall, eds., www.ronaldbirdsall.com .

Simon et al. Astronomy and Astrophysics, 1994.

Sprajc, Ivan. Latin American Antiquity 11:402-415, 2000.

Strasbourg University. VizieR online catalog service.

Zhongolovich, 1952. In: CRC Handbook of Chemistry and Physics, 1987-1988.

Hi Dr Joe, I read something about dna being found in American Indians that matches dna found in the middle east and they say it arrived in the Americas at least 3,000 years ago. If dna from the middle east migrated from there to America more than 3,000 years ago I wonder if many other unknown events occurred way back when Stonehendge was an active human settlement. How far did people travel and how much did they know? Have you seen any of the dna stuff being done at this time, Dr. Joe?

<blockquote id="quote"><font size="2" face="Verdana, Arial, Helvetica" id="quote">quote:<hr height="1" noshade id="quote"><i>Originally posted by Jim</i>
<br />Hi Dr Joe, I read something about dna being found in American Indians that matches dna found in the middle east and they say it arrived in the Americas at least 3,000 years ago. ... <hr height="1" noshade id="quote"></blockquote id="quote"></font id="quote">

Excellent point. There is the beautiful giant stone head found in Mesoamerica, nicknamed "Baby Face" who is a typical West African.

Quetzalcoatl himself might have been a visiting European, like the navigator, Adams, in the true story that became the movie "Shogun". The story of "ascending to heaven on a rope" might be just how centuries of the game of Post Office altered the true memory of the impressive ropes holding the cloudlike sails of Quetzalcoatl's European style sailboat.

Dr. Joe, This is a good example of how people were able to organize way back when sea level was much lower than it is today and only a few million people lived on the planet. They had natural abilities and intellect as well as everything they needed to build stuff that would decay in a few centuries. We have very few clues to what they did or why they did it. We have no clue about what life was like for a population smaller than a modern major city with the whole planet available tax free.