I've been improving my calculations, and the latest edits to my posts contain those improvements, but here's another summary that I think is my best yet:

1. Torque-free precession is when there is practically no external torque (i.e. the tiny torques due to Luna & the Sun are negligible) . Because there is no external torque, Earth's spin angular momentum vector is constant. But that doesn't mean that the same piece of turf has to be under the pole all the time. Earth can move so that different pieces of turf are under the pole. The "pole" (i.e. latitude & longitude of the piece of turf) can move (i.e. "precess") a lot, relative to Earth coordinates, even though the angular momentum vector is unchanged in "absolute" coordinates. (When this happens there is also a slight daily wiggling of Earth's spin vector in absolute coordinates, due to Earth's slight asphericity and Earth's being somewhat askew, which causes the spin vector to have a direction slightly different from the angular momentum vector, but that's pretty negligible.)

2. Earth has almost perfect symmetry about the z-axis of its spheroid. So, two of Earth's moments of inertia are equal, both slightly smaller than the z-axis moment of inertia. The equality of those two x & y-axis moments, simplifies "Euler's equations" (Euler's equations amount to a statement of conservation of angular momentum, expressed in rotating Earth coordinates by way of an expression for "Coriolis force" applied to the angular momentum vector itself). Then it's pretty simple to rearrange Euler's equations to show that Earth can "precess" so that the pole moves slowly along a line of Earth latitude. Even without Euler's equations, just considering conservation of energy and angular momentum, it is evident that two variables, (1) the latitude of this precession line, and (2) the period of Earth's rotation, are determined by two equations involving angular momentum and kinetic energy of rotation and Earth's moments of inertia. Angular momentum is a known constant, the same as it is today, but I need to know the kinetic energy (see #4) and the moments of inertia (i.e. the flattening) (see #5).

3. As long as the changes in kinetic energy and angular momentum are small, the instantaneous period of Earth's sidereal rotation changes only by a few seconds: e.g. 56 sec shorter, in the 18.90deg precession solution discussed below. But the latitude of the precession line, i.e. angle of precession, can become large with only small changes in kinetic energy and no change in angular momentum. Basically this is because Earth's flattening is small. When Earth's kinetic energy is exactly what it is now, all Earth can do is rotate with nearly perfect symmetry about its z-axis, as it does now; but if Earth somehow acquired more kinetic energy of rotation, it could precess at a large angle.

4. Earth's gravitational self-energy is large; about 1000x Earth's kinetic energy of rotation. The energy released by, say, a major decrease in Earth's flattening (like energy released by a falling weight or a skater pulling in his arms) is of the order of magnitude to enable large-angle torque-free precession. My best effort to calculate this energy of flattening, is that it is

8/5 * (k/3)^2 * Earth's total gravitational self-energy

"k" is the flattening ratio, 1/297. (In practice I make small adjustments to account for my belief that the core does not participate in this torque-free precession.) Half of the above energy, a "4/5" is simply due to generally greater distances between mass particles, when Earth is flattened. Half of the energy, i.e. a "4/5" is a "monatomic adiabatic perfect gas response" to the other half, assuming Earth isn't truly "incompressible": when Earth is less flattened, forces are greater, pressures are greater, and Earth shrinks slightly to a new equilibrium, releasing yet more potential gravitational energy, namely an "8/5" but due to "the virial theorem", half of that energy goes into heat, and half, another "4/5" is available for mechanical work, so 4/5 + 1/2 * 8/5 = 8/5 is the answer.

5. I also need to know how much Earth flattens. I simply averaged all the spheroids one would get with the poles at different points along the line of latitude. The answer is that the flattening is lessened by the factor

1 - 3/2 * sin^2 (alpha)

where alpha is the colatitude of the precession, i.e. the central half-angle of the precession cone.

6. Using my estimates of flattening and gravitational energy release, I have all that I need to solve those two equations in two unknowns, for Earth's spin rate and precession angle. I find that there are two solutions:

I. What is happening today.

II. Torque-free precession along the 71st (lat 71.10) parallel, with less than a minute difference in day length.

Between these two solutions are "energy trough" states, i.e., states that are easily traversible. I & II are like the two extreme swings of the pendulum.

7. Now here are the two reasons to believe that I am right in #4 and #5:

a) There is another equation, obtained from Euler's equations, that gives the period of the torque-free precession. The period turns out to be a year; if my estimate of the gravitational potential energy release, is increased by only 2% (believable, considering the crudity of my approximations) then the precession period is exactly one tropical year.

b) The torque-free precession is at such a latitude that it would explain Hapgood's estimates of the poles. With a precession period of one year, having the pole at Hudson's Bay in winter has about the same effect on ice cap position, as having the pole at Hudson's Bay year-round. Hapgood estimated the last three Ice Age pole latitudes (in today's Earth coordinates) as, most recent to least recent, 60, 72, & 63deg latitude.

The Cholula pyramid in Mexico is the world's largest manmade structure, more voluminous than the Great Pyramid of Giza, though only half as tall. Today it is covered in soil and vegetation and resembles a hill with a Spanish church on top. Arcturus is the brightest star in the northern hemisphere (though some authorities say Vega, depending on photometric details).

I used VizieR's online Bright Star Catalog for the J2000 coordinates of Arcturus, got the Proper Motion (-1.1"/yr RA & -2.0"/yr Decl) from Wikipedia, and used the online NASA Lambda utility to convert to 2013.0 celestial coordinates (mean equinox and ecliptic of date). I neglect Earth nutation and the aberration of starlight; neither effect amounts to more than a few arcsec. I find the Declination of Arcturus thus to be 19deg06'55".

According to Wikipedia, the geographic latitude of the Cholula pyramid is 19deg03'27"; another online source gives 19deg03'29.8". 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 "geodetic" surface) commonly deviates from the perpendicular to Earth's reference ellipsoid by up to 50" in mountainous regions, the "deflection of the vertical"; this phenomenon might explain 1' of the 3' error above. Also even with perfect knowledge of Earth's precession, the presumed 3.5' error in Arcturus' Declination could be caused by only a 5% overestimate of Arcturus' Proper Motion in Declination over 2200 yrs (the time since the pyramid's construction began, according to Wikipedia, though maybe there were even older predecessor structures)(2" * 5% * 2200 = 220").

The modern distance and radial velocity estimates for Arcturus, 11pc and 5km/s, imply that Arcturus' motion is so nearly tangential that its Proper Motion never has been more than 0.2% greater than it is now. On the other hand a 1' Declination observation error in 600 years would be 1*60/(600*2) = 5%.

According to Wikipedia, the pyramid is part of a flood myth:

"According to myth, the pyramid was built by a giant named Xelhua of adobe bricks, after he escaped a flood..."

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 difference in their right ascensions then is 2.670388deg, neglecting aberration and nutation, which have < 1" effect on this difference.

The midpoint of the declinations, is 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

In practice, this might be measured by observing the zenith through an orthogonal grid overhead, as Castor and then Pollux crossed the meridian. The angles would be replaced by their tangents, giving

x1/(x1+x2) = 0.519944

The Great Pyramid of Cheops (Khufu) corresponds to Pollux, Mycerinus (Menkaure) to Castor, and Chephren (Khafre) to the observer. Let y1 and Y denote the distances Cheops-Chephren and Cheops-Mycerinus measured by Petrie.

y1/Y = 0.520079

The difference, 0.000135, corresponds to a mere -2" declination shift for the Castor-Pollux midpoint.

(Birdsall's online edition of Petrie's "The Pyramids and Temples of Gizeh", ch. 13 sec. 92, gives an erroneous slope for the Chephren-Mycerinus line, caused by a sign error in the rotation, from Petrie's temporary station mark system coordinates, to his approximate ancient north. I find the true slope directly from Petrie's pyramid center coordinates, which are corroborated by his pyramid corner coordinates, in ch. 5 sec. 19.)

The foregoing, based on declinations, is most obvious to us, but requires measurement at two points in time. Another way to measure the split across the 30th parallel, would be at one point in time, when the stars' EW discrepancies from the zenith are equal, sighting upward through an orthogonal grid. The split under this condition, is found using position vectors, which must be lengthened slightly to extend to the tangent plane at the zenith. The condition of equal EW discrepancy on the tangent plane, gives an equation for the meridian of, say, Pollux, and then the NS distances, x1 & x2, on the grid can be found. This method gives

x1/(x1+x2) = 0.518198

Let y1 and y2 denote the distances Cheops-Chephren and Chephren-Mycerinus distances. Then

y1/(y1+y2) = 0.517484

The difference, 0.000714, 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, so the opposition of 2014AD fits even better.

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:

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 explanation, that Flamsteed's observations began in 1689 and most were made during the early 1690s. If Flamsteed's actual mean epoch were 1690AD and Ptolemy's 138AD, then a third order change in Vega's proper motion, mas/yr^3, sufficient to move the date of best fit by +40yr in 310yr, would move it 40*(1862/310)^3 = +8668yr in 2000-138=1862 yr, vs. observed 5225-138 = +5087yr above.

Unlike Flamsteed, Tycho Brahe's and James Bradley's observations were reduced and published by others, after estate contests. Kepler reduced Brahe's; with my star identifications in Kepler's Rudolphine tables (uncertainty boxes 1'x1') I find that none of the four triangles fit the foregoing pattern regarding fits to proper motion. For Bradley's positions (which I have as corrections from Flamsteed to Bradley, included in Flamsteed's table by the editor Baily; my arbitrary uncertainty boxes are 0.5'x5") the triangles containing eta Lyrae correspond to 1695AD & 1690AD with 0.08' or 0.03' minimum rms error; but the triangles containing theta Lyrae correspond to 1770AD & 1780AD, both with 0.06' minimum rms error. With an observation epoch of 1755AD for Bradley (Pannekoek's history says Bradley's equipment upgrade was in 1750) and based on Flamsteed's results, the expected displacement would be +40*((2000-1755)/(2000-1690))^3 = +20yrs (corresponds to 6.5" PM for Vega at the present speed) and indeed 1755AD+20=1775AD observed, +/- 5yr, for the triangles containing theta Lyrae. So, there might be a large cubic term in Vega's proper motion which hindered the builders' accuracy.

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 24% underestimate of its predicted proper motion in declination, so Cahokia was misplaced by 0.1degree latitude. Ptolemy's and Flamsteed's catalogs suggest large higher-order terms 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 70" S of my latitude for the main Bosnian pyramid, Visoc^ica. That would be only 55-70 = -15" 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 Buc^ki Gaj & 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 (exchanging NS & EW) the Moon in Teotihuacan.

Robert Temple has written about the omnipresence of arctan(0.5) in Giza architecture. Because this angle is formed simply by a "T" of equal strokes, its omnipresence might be accidental. On the other hand, let's use geocentric latitudes based on the coordinates I use for this paper, and draw two great circles on a sphericized Earth, one through the Great Pyramid and Visoc^ica, and another through the pyramid of Cholula and Monk's mound. These circles intersect at longitude W75.8deg, geocentric latitude N56.7deg. This is near Prof. Charles Hapgood's estimate of W83, N60 for Earth's last Ice Age geographic pole (Hapgood, "The Path of the Pole", map on inside cover and also ch. 4, pp. 90, 106; Fig. 21, p. 95, says 73W and Wikipedia repeats this misprint). If I take the Ice Age pole's longitude from the longitude of the farthest southern extent of the late Laurentide ice sheet, and the latitude from the latitude of the farthest western extent, then maps (Dawson, "Ice Age Earth", 1992, penultimate and antepenultimate Laurentide ice sheet maps) indicate that the pole was a few degrees southeast of Hapgood's coordinates.

The angle of intersection of the great circles is arctan(0.500680) = arctan(0.5) + 112". Enver Buza's coordinates ("Bosnian Valley of the Pyramids - the Analyses of the Landscape and Topography", 2007, online) for Buc^ki Gaj, are equivalent to W18d10'56" N43d57'27". These fit even better than Visoc^ica: arctan(0.500481) = arctan(0.5) + 79".

To find the intersection angle using geodesics on Earth's spheroid, I first find the angle for great circles using geographic, not geocentric, latitudes: the angle is arctan(0.5)+404" & +370" for Visoc^ica & Buc^ki Gaj, resp. My elementary calculation exact to first order in 1/297, is that for the actual geodesics, the angle is smaller than these, by 381", i.e. arctan(0.5)+23" & -11", resp. As with Capella's horizon grazing (after adjustment for the pole shift and for the builders' inexact ecliptic plane motion estimate) perfection is intermediate between Visoc^ica & Buc^ki Gaj, but nearer the latter.

Here are some details of my geodesic calculation. Let the auxiliary sphere be plotted with pyramid sites using geographic latitudes, and great circles drawn for the Old and New World pairs of pyramid sites. Let P0 be the point on the spheroid, with the same geographic coordinates as the intersection of the circles on the auxiliary sphere. Let g be a geodesic drawn from P0 toward the same azimuth as for one of the great circles on the auxiliary sphere; g misses the pyramids, because of southward sideways deflection by the spheroid's normal. To first order in 1/297 this sideways deflection in radians per unit arclength is

f = 1/297*sin(2*phi)*cos(alpha)

where phi is latitude and alpha is the angle at which the great circle (and hence, approximately, g) cuts the parallels. The displacement at, say, pyramid site P1, is the arclength integral from P0 to P1 of

sin(s1 - s) * f ds

which by Napier's rules (conveniently measuring s, from the great circle's point nearest the pole) becomes a function solely of s, with an elementary indefinite integral. For the geodesic to pass through P1, this error must be canceled by the sum of two adjustments: an adjustment v0 to the initial azimuth, which causes displacement

v0 * sin(s1-s0)

and an adjustment w0 to the location of P0 (the sphere is two-dimensional, so w0 can be chosen independently for the Old and New World circles) which causes displacement

w0 * cos(s1-s0)

Both pyramid sites, say, in the New World, P1 & P2, must intersect g; this gives a system of two linear equations in two unknowns which I solve for v0 (146.4" & 234.4" northward for the New & Old World, resp.).

According to Buza's coordinates, the layout of the three Bosnian pyramids also evokes the angle arctan(0.5). In the triangle they form, the angles at Visoc^ica, Pljes^evica and Buc^ki Gaj, are arctan(2.0323), arctan(1.8040) and arctan(1.4389), resp. For the convex hull of a "T" of equal strokes, the angles would be arctan(2), arctan(2) and arctan(4/3).

Summary of addendum re Bosnia. Regardless of how the Bosnian pyramid might have been produced, 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. The Bosnian pyramids lay directly north from Giza during the last Ice Age, as did Cahokia from Cholula, because of Hapgood's pole shift.

The "17degree" (or "15degree") angle(s) at Teotihuacan: proof of torque-free Earth precession

Many authors discuss the offset angles from true north, of the buildings and streets of Teotihuacan and some other Central American pyramids. These angles are variously given as 15, 16, or 17deg, sometimes plus fractions. Four different angles are being discussed:

According to Giulio Magli of Milan (ArXiv.org) the Teotihuacan angles all are either about 15deg25' (15.417deg)(angle "Nord") or 16deg30' (16.500deg)(angle "Est") east of true geographic north. (The former angle also is attested by Rene Millon's 1970 photographic map of Teotihuacan's 1962 appearance.) This makes two angles under discussion. I would add, that the line between the pyramids of Moon and Sun, is offset west of true north, according to the coordinates found at Wikipedia, by roughly arctan(0.002/0.0071) or, precisely accounting for geocentric latitude, by arctan(0.0002*cos(19.6)/(0.0071*(296/297)^2) = 1.530deg; the 1-sigma rounding error is 0.00005/sqrt(3)/0.0002*1.53 = 0.22deg. This interpyramid line, may well be more important than the present true north. Teotihuacan's prominent lines are rotated clockwise from the interpyramid line, by about 15.417+1.530 = 16.947deg, and 16.500+1.530=18.030deg.

Suppose the interpyramid line is, really, exactly 1.68deg west of north (this is consistent with the given 1.53 +/- 0.22; though measuring with a ruler on the large paper map of Rene Millon & Armando Cerda, from the centers of the topmost rectangles drawn on the pyramids, I find 2.106deg). Maybe this is the line to the ancient pole of torque-free precession (i.e. spheroid pole); suppose Teotihuacan's latitude relative to this ancient pole, was about the same as now (the expression below is an order of magnitude less sensitive to this factor). Let a great circle extend from Teotihuacan, in the direction 16.500+1.68deg east of this ancient north pole, tangent to the latitude circle of torque-free precession. Then by Napier's rules for spherical triangles, the opening angle of the torque-free precession cone is

But where else do we see this angle? We see it in 15.417+1.68=17.097. So, the interpyramid line signifies the direction to the (Ice Age) torque-free precession pole. The 16.5deg E of N angle, signifies the line to the farthest eastward deviation of the ancient pole which was precessing yearly. The 15.417deg E of N angle, signifies that the precession occurred along a cone of opening angle 15.417+1.68 = 17.1deg, not much less than the 18.9deg I calculated.

Go:bekli Tepe Vulture Star Map dates catastrophe 12900 years ago

Turkish archaeologist Haldun Aydingu:n, is credited for a photo of the Vulture Map on a column at Go:bekli Tepe. The version from which I measure, I printed from an online photo of that column, presumably Haldun's, found by Google Images on the website timothystephany.com (Timothy presents an alternative star map theory).

The bundles and baskets at the top of the column, represent a calendar [note July 6: a number; see addendum below]. The vulture on our left, holds an egg on his left wing, signifying a catastrophe which produced many vultures.

The large hole or eye in the center of the vulture's circular head, is Denebola (beta Leonis, a magnitude +2.14 Vega-like star in a multiple star system). The line between the horizontal and vertical parts of this T-shaped column, is the celestial equator of epoch. This celestial equator is shown most precisely by the horizontal straight line between the vulture's legs. The large long-necked bird at the column's bottom, has a profile resembling a parabola opening downward; the hole or eye in its head is near the focus of that parabola. This hole is nu Hydrae, a magnitude +3.11 orange giant.

Denebola and nu Hydrae define the epoch of this map, by having equal right ascensions. I measure 89.8deg counterclockwise from the celestial equator defined by the line between the vulture's legs, to the straight line between Denebola and nu Hydrae on the map. Using the proper motions given in the online Bright Star catalog (without correction for coordinate curvature; the proper motion arcs are < 2deg) and the rigorous precession formulas from the 1990 Astronomical Almanac, this angle occurs in the sky, 12,900 yr (to the year!) prior to 2000AD. An exact 90deg angle occurs 12,928 yr prior to 2000AD. Because Denebola & nu Hydrae move tangentially (their radial velocities are 0 & -1 km/sec, resp.) the geometric change in the angular speed of their proper motion amounts to < 1 mas/yr in 13kyr.

I measure the ratio of upper to lower distances from the equator, conveniently marked by the vulture's legs, as 1.0846. This ratio of declinations occurs 12,331 yr prior to 2000AD. There seems to be some minification at the bottom of the photo, because a bulky "view camera" apparently was not used. Because the photographer found it impractical to aim exactly perpendicularly, the film was not parallel to the stone. If the lower segment is minified 10% relative to the upper and this were corrected, then the date becomes 12,443 yr before 2000AD. The distance ratio is much more time-sensitive than the angle with the equator: at 12,400 yr before 2000AD, the angle with the equator is still 85.76deg; but at 12,900 yr before 2000AD, the upper::lower ratio is only 0.60865.

The hole or eye in the center of the circular bird's head in the lower right corner of the upper part, is Regulus (alpha Leonis). Above and to the right of Regulus is another pair of bird's legs with a long straight line between them; this is the ecliptic. Because this star map is for 1/2 precession cycle ago, the tilt of the ecliptic here, near longitude 180, was the reverse of what it is now. Though the ecliptic presently is slightly south of Regulus, 12,900 yr ago it was about one degree north of Regulus, according to the dubiously convergent formula in the 1990 Astronomical Almanac which I confirmed by extrapolating NASA Lambda values. The bird to the right of the ecliptic-leg bird, whose neck, head and breast appear near the right edge of the column, has a hole or eye above its cheek which represents gamma Leonis (Algieba). On the map, the ecliptic is about three degrees from Regulus, if my identification of Algieba is correct.

The bird whose legs form the ecliptic, lacks obvious eyes. Instead there is a manifoldly right-angled shape resembling a bold "I" in the upper right. Such an abstract shape, rather than a realistic bird as elsewhere, would imply that a hole signifies not a real star, but only the direction to a star. The lower left corner of the upper capital of the "I" is collinear with Regulus and Algieba, but was deliberately defaced to signify that this perfect collinearity was merely an earlier situation. Near the center of the upper capital itself is a hole which signified the then-current position angle of Adhafera (zeta Leonis). Using proper motions from the online Bright Star Catalog, I find that the line Algieba-Adhafera, was 2.87deg (my measurement to about 10% accuracy) clockwise from perfect collinearity with the line Regulus-Algieba, 12,615 yr before 2000AD; perfect collinearity occurred 13,761 yr before 2000AD. Here, the geometric correction for time change in proper motion, mainly for Algieba, is a few mas/yr; this changes the former date to 12,713 yr before 2000AD. The actual distance 12 kyr ago, between Algieba & Adhafera, in proper proportion to the distance between Regulus & Algieba, would have been about halfway between the defaced corner, and the hole in the capital "I".

The smaller eccentric hole or second eye, in the vulture's head, at about 2:20 position from Denebola, is 95 Leo. This magnitude +5.53 line-of-sight neighbor has the same spectral type as Denebola, main sequence A3V, but is much more distant. With proper motion to 12,900 yr before 2000AD, 95 Leo would lie 40.68' from Denebola, at azimuth (clockwise from north) 13.13deg in J2000 celestial coordinates; after rotation to the celestial coordinates of date, 12,900 yr earlier, the azimuth is 62.55deg; with a protractor I measure the azimuth according to the vulture's legs' equator, as 72deg; but I can do much better:

Earth's obliquity at that epoch was 24.03deg, which multiplied by cos(right ascension) = cos(-12.08) gives approx. 23.50deg as the local inclination of ecliptic to celestial coordinates. So the expected azimuth from ecliptic north, is 62.55-23.50=39.05. The straight line separating the upper and lower parts of the vulture's beak, lies parallel to the birdslegs' ecliptic depicted near Regulus, so I assume it is a local line of ecliptic latitude. Thus with a protractor, I estimate the ecliptic azimuth of the line Denebola - 95 Leo, as 40.1deg and by orthogonal ruler measurements, as 39.8deg, averaging 39.95. Conveniently, the proper motion of Denebola relative to 95 Leo, 0.502"/yr, is almost perfectly perpendicular to their line 12,900 yr ago; the additional 39.95-39.05=0.90deg azimuth requires only 76 yr, giving a date 12,976 yr before 2000AD.

Let's compare actual inter-star great circle distances 11,000 yr prior to 2000AD, to distances on my photo of the Vulture Map. The distances are normalized to the Denebola-nuHydrae distance. Vulture Map distances are in parentheses:

In sum, Regulus was scooted left, to make room for a magnified inset of western Leo. The scorpion in the southern hemisphere, assuming that it is part of the larger map including Denebola and nu Hydrae, coincides roughly with the positive Cosmic Microwave Background dipole.

Aside from the Vulture Stone Star Map, the Younger Dryas catastrophe date also is corroborated by Earth's rate of precession. Newcomb's cubic formula (as cited by Clemence, Astronomical Journal 52:89, p. 90, 1946) implies that 1/2 precession cycle before 2013.0AD, is 13,257 yr before 2000AD. Newcomb's successors have proposed small corrections to the linear term: one typical correction (Van Flandern, IAU Colloquium #9, Heidelberg, August 1970) changes this to 13,253.

Listing all the date estimates:

1/2 precession cycle: 13,253 yr before 2000AD

from Vulture Map angle of Denebola - nu Hydrae line to equator: 12,900 +/- ? ratio of upper and lower segments of that line: 12,400 +/- ? angle Regulus-Algieba-Adhafera: 12,713 +/- 100 hour angle Denebola - 95 Leonis: 12,976 +/- ?

The mean of the four Vulture Map dates, is 12,747 +/- SEM 128.

By virtue of the high redundancy of its map scheme, only the 12,713 +/- 100 yr value, is likely to be nearly independent of whatever distortions might have been caused by photographic angle and internet image processing. Though differing from the usual 12,900 yr Before Present date given for the onset of the Younger Dryas, it is near the date, 12,670 yr before 2000AD, given by Brauer et al, as the exact year, indicated by European lake sediments, of onset of the Younger Dryas.

Addendum July 3, 2012

The foregoing was composed and posted, about 1/3 of it each day, June 30 - July 2. This addendum was composed and posted July 3 - 6. I discovered the night of June 2:

The geographic latitude of Go:bekli Tepe, is not the declination of Arcturus then, nor is it (like Cholula) the declination of Arcturus now. It is what the declination of Arcturus would have been then, if Arcturus had been at the galactic coordinates it has now. Thus two dates - then and now - were encoded by the latitude, without need of accurate prediction of equinox and ecliptic precession 13,000 yr into their future.

More precisely: according to the 1990 Astronomical Almanac's rigorous precession formula, the online Bright Star Catalog's J2000 celestial coordinates of Arcturus for epoch 2013.0AD (obtained from their epoch 2000.0 coordinates by including proper motion from 2000.0 to 2013.0) equate, for the mean equinox and ecliptic of 12,761.5 yr prior to 2000.0AD (neglecting nutation and aberration) to a declination equal to the German Archaeological Institute's current geographic latitude of Go:bekli Tepe. The rate of change of the declination of this point in the sky, then, was +23"/yr due to coordinate change.

The pole shift indicated by the position and orientation of the Giza pyramids, would have caused nowhere more than about 5' change in latitude. Even if, during the almost three times longer interval, Go:bekli Tepe's latitude had changed 15', this would change the date only 15'/23" = 39 yr. If the builders were unaware of the acceleration of Arcturus' proper motion, due to negative radial velocity, their underestimate of that proper motion would have made Go:bekli Tepe's latitude consistent with a date only 6.6'/23" = 16 yr earlier.

The convergence of the precession formula is dubious for this remote epoch; the last term in the polynomial for the "theta" of the precession matrix (theta is the angle between old and new poles) is 1/3 as big as the leading term! Also, the Vulture Star Map itself, according to the discussion above, suggests that the Astronomical Almanac's formula for the ecliptic, is off by two degrees at that epoch. So, it is difficult to be sure of exact estimates when knowledge of precession to that epoch is needed. Woolard & Clemence, "Spherical Astronomy", 1966, pp. 277-278, say, "Even with rigorous formulas, however, only approximate coordinates can be determined for remote dates,...".

The rigorous precession formula in the 2009 Astronomical Almanac uses 5th degree polynomials for the precession angles, instead of 3rd degree as in 1990. The last terms are somewhat smaller. The result with the 2009 coefficients, for Go:bekli Tepe's latitude equaling the declination, at the builders' epoch, of the 2013.0AD Arcturus position, is 13,076 yr before 2000.0AD, instead of 12,761.5 yr, and +20.5"/yr change, instead of +23". Though the ecliptic formula in the 2009 Almanac also uses 5th instead of 3rd degree, its 5th degree term, 130 cyr ago, isn't even any smaller than the 3rd.

The difference between the coefficients given in the 1990 and 2009 Almanacs, as far as they are given in both almanacs, is small. To the cubic polynomials of 1990 for the classical precession angles zetaA & thetaA, I fitted a sine wave of period equal to twice Newcomb's estimate of the time for precession 180 deg back from 2013.0, plus a linear term, to agree to third order at T=0. This gave a reasonable zetaA, but a thetaA 10 degrees too large, so I arbitrarily set thetaA equal to the sum of the obliquity now and the obliquity 1/2 precession cycle ago, plus the effective amount the ecliptic had tilted (according to the 1990 Almanac): the result is 13,122 yr before 2000AD, agreeing with the straightforward application of the 2009 Almanac formula.

However, as explained above, the Vulture Star Map contains date information dependent only on proper motion, not precession. One such indication of the present time, is the semidiameter of the vulture's head, drawn through the two eyes. According to my measurement, it is in about the same ratio to the distance between the two eyes, as the distance between Denebola and 65 Leo today, is to that distance 12,900 years ago when the stars were at their apparent nearest to each other. See Addendum A, July 6, for another precession-independent feature of the Vulture Map, which seems to indicate the present time.

[July 5] Yet another precession-independent indication of the present time, is the hole or star just off the tip of the vulture's left wing. The location off a wing, rather than in a bird's head, might indicate a future position. If this star is rho Leonis, a distant B1I star with small proper motion, then the large proper motion of Regulus & Denebola lets us test, which epoch fits the map better. Let the actual distance of epoch be divided by the map distance, and this ratio normalized to one for the alpha-beta Leonis distance. Then

alpha-rho = 0.998 beta-rho = 0.940

at epoch 2000.0; but at epoch 2000.0 - 12,900,

alpha-rho = 0.851 beta-rho = 0.987

In other words, the beta-rho map distance is correct for 12,900 yr ago, but the alpha-rho map distance correct for today.

Addendum A, July 6, 2012

The smaller bird at the lower right of the upper hemisphere, which contains Regulus at the center of its circular head, has R Leonis & omicron Leonis beneath its bill. The bird above it, whose legs show the ecliptic, has eta Leonis & HD87500 (Vmag = +6.37) at the upper ends of the viewer's rightward & leftward legs, resp. These four stars are nearly collinear on the Vulture Map, and, due to proper motion, much nearer collinearity at our own epoch than the epoch of the builders; but the relative distances more closely resemble those of the builders' epoch.

The hour angle of R & omicron Leonis, has changed grossly and should give a good date estimate: assuming that the birdslegs ecliptic is the ancient ecliptic, and including the modern estimate of the change in the angle of the ecliptic there, I find that these stars' map azimuth from modern celestial north, is

139.9 (measured on map) - 19.43 (local inclination of modern ecliptic to celestial coords) - 1.04 (approx. change in local ecliptic inclination to 12,900 BP) = 119.43deg

However, the actual epoch 2000.0, J2000 azimuth is 134.305deg, decreasing 0.00074 deg/yr.

The bent line between the legs of this Regulus bird, might signify, proximally, the slope of the modern ecliptic to the modern equator; and distally, the slope of the modern equator to the builders' equator.

Addendum B, July 6, 2012

The number encoded at the top of the Vulture Map, is (59-5)*(59-3)*(59-2) = 172,368. The vulture is holding a disk, perhaps Luna; this many current sidereal months, equals 12,893.58 Julian yr (most scientists date the sudden Younger Dryas onset as 12,900 yr BP).

If this were exactly half the precession cycle, then the precession rate would be 50.25756"/Julian yr, exactly the precession rate of 1937.4AD (according to Newcomb's quadratic formula as cited by Clemence, 1946; using the current length of the sidereal month, the change in which is barely significant even at this precision). In other words: the large number on the Vulture Map, tells what length half-cycle of precession (in mean sidereal months) corresponds to our present precession rate, and so marks our present time with only 76 yr error. Let's seek even greater accuracy.

The foregoing, finds t such that a+b*t+c*t^2 = A, where the polynomial on the left is Newcomb's precession as cited by Clemence, and A is the precession rate needed for 1/2 cycle in 12,893.58 yr. Though it makes only a fraction of a year's difference, let's use henceforward, sidereal months of 12,900 yr ago, per Simon et al, Astronomy & Astrophysics 282:663+, 1994, sec. 3.4.a.1, p. 669. A more general function, of period 12,893.564 Julian yr = 2*pi/w, and mean A = w/2, is

A + B*sin(w*t+phi) + C*sin(2*(w*t+phi))

so set this = a+b*t+c*t^2 to second order at t=0; the three equations may be solved by row elimination for B*sin(phi) & C*sin(2*phi), then for cot(phi) by a double-angle identity and quadratic equation. Van Flandern (op. cit., p. 183) cites a +1.36"/cyr correction to Newcomb's precession, from a study by Martin & Van Flandern; and corrections from +1.08" to +1.38"/cyr, from various studies by Fricke. DeSitter & Brouwer, Bulletin of the Astronomical Institutes of the Netherlands 8:213+, 1938, have updated Newcomb's quadratic formula, retaining 1850.0AD as the reference year (they use tropical years, so, to follow Clemence, I converted their coefficients to Julian years). Their zeroth order term exceeds Newcomb's by an amount, 1.171"/cyr, consistent with Fricke's estimates. Andoyer's definition of precession (used by deSitter & Brouwer) differs from Newcomb's by only 0.001"/cyr/cyr (Leiske et al, A&A 58:1+, 1977); I'll disregard this discrepancy because the uncertainty in deSitter & Brouwer's leading term is 0.10"/cyr, and the Andoyer definition is more fundamental physically. The result is that the node is at 1884.6AD. The amplitude of the second harmonic is 10.0% that of the first, and is opposite in sign.

The node would be at 2013.0AD, if the zeroth order term of deSitter & Brouwer's general precession, were 2.869"/cyr smaller (the relationship is very linear in this range). Here the amplitude of the second harmonic would be -12.0% that of the first. Curiously, this alteration of the general precession, equals the pole shift that has occurred since the Giza pyramids, approx. sqrt((340"*sin(60))^2+75"^2), divided by twice the Mayan Long Count (51.2526*2 centuries), i.e. 2.964 +/- 0.082" / Julian cyr, using Petrie's error bar.

Even more suggestive, is the Thomas precession of a gyroscope, at an observatory on Earth's surface on the Arctic circle, and initially parallel to the ecliptic pole. At the equator, -1/2*v*a/c^2 = -16.47321"/cyr. I use the equatorial acceleration of gravity from the table on p. F139 of the 1987 CRC Handbook of Chemistry & Physics, and Earth's rotational frequency from p. F134. The speed of light and Earth radius, I use from the 1981 CRC Math Tables, p. 5.

At the Arctic circle, this Thomas precession would be multiplied by sin^2(obliquity). By Simpson's rule (exact for cubic polynomials such as the 1990 Astronomical Almanac's) I find the mean obliquity over the 128.94 centuries prior to 2013.0AD, as 1/6*(4*24.1415 + 24.0345 + 23.4376) = 24.0064deg; -16.47321"/Jcyr * sin^2(24.0064) = 2.7266"/Jcyr. The minus sign signifies that the Thomas precession is in the opposite sense to Earth's rotation: the gyroscope precesses retrograde around Earth's axis, and Earth's axis precesses retrograde around the gyroscope, giving a larger value for the general precession, if the gyroscope is used in place of the real ecliptic pole (and the gyroscope is subject, in addition, to all the same motions as the ecliptic pole). Maybe the smaller value of the precession, needed to indicate the date 2013.0AD, arises from subtraction of some error like that of my hypothetical gyroscope.

The multistroked "V", alternating rightside up or upside down, triangle-like objects, are maximally redundant, rugged unit symbols. The builders wanted future people to read this number accurately. The five large Vs at the bottom, sitting on a long rectangle (perhaps a plate seen edge-on, or perhaps a one-dimensional line) signify a base five number system: the most obvious base. Above this is a row of eleven squares (perhaps the same plate seen face-on, or perhaps equal line segments) and eleven Vs atop the row of eleven squares. These signify 11*5 = 55. To the left of this row, are four more Vs unassociated with squares; these signify 4*1; so the whole row, implies 55+4 = 59.

The large, handled baskets above, evoke three dimensions; baskets are three-dimensional, not flat. To make the concept more obvious, there are three of them, perhaps views along the three axes of the same basket. So the number 59 is to be cubed, but wait! The rightmost basket has two Vs above it, and a slash at its bottom marking where this short row of Vs ends. The leftmost basket has at least four Vs, and hints of a fifth V at the left edge; there is no slash at the bottom of the leftmost basket, so there must indeed be a full row of five Vs. The middle basket has at least a V at each end, no slashes at the bottom, so by the alternation of the Vs (up & down) an odd number of Vs filling in between, but three would hardly fit; therefore 2+1=3 Vs. We expect a monotonic sequence.

These numbers - 5,3,2 - signify the number taken away (taken out of the basket) in each dimension. At least the assumption that they are subtracted, gives the result, above, that is most suggestive.

I find 172,368 / 2 = 86,184 = 19*7*81*8. Hindu mythology gives the height of "Mount Meru" as 84,000 = 7*125*3*32 "yojanas". Buddhist mythology gives the same mountain, called "Sumeru", as 80,000 = 625*128. Both numbers might be roundings.

Another notable interval is Brauer's lake-varve onset of the Younger Dryas, 12,683 yr before 2013AD; 12,683 / 2 = 6341.5 yr. The Atharva-veda gives 6333 as the number of Gandharvas; 6333 sidereal yr = 214.99 Saturn sidereal periods and so might represent a rounding of the half-Brauer interval. Also, 6333 sid. yr. = 84,664 sid. mo. = 557*19*8 = 557/567*86184; that is, it might represent a mistranscription of the half-Gobekli interval.

Petrie's height for the Great Pyramid (Sec. 25, Birdsall's online edition) is 146.71 +/- 0.18 meters. The 1990 Astronomical Almanac, p. F3, gives Luna's radius as 1738km (1738.1km on the current online NASA Fact Sheet). The ratio is 11,847. On the other hand, Hipparchus' apparent lunar diameter was 33'14" (Neugebauer, "History of Ancient Mathematical Astronomy", Part II, Sec. IV B3,4A, p. 659). If this were combined with the correct lunar distance, mean 384,400 km according to the 1990 Astronomical Almanac (same as semimajor axis given on NASA Fact Sheet) then Luna's radius would have been 1858.0km. The ratio then is 12,665.

John Major Jenkins popularized the phrase "precessional alarm clock". Thus I was led to the right trees, by other hounds who, though barking up the wrong trees (Venus, the Pleiades, the galactic center) at least were in the right forest.

In my previous article, I noted that Arcturus' actual declination, including nutation, aberration and proper motion, at its opposition in the spring of 2013AD, equals the geographic latitude of the pyramid of Cholula with an error of only 194". This error decreases to 14", if I follow Petrie in believing that the small, consistent misalignment of the Giza pyramids indicates a small pole shift, and if I also assume that the pole shift was such that the Great Pyramid originally was at exactly 30N.

Most authorities list Arcturus as the brightest star in the northern hemisphere, though some list Vega, depending on photometric details. Likewise Cholula is not just any pyramid; most authorities call it the largest pyramid, by volume, ever built. The chance that the brightest star in the northern hemisphere would happen to lie within 194" of the 19.06deg latitude of (according to most authorities) the most voluminous pyramid ever built, is p=0.18%. If correction for the pole shift suggested by the Giza layout is admitted, this becomes 14" and p=0.013%.

As I explain in my previous article, the analogous alignment of Algieba (the brightest star in the "sickle" of Leo and one of the 50 brightest stars in the sky) over the pyramids of Teotihuacan, at its own opposition in late winter 2013, is almost as accurate as that of Arcturus over Cholula. The pyramids of Cholula and Teotihuacan are archaeologically similar. Arcturus and Algieba also are similar: both are orange, high proper motion stars.

If the Great Pyramid of Giza were intended to mark the pole by lying at exactly 30N, it hardly could lie exactly under any very bright star at a specific time like 2012AD. So, the pyramid builders resorted to the next best thing. The most prominent pair of bright stars in the northern hemisphere, Castor and Pollux (no pair of stars in the northern hemisphere are both brighter and nearer together) happen to straddle the 30th parallel now. In an addendum to my previous article I note that the ratio of distances Cheops-Chephren : Chephren-Mycerinus (sometimes with and sometimes without projection of the latter onto the line of the former) nearly equals various practical, precise definitions of the current break ratio of Castor and Pollux over the 30th parallel.

This leaves another degree of freedom: if the relative distances and break angle at Giza are determined by Castor, the 30th parallel, and Pollux, still the Cheops-Chephren slope still can be chosen freely. The Cheops-Chephren slope has been chosen to equal the azimuth of the rising of Vega: not then, but rather, now. It is another "precessional alarm clock".

The rising of Vega.

Vega is the second brightest star in the northern hemisphere, and according to some authorities, the brightest. At its opposition in the summer of 2013, Vega's actual observed declination exceeds the geographic latitude of Monk's Mound (near Cahokia, Illinois) by 523". The pole shift correction used above, reduces this to 309". Ptolemy's star positions within the constellation Bootes, are consistent with modern estimates of the proper motion of Arcturus; but his positions within Lyra, are not consistent with the modern theory of the constant rectilinear proper motion of Vega. In my previous article I further found similar evidence of a large higher order term in the proper motion of Vega, from positions of Flamsteed and Bradley. So, maybe the inaccurate placement of the Cahokia Mounds is due partly to the pole shift indicated at Giza, and partly to poor prediction of Vega's proper motion.

There are three competing definitions of the azimuth of rise of a star:

Definition 1) the azimuth as would be observed from the surface of a perfect airless spheroid. This is competitive, because it can be calculated from the great-circle path of the star, observed far above the horizon, where atmospheric refraction is negligible. Vega's actual observed declination, including nutation, aberration and proper motion (but after correction for atmospheric refraction, if any) at its opposition in the summer of 2013, is 38deg47'57". This implies an azimuth of rise, from geographic latitude 30.0deg on the surface of an airless 296/297 spheroid, of

43.653651deg.

If the builders wished to notify future people of an important date (what other motive could they have had?) it seems they would not have chosen a method which could not be directly observed. So, definitions (2) or (3) are likelier.

Definition 2) the actual observed rising, when the star first appears above the ground below, as seen from the top of the Great Pyramid. This is impaired by the faintness of the star at rising, the large atmospheric refraction, and the likelihood of considerable erosion of the hillside across the Nile, over millenia. Even modern astronomers hardly can measure accurately, apparent altitudes less than one degree (Greenbaum, Astronomical Journal 59:17-19, 1954, Fig. 1, p. 19). I use the original Great Pyramid height of 146m above its base (Fakhry, "The Pyramids", p. 115), its base elevation above sea level of 59.85m (Butler, "Egyptian Pyramid Geometry", p. p. 125, as cited on a messageboard; citing Maragioglio & Rinaldi, citing Vyse), and the elevation of 100m (Manley, "Penguin Historical Atlas of Ancient Egypt", map, p. 29) of the north slope of the bluff across the Nile, on which Vega would rise about 30 miles to the northeast. I assume that the line from Cheops' peak, to the bluff, is perpendicular to Earth's radius, where it intersects the bluff, and use the radius of curvature of Earth's spheroid, along this approximate azimuth, to find the angle of depression at Giza neglecting refraction. By the formula 1.24/10^6 deg/meter (converted from civil engineer Haseeb Jamal, teaching website www.aboutcivil.com) I find half this, as the angle of depression of the light ray where it grazes the bluff. A slightly different alternative would have been to use 1.17/10^6 deg/m, the practice of British surveyors in Egypt in the 1920s, who prescribed 0.13 times Earth's curvature and made observations "in the afternoon hours when refraction is at its minimum and steadiest value..." (Ball, Nature 109:8, 1922).

Then I used the time-honored Astronomical Almanac formula for astronomical atmospheric refraction (appears in most if not all of the 1985 through current Almanacs) with atmospheric pressure appropriate to 100m elevation according to the standard atmosphere table in the 1987 Handbook of Chemistry & Physics, and temperature 37C roughly appropriate to a sunset in contemporary Cairo in July (and a natural reference temperature for summer, because it equals human body temperature). Then I added all three terms to find that this rising of Vega seen from the top of the original Great Pyramid, would occur when Vega was geometrically 90.872937 deg from the zenith. This implies an azimuth

42.911458deg.

Definition 3) this definition is most practical because it is observable, yet the star is brighter at "rising" and can be seen for awhile prior, the refraction is less, and the "horizon" is indestructible: the star appears to cross, not the dirt horizon, but rather the horizon circle 90.0deg from the zenith. The Astronomical Almanac refraction correction (using now, standard atmospheric pressure at the original top of the Great Pyramid, 206m above sea level; and still 37 C) implies a geometric zenith angle of 90.508584. This implies an azimuth

43.224214deg.

The actual Cheops-Chephren azimuth, relative to Petrie's estimate of ancient Giza's true N (5'40" W of modern true N) is

43.394590deg

for an error of 10'13".

The pole shift was thought by Petrie to result from climatic change in ocean currents; astronomical predictions might have been possible thousands of years into the builders' future, but climate predictions less so. Adjustment for Petrie's pole shift much improves accuracy at Cholula (Arcturus), Teotihuacan (Algieba), and Cahokia (Vega), raising the question: if the Giza builders could predict the pole shift, why didn't they tell their American colleagues, who undertook similar projects? Also, why would the Giza builders align their pyramids with the ancient pole, if they knew what the modern pole would be? So, though the difference is small, the builders' accuracy should be tested against the presumed ancient pole, not the modern one.

Also, maybe Giza made the same error in predicting Vega's declination, that seems to have been made at Cahokia. If the Cahokia declination prediction had been correct (I use here the Wikipedia latitude of Cahokia, 38d39'14", which fairly represents the middle of the site, about halfway between Monk's and Harding's mounds) the azimuth of Vega's rising, according to Definition (3), would have been

47.336955deg,

for an error of 43.394590 - 43.336955 deg = 3'27.5", a factor of three less, than the error found without the Cahokia-based correction.

If I improve the Astronomical Almanac formula by heeding some of the best available astronomical literature on near-horizon refraction, the error is reduced by yet another factor of three. The biggest improvement to be made, is to realize that the refraction is only very roughly inversely proportional to the absolute temperature at the observatory. An ideal gas law indeed relates density to temperature and pressure, but the temperature on the ground is only vaguely correlated with the temperature at the important much higher elevations (even the height of the tropopause, at which the derivative of temperature with respect to height changes sign, varies greatly).

So, the best refraction formula involving temperature & pressure, is likely to have factors with exponents somewhat different from -1 & +1. Newcomb, "Compendium of Spherical Astronomy", 1906, Table XXII, p. 433, gives linear temperature and pressure corrections at 0 degrees apparent elevation, which amount to a first order approximation to a formula with T and P exponents of -5/3 and +10/9, resp. I found six authoritative formulas to which I could apply Newcomb's linear corrections:

1. The Astronomical Almanac formula above, appears to have reference temperature 0 C and reference pressure one standard atmosphere, 1013.25 mbar. For 0 deg elevation, using Newcomb's coefficients to correct the 0 C, 1013.25 mbar value, to 37 C and 989.09 mbar, the refraction is 0.449549deg instead of the 0.50854deg found above with the formula as printed.

2. Newcomb's chart itself refers to 50 F and 30 inches of Hg; with his correction coefficients, it gives 0.466139.

3. Mueller, "Spherical and Practical Astronomy", 1969, Table 4.8, p. 107, refers to 10 C, 760 mmHg, and 60% relative humidity. (Humidity is seldom specified; an author writing in the Astronomical Journal, doubted the importance of considering it.) Newcomb's correction coefficients, applied to Mueller's table, give 0.464312.

4. Bennett, Journal of Navigation 35:255-259, 1982, as cited by Wikipedia, refers to 10 C and 1010 mbar. Correcting it per Newcomb, gives 0.468894.

5. The modified Garfinkel values ( = 1953 Air Almanac)(exactly verified experimentally by Greenbaum, Astronomical Journal 59:17-19, 1954) refer to 80 F, 30 inches Hg, and an observer specifically at sea level. Correction per Newcomb gives 0.469222. Greenbaum's Fig. I, p. 19, shows that the standard deviation of individual observations is about 0.005deg throughout his range of apparent altitudes: < 6deg down to an observational limit of less than one degree.

6. The 1952 Astronomisch-Geodaetisches Jahrbuch, cited by Greenbaum and referring to the same conditions as (5), corrects in the same way to 0.502556.

The above, shows that (6) and the Astronomical Almanac formula as printed, give outlier values. The others, (1)-(5), average 0.463623 +/- SEM 0.003640. Using the builders' presumed prediction of Vega's declination implied by the latitude of the Cahokia Mounds, and the presumed ancient orientation of true N inferred by Petrie, this value of refraction gives an azimuth, for Vega rising at zero altitude and observed from the top of the Great Pyramid, of 43.375169 +/- 0.00309, only 1'10" +/- 11" less than the azimuth of the Cheops-Chephren line.

The error is reduced again by a factor of three, if I use the latitude of one specific mound at Cahokia: "Harding's" (sometimes misnamed "Rattlesnake") mound, #66. This mound, one of the largest, delimits the southern border of the Cahokia site (Fowler, Illinois Archaeological Survey Bulletin #7, 1977, Fig. 11, p. 16). Using Harding's mound's geographic latitude, 38d38'39.1" as I measure from the US Geological Survey map (and then corrected for Petrie's pole shift) as the builders' predicted declination for Vega, I find that the azimuth of Vega's rise as observed from the Great Pyramid, as described above, would have been

43.387887 deg,

only 24" +/- 11" less than the azimuth of the (ancient north corrected) Cheops-Chephren line.

Cahokia.

"And the purpose of #66? It is too narrow on top for large wigwams or temples. The augur tests - nearly 200 to the base - did not indicate the...floor common in mortuary tumuli. ...There appear to be no burials, altars, or distinct stratigraphy in the body of the mound."

- Moorehead, "The Cahokia Mounds", 1929 (reprinted 2000), p. 293 (p. 105 in original)

A smaller sister mound to the east, #65, and #66, were noted by Moorehead to be constructed of "gumbo" (clay, sometimes hard clay called "shale" locally): "gumbo...is difficult to handle - even...steel shovels...baskets...a herculean task." (p. 105 in original). Mound #65 had "no remains of any kind" (p. 83 in original); though #66 had superficially buried ancient human remains, usually in "bundle-like groups", artifacts were almost nil: "...a red sienitic granite discoidal, 3 in. in diameter and 1 in. thick, almost as perfectly fashioned as if it had been turned on a lathe, lay at the point of the lower jaw of one skull. This, and a flint scraper, constituted the sole artifacts recovered from #66..." (p. 72 in original).

Moorehead found #66 to be amazingly regular in shape, except for some modern damage (see foldout topographic map, "Fig. 8", between pgs. 268 & 269 in reprinted edition; from this map I measure that the steepest and most regular N & S slopes are about 18deg). My measurement on the USGS map indicates that the azimuth, to the center of the top platform of Monk's mound (#38; the largest, and delimiting the northern boundary of the site) from the center of the base of #66, is 6deg29.4' E of N; to the approximate center of the base of #38 (precisely, the midpoint of the EW segment between 430 ft contours, through the center of the top platform) it is 5deg50.4'. Moorehead et al found the orientation of #66 to be (broadside to) 5deg30' E of N.

Most strangely, the local magnetic declination was also 5d30' E of N as "ascertained by local surveyors" (pp. 274, 276 in reprinted ed.) and "used by the county engineer in his surveys in this locality" (p. 68 in original). As if to refute any argument that this happened by chance, the sister mound #65 was oriented to 5d30' W (not E) of N (pp. 269, 276 in reprinted ed.).

According to Svendsen, "U.S. Magnetic Tables for 1960", Table 4, p. 32, the magnetic declination at 38N (resp. 40N) 90W in Illinois, moved only 05' (resp. 07') westward from 1925 (approximately Moorehead's epoch) to 1960AD, but moved 26' (resp. 41') westward from 1900 to 1960AD, and 165' (resp. 184')westward from 1800 to 1960AD. (The implied precision of Moorehead's azimuths seems to be 10'.) This suggests a recent near-constancy of magnetic declination, which raises the farfetched (to some) question of the builders predicting Earth's local geomagnetic field wanderings. Values usually given for magnetic declination and its time derivative, might be interpolations on grids too coarse.

The risings of Castor and Pollux.

The "Barrington Atlas of the Greek and Roman World", 2000, p. 74, shows relevant walls (or rather, a wall with a gap) along the northwesterly-protruding ridge, like a natural wall, which extends from the hills east of the Nile opposite Giza. This wall is attributed to multiple periods, hence probably dating to the Old Kingdom. The eastern part of the wall lies on the 500-foot contour; I will guess that it was 3m high, half of it was above the contour, hence its height was 153.9m. Though the western part of the wall lies completely within the 500-foot contour, I'll guess its original height was the same, 153.9m. I use the Great Pyramid height above; and Fakhry's original height for Chephren's pyramid, 143.5m, plus Butler/Maragioglio/Vyse's elevation of its base, 71.13m.

I use Definition (2) for the rising of Castor and Pollux, again with the surveyor's vertical refraction formula and with the Astronomical Almanac formula for refraction (this time, because the oppositions are in winter, with T = 0 C, and a slightly higher atmospheric pressure, namely the sea-level standard atmosphere; though the average January temperature in modern Cairo is 14C, ancient Egyptians might have had a colder winter). I found the coordinates of the west end of the west wall, and the west end of the east wall (ends as viewed from Giza) to about 0.3 arcminute error, with a ruler. Using the radius of curvature of Earth along this azimuth, and the angles between Cheops (resp. Chephren) and the west end of the west (resp. east) wall, I used the law of tangents (Brink, "Trigonometry", p. 172) to solve the triangles without assuming the pyramid-wall lines, at either end, to be necessarily perpendicular to Earth's radius.

The azimuths to the rise point are:

Cheops-Castor 51.81deg Chephren-Pollux 56.57deg

The azimuths to the from these pyramids to their respective wall ends, relative to Petrie's ancient true N, are:

Cheops- W end of W wall 51.48deg Chephren- W end of E wall 57.14deg

Assuming that the map is perfect, still an 0.3 arcminute error from my map measurement, corresponds to 0.8 deg in azimuth. So, to the accuracy of my measurement, these walls are placed so that Castor and Pollux rise at their west ends as viewed from Cheops and Chephren, resp. This condition pertains, at least within a degree or so, at the present time, and not at all at the time of the builders.

July 25, 2012 crop circle (Windmill Hill, near Avebury, Wiltshire) says doomsday date Oct. 8, 2012

As others already have suggested, the 16 overlapping disks signify days. (Nyako Nakar and also Chuen Xul show 16 on their diagrams, and that's my count from the online photos; but Bertold Zugelder shows 17 on his diagram.) The non-overlapping disks each signify consecutive "afternoon half moons" (in more exact astronomical language, the "first quarter").

The biggest disk, the one near the center, signifies the afternoon half moon of July 26 (9h GMT), the day after discovery of this crop circle. The long, monotonically diminishing sequence of seven disks, on one side of the largest disk, signifies the past seven "first quarters", those that have occurred since the winter solstice of 2011.

On the other side of the biggest disk, the next disk signifies the afternoon half moon of August 24 (14h GMT). The thin circle next to (just outside) it, signifies the ascending node, which the moon reaches that same day, Aug. 24 (12h GMT).

The next and final disk signifies the afternoon half moon of September 22 (19h GMT). That very same day, at 15h GMT, is the equinox, which is symbolized by the thick cresent-like circle just outside the moon disk.

Summarizing, we have:

1. The afternoon half moon of the day after the crop circle, shown by the biggest disk.

2. The seven prior first quarter moons since the winter solstice, shown in diminishing size behind it.

3. The next afternoon half moon, which the thin line shows to be near (only two hours after) the ascending node.

4. The final afternoon half moon, which the thick line (crescent) shows to be near (only four hours after) the autumnal equinox.

The 16 overlapping circles then show us how many days until the big event. Sixteen days after Sep. 22, is Oct. 8. That seems to be the day. Prepare as you would for an earthquake, wildfire, flood or other natural disaster.

Others (e.g. Zugelder's diagram) have noted the break in the axis of symmetry: the 16 overlapping disks lie on a greater azimuth than the non-overlapping disks. These azimuths correspond to the azimuths of July 25 & 26 moonrise, with the later rise (July 26, azimuth 117.1820deg at the coordinates of Stonehenge and zero altitude, according to the airless model of the online JPL ephemeris) corresponding to the larger azimuth of the disks representing later time (the overlapping "day" disks). The July 25 moonrise (azimuth 109.9716deg) corresponds to the azimuth of the disks representing earlier time (the non-overlapping disks).

The most nearly perpendicular aerial photo I've found, is Bert Janssen's, posted on www.cropcircleconnector.com . Tracing it from the screen and comparing its two axes, to the nearest (presumed NS) tractor lines, I find that the overlapping disks are at azimuth 125.70 and the non-overlapping at azimuth 120.365. The camera angle should increase these angles, so this constitutes at least rough agreement with the azimuths of moonrise.

July 28 crop circle (Jubilee Copse, near Hannington, Wiltshire) confirms Oct. 8 +/- 7 date

This crop circle contains a 12x12 grid; the crop in 65 of the squares in unflattened, and 79 are flattened. This indicates 144/2 = 72 +/- 7 days. July 28 + 72 days, is October 8.

July 22 crop circle (Aldbourne, Oxfordshire) confirms Oct. 8 date

This is a square inscribed in a circle, but altered so one side of the square is short. If each side of the square should have been 365.25/4 days, then as I measure the slanted aerial photo on the screen, the short side is 77.26d. July 22 + 77d = October 7. The three dots draw attention to the "moving" end of the short side: they signify past, present, and, the biggest, fanciest dot, the future.

Correcting for Earth's eccentric orbit using the trapezoidal rule for integration, a more exact time for the short quadrant, which if complete would have ended about Oct. 7 + (91.31 - 77.26), is

June 12 (reported) crop circle (Silbury Hill, near Avebury, Wiltshire) confirms Oct. 8 date

This small formation was discovered June 12, but may well have been formed a day or two earlier. It shows a "sun" with five Earth / full moon alignments in a regular pentagon. There is a "3rd quarter" moon (geocentric) at 11h GMT on June 11, and another at 08h GMT on Oct. 8. This suggests that these circlemakers had some familiarity with our calendar and clocks. Be that as it may, the interval for formation of the crop circle, June 11 (?), to Oct. 8, thus covers five "3rd quarter" moons exactly.

Werewolves of London again?

Even before I made this second post about crop circles, a Boston banker who reads this messageboard, emailed me that he thought it might be a signal to co-conspirators in a rigged stock market crash. The circles I discuss here, not only indicate Oct. 8, they refer to the number of days in a week, dates of the month and hours of the day; if extraterrestrials knew about those details of our language, they wouldn't need to be cryptic.

This conspiracy would have to be London-based with inside men in the RAF. October is the month most associated with big stock market drops, and Monday (e.g. Oct. 8) is the day of the week most associated with them. The crop circles might be made at night by aircraft emitting microwave beams.

Jubilee Copse July 28 crop circle confirms October 8 date

This 12x12 checkerboard crop circle has 12 words of 12 "bits" of information each. We know which are the rows and which the columns, and which end begins the word, because the first "bit" of each word is a broader rectangle. These broader rectangles are found along one side of the checkerboard, the righthand side as the circle usually is shown on the computer monitor.

Each row is thus read from right to left, and is a twelve-digit binary representation of a real number between 0 and 1. The rounding error cannot exceed 1 part in 2^13 = 8192. This corresponds to 1.07 hour if the unit is a year, or 0.044deg if the unit is a circle.

The seventh row (number) is the easiest to interpret. This number, multiplied by the 365.24219 days of the current tropical year, gives 291.498 days. The latest winter solstice was Dec. 22, 2011, at 05:30 GMT. Adding that many days, gives Oct. 8, 2012, 17:27 GMT.

Seven and eleven are the largest prime numbers < 12. This might be a hint that the 11th row (number) is akin to the 7th, and indeed they are nearly equal. However, the 11th number does not seem to correspond to the great circle arc swept by the sun between 05:30 GMT Dec. 22, 2011 and Oct. 8, 2012. Instead, it seems to correspond to the distance traveled by the sun in space (relative to the Earth) as a fraction of the distance covered in one sidereal year.

My first-order approximate calculation gives an end date Oct. 8, 2012, 15:48 GMT. Because the rounding error from these 12-digit binary numbers can correspond to more than an hour, this and the previous date are consistent with each other, up to rounding. In my approximation, I found the arc from the JPL ephemeris apparent RA & Declination, which refer to the equinox and ecliptic of date, then subtracted the precession. I then approximated the Earth-sun radius variation, with perihelion at 11deg longitude, as proportional to the negative cosine of (longitude minus 11deg), and integrated the tangential motion, neglecting the radial motion.

My second-order calculation uses eccentricity 0.016723 (from data in the 1987 CRC Handbook of Chemistry & Physics) and a perihelion at sun longitude 13.56+270 = 283.56, which is the mean of the apparent geocentric sun longitudes, at the perihelion times given to the nearest hour by the USNO for 2011 & 2012. I retained all terms of second order in the eccentricity, and found that the 11th number corresponds to the fraction of distance traveled to 16:41 GMT Oct. 8, less than half as far from the other value, as the first-order calculation was.

The 12th number, seems to correspond to the change in Luna's right ascension, between Luna's crossing the ecliptic Oct. 4, and "the event", Oct. 8. This number is a fraction of 180deg, not 360deg; this halves the rounding error. The difference between ascending and descending nodes is arbitrary; one or the other always will lie within 180deg. Because of the thirteen times greater angular speed of Luna vs. the sun, and the halving of the unit interval, this number could give the time of "the event" to within two minutes, instead of an hour. The time given by this twelfth number is Oct. 8, 2012, 17:23, differing only four minutes from the time implied by the seventh number. Maybe this is luck; or maybe even the circlemakers don't know the time to within minutes, but decided to make the twelfth time close to the seventh, rounding error and all, to impress us with their skill and with the importance of the message.

The numerology of the June 12 Silbury Hill crop circle (see previous post) suggests that the circlemakers know about the hours of our day and the days of our month. On the other hand, the numerological relation could have been mere luck, or Dr. Jung's "synchronicity" (an organizing principle perhaps above the circlemakers). Logically, the June 12 crop circle message stands, even without the numerological extras.

Today, August 2, I find that the 8th of the base-2 numbers in the crop formation, again assuming that the unit is 360degrees, equals the difference in right ascension (apparent; equinox of date, per JPL Horizons) between Luna's center and Mars, at 16:51 GMT on Oct. 8.

Assuming they signify fractions of a 180deg arc, the 1st & 2nd numbers differ by 1741". The geocentric angular diameter of Luna at its apogee Oct. 5, is 1769". Also, the mean of the 1st & 2nd numbers, is the reciprocal of 2.71888, close to the base of natural logarithms, 2.71828... .

At 2012.8AD, according to the 1990 Astronomical Almanac, p. B18, Earth's mean obliquity is 23.43763deg. According to Simon et al, Astronomy & Astrophysics 1994, Luna's orbital inclination then, to the ecliptic of date, is 5.15668deg. So Luna's maximum declination would be 28.5943deg. The 3rd number, assuming it is a fraction of a 360deg arc, is 28.652deg.

For the angle between Mars and Venus on Oct. 8, the light-time correction happens to be small, only 4". The 5th binary number in the Jubilee Copse crop formation, if it represents a fraction of 180deg, would represent 84.7265deg. The JPL ephemeris gives this as the difference in ecliptic longitude (apparent, using ecliptic of date) between Mars and Venus, at 19:42 GMT, Oct. 8.

More natural, would be to project Mars' position onto the great circle tangent to Venus' sky path (tangent at, say, 17:23 GMT) rather than Earth's ecliptic. The 5th number (times 180deg) equals this difference in "longitude", at 18:22 GMT.

*********

The August 2 crop formation at Chalk Pit, near Wootton Rivers, Wiltshire, is an Archimedes spiral of approx. six turns. This crop formation would have been made at almost exactly the time of the full moon, 03:27 GMT. Two full moons in the future, would have been Sept. 30.

The arclength of an Archimedes spiral is given exactly by an elementary integral (CRC tables #156). The ratio of the length of six turns, to the length of one turn, is 33.5441. What if the mean synodic month, 29.5306 d, were replaced by this number of days? We would have Aug. 2 + 2*33.5441 = Oct. 8.

August 1 crop circle (Avebury Stone Circle, Wiltshire) gives Oct. 12 date; July 31 (Windmill Hill, Wiltshire) gives Oct. 8; July 29 (Andechs Abbey, Bavaria) gives Oct. 9 (8?)

This formation shows 13 disks trailing Jupiter, 7 trailing Venus and 6 trailing some other planet. By interpolating in the JPL ephemeris (which gives positions apparent at the Sun) I found that this crop circle indicates, by one method of analysis, the time T = 19:04 GMT, Oct. 12, 2012.

A planet's "synodic period" is defined as the time for recurrence of the same Earth-Sun-Planet angle. I approximate this angle by its projection onto Earth's ecliptic of date. Thirteen synodic Jupiter periods after the above specific time T, Jupiter's longitude, corrected to a fixed equinox, has changed the same amount, as has Venus' longitude after seven synodic Venus periods. That is, Earth's longitude is the same after 13 synodic Jupiter periods as after 7 synodic Venus periods.

The angle by which both the Jupiter's and Venus' and Earth's longitudes change, is 70.939deg. This is close to the 69deg angle given by Jocelyn Cazes on her diagram, which is posted on the website

cropcircleconnector.com

Addendum #1, Aug. 7: If I choose T = Nov. 17, approximately, then the change in Earth's longitude after six synodic Mars cycles, equals that after seven synodic Venus cycles, except for sign. So, the smallest planet depicted in the crop formation, might represent Mars. Venus was placed between Mars and Jupiter, to hint that it is the planet involved in both comparisons.

Addendum #2, Aug. 7: The July 31, Windmill Hill, Wiltshire crop circle, shows five binary numbers of eight digits each, which I list from the inside to the outside of the spiral:

The first "0" of flattened crop, is to enhance the legibility of the second (first meaningful) digit, always a "1". The only number which appears twice, is 69 and

July 31 + 69 = Oct. 8

The full moon is Aug. 2, so 67 might be the count from then to Oct. 8, a backup in case the formation was not discovered the day it was made.

The following times of that month also are noteworthy:

max speed of Luna toward Earth: July 19, 17:27 GMT max " away from " : Aug. 4, 16:00 GMT

If Luna's orbit were not perturbed by the sun, that is, if Luna's orbit were a perfect ellipse, these times would be for the latera recta, 90 deg from the perigee and apogee, as measured from Earth's center. The former time is 81 d (not 80 d) before Oct. 8; the latter time is 65 d before Oct. 8. The fastest speed of the Earth-Luna barycenter toward the sun, occurs Oct. 5, 23:39.

Addendum #3, Aug. 8: The Andechs Abbey, Bavaria crop formation is listed as discovered July 29, but always there is the possibility that it was formed slightly earlier, maybe before midnight July 28, or even before dawn July 28 and not noticed. July 28 + 72 = Oct. 8. This formation consists of 12 nested tetrahedrons. Some faces of the tetrahedrons are missing, to allow the nesting. This suggests that the meaning is in the number of edges (6) not the number of faces (4). The bold lines on the intact faces, also suggest attention to the edges. Twelve tetrahedrons times six edges per tetrahedron = 72.

Have you ever noticed that when two objects that have the property of mass approach each other the space between them shrinks? Could it really be that simple? Does space condense in the presence of mass?

I'd procrastinate, but I can't seem to find the time

August 13, 2005 crop circle, Woolstone Hill, near Uffington, Oxfordshire, indicates Oct. 7 or 8, 2012

I'll explain this one according to the train of thought, that the circlemakers seem to want us to have. The most emphasized number in this crop formation, is six (which can be seen either as flat, or as faces of stacked cubes).

The next most emphasized number, is sixteen (the number of small squares forming the large perimeter square). There is a subtle ambiguity, though: 16, or 15? Each of the small squares contains a spiral-like sequence of monotonically lengthening tiny rectangles. The longest tiny rectangle is on a side of the small square, which rotates 90 degrees from one square to the adjacent square, except for one step, where it does not rotate (the next rotation is 180 deg to compensate). So, there are 16 small squares, but these rotate only 15 times.

This ambiguity, "15 or 16?", parallels the ambiguity, "sidereal or synodic month?". For the length of the mean sidereal month, I use polynomial 3.4.a.1 of Simon et al, Astronomy & Astrophysics 282:663 (1994) p. 669; it gives 27.3216616 d during the entire interval Aug 2005 - Oct 2012. For the mean synodic month, I modify the mean sidereal month according to a sidereal year of 365.25636 d, getting 29.5305889 d. The ratio of synodic to sidereal month is

16/15 * 1.0133 = 16/15 * (1 + 1/75)

so the "15 or 16?" ambiguity hints that the circle deals with both sidereal and synodic months.

There is another kind of month often used by astronomers, especially ancient astronomers predicting eclipses: the draconic month, averaging 27.21222085 d during the interval. Also, there is the anomalistic month, averaging 27.5545498 d.

Since the 16 and the 6 in the crop formation are qualitatively different in design, we are supposed to multiply them as factors, rather than add them as we would objects of the same kind.

16*6 = 96 draconic months = 2612.373 d = Aug 13.0, 2005 + 2612.373 d = Oct 7.373, 2012

Repeating the above calculation with sidereal months, gives

Oct 17.880

With anomalistic months, it gives

Nov 9.237

and repeating it with synodic months, but using 15 instead of 16, gives

Nov 21.753

What about the fine radial segments, called "feathers" by many who study this crop formation? Of the innermost group of feathers, I count blocks of 11, 11, 11, 10, 10, 10. Of the outermost group of feathers, I count 18, 18, 18, 19. Of the middle group of feathers, I count 11, 11, 11, 11.

The average value of blocks of the innermost group of feathers, is 10.5. For the time estimate based on the sidereal months, I have

Oct 17.880 - 10.5 = Oct 7.380

perfectly agreeing with the time based on the draconic month. The total number of feathers in the middle group is 44. Thus based on the synodic month, I have

Nov 21.753 - 44 = Oct 8.753

and finally for the anomalistic month, using half the total number of feathers in the inner group, 63/2 = 31.5, I have

July 15, 2008 crop formation near Avebury (the Mayan End Date solar system): a fake?

I've discussed this crop formation before on this thread, but today I used my old measurements from the photo by Nick Nicholson which I printed out from www.ConspiracyPlanet.com, and my old corrections for the obliquity of the photo, together with accurate quadratic interpolations of ecliptic longitudes given by the JPL online ephemeris. The longitude differences between the planets and Earth are consistent with

Mercury Dec 20, 2012, 17h GMT Venus Dec 14 15h Mars Jan 08, 2013, 10h Jupiter Dec 24, 2012, 15h Saturn Dec 27 1h Uranus Dec 20 19h Neptune Dec 27 15h

The other planet isn't Pluto: it's a schematic indication of our moon, Luna, with geocentric longitude equal to the heliocentric longitude shown for the object in the crop formation. Simon (sec. 3.4.a.2) gives the mean lunar perigee in coordinates of date, and the lunar perigee shown in the crop formation is consistent with Jan 13, 2013, 07h. Luna ("Pluto") itself is shown at longitude consistent with Dec 16, 2012, 07:30 GMT.

Such inaccuracy is not characteristic of the circlemakers. Luna's rapid change in longitude would have made it easy to depict Dec. 21 rather than Dec. 16 (60 deg difference). Be that as it may, the mean of the nine indicators is

June 2, 2012 crop formation at Manton Grove, near Marlborough, Wiltshire: Oct. 8 again

This crop formation, sometimes called the "polar clock", has been noted by crop circle experts to show signs of authenticity in its details of construction. I also show here that in the ingenuity and precision of its astronomical plan, it is typical of other excellent crop formations. Indeed it is an astounding astrometric tour de force.

It indicates the date Oct. 8, 2012. June 2 is 128 days prior to Oct. 8. This is a round number, 128 = 2^7, in the most universal arithmetic, binary arithmetic, so should be dignified by a crop circle. The number 128 participates in these astronomical resonances:

128 tropical Earth years (of 365.242188 d, determined for 2012.6AD from a sidereal Earth year of 365.25636 d, together with Clemence's version of Newcomb's quadratic precession rate polynomial, with Van Flandern's suggested dp1 = +1.26"/century correction)

= 68.053 sidereal Mars years = 208.07 sidereal Venus years = 46751.00006 d

On the other hand, 128 sidereal Earth years (of 365.25636 d)

= 46752.8141 d = 46753 - 66.9/360 d

There is another kind of Earth year, analogous to the draconic month: the time for Earth to make one revolution relative to the regressing node of Earth's orbit on some fixed copy of Earth's orbit such as the J2000 ecliptic. I approximate the frequency of revolution of this node, by numerically differentiating the formula for the node of the ecliptic of date on the 2000.0AD ecliptic (cubic forward, or backward, difference formula from 2000.0AD at 2 yr or smaller intervals) at the bottom of p. B18 of the 1990 Astronomical Almanac. I find that this "nodical year" is

365.25636 * 360deg /(360deg + 8.700") = 365.253908 d

and 128 such years is

46752.5002 d = 46753 - 179.9/360 d

Earth's year relative to its perihelion, i.e. the anomalistic year, is

365.259635 d (resp. 365.259549 d)

for 1994-2000AD according to Weisstein's online World of Astronomy (resp. adding 25 min to the tropical year, according to the USNO website); 128 such anomalistic years is

46753.2333 d (resp. 46753.2223 d) = 46753 + 84.0/360 d (resp. 80.0 d) = 46754 - 276.0/360 d (resp. 280.0 d)

and I'll adopt the mean of Weisstein and USNO, 278.0.

The actual perihelia and aphelia of the Earth-Luna barycenter vary considerably due to planetary perturbations; let's find them from the JPL online Horizons ephemeris for Jan 2012AD, Jan 2012-128, July 2011AD and July 2011-128:

perihelia 1884AD Jan 01 23:38.9GMT 2012AD Jan 04 02:59.4

At the time of this crop formation, the immediate past block of 128 actual (i.e. according to actual Earth-Luna barycenter) perihelion-to-perihelion anomalistic years 1884-2012AD equals

46754 - 309.9/360 d

and the immediate past 128 actual aphelion-to-aphelion anomalistic years 1883-2011AD equals

46754 - 259.9/360 d

Now let's compare these angles to the observed. I measure on a printout of the most perpendicular aerial photo I found online, a photo stamped "WCCSG.COM" and with the crescent "APS" logo. One way (method A) to measure the arcs, is to center the protractor on the center of the central disk, and measure to, say, the centers of the end-segments of the arcs. Another way (method B) is to extend lines from the end-segments of the arcs, and with a protractor measure the angle between those lines. Evidently these two methods are not equivalent: for example, the best fitting line through the common end segments near 5:00 in my photo, considerably misses the center of the central disk.

I used both methods, and will denote the measurements A and B. In order from inside to outside, the lengths of the arcs, in degrees, are

inside the whole, third, arc (symbolize less than 46753 d):

A 210 B 220.5 expected 179.9 ("nodical")

A 61.67 B 69.67 expected 66.9 (sidereal)

outside the whole, third, arc (symbolize less than 46754 d):

A 247 B 250 expected 259.9 (actual previous 128 aphelion-to-aphelion)

A 273.5 B 274 expected 278.0 (approx. mean anomalistic)

A 320.5 B 319.67 expected 309.9 (actual previous 128 perihelion-to-perihelion)

The "expected" values are according to the various kinds of years giving something less than a whole number of days, when multiplied by 128. The third arc, or ring, of the crop formation symbolizes the tropical year, which gives a whole number of days, when multiplied by 128. Four of the five arcs agree closely with theory, within my evident error (or perhaps rather, ambiguity) of measurement. I hope to make more accurate measurements soon, and to compare my arc measurements with those of other investigators.

Not only are crop circle messages generally vulnerable to random degradation, they are vulnerable to intentional jamming and destruction by adversaries of the circlemakers. The complexity of the codes gives redundancy, and also makes it difficult for adversaries to guess quickly enough, while the circle is being made, how to damage the message efficiently.

Thanks for asking! Unfortunately, I don't have the time or money, and no one else is looking to my knowledge. Of course there already are the sky survey photos of it.

So you are confident it exists and that you have its orbit pretty well figured out? I'm just an interested follower of the story - no ability to search for it myself.