The Pyramids of the Giza Plateau: Astronomical Observatories Based on a Mathematical Model of Vision

[Note: This article was originally posted on the blog 'Luck-e Jake']

December 6, 2015
Michael J. Ajemian

The Pyramids of the Giza Plateau:

Astronomical Observatories Based on a Mathematical Model of Vision

[updated intro 12/21/15; added background description and fixed a typo.]  This concept was an accidental discovery on 6/18/95.  I'd read a book about the Great Pyramid, kept wondering why all the precision construction, then noticed something in the reflection of a marble table as I moved my head - I realized I could see things using reflection that I couldn't see directly as I looked at the view out the window in the table.  The math wasn't difficult to understand.  But for a head injury I briefly died in just prior to learning how it worked, this would have been written years ago.  I hope you enjoy it, because to me, it's really exciting!

The pyramids of the Giza plateau represent perhaps the most recognizable architecture in the world.  The Great Pyramid is an engineering marvel and enigma.  The entire structure inspires a sense of awe for it's dimension, sringent specifications, and the incredible problems in engineering that had to be solved just to build the hulking structure.

At the time of construction, the pyramids were covered in highly polished limestone.  Evidence of the limestone casing is seen around the base of the pyramid of Cheops.  Sir Flanders Petrie noted the precision of the casing stones as being "equal to opticians' work of the present day, but on a scale of acres" and "to place such stones in exact contact would be careful work; but to do so with cement in
the joints seems almost impossible". (Romer, 2007, p. 41) Pretty cool stuff to look at.  From an engineering perspective?  It's mind-blowing.

While most people think the pyramids are monstrous tombs, the unbelievably surreal precision of the faces suggests a purpose to the structure beyond simply shining like a jewel in the sun.  In fact, it seems to me they were built for astronomy by some people who understood the mathematics of vision.

The faces of the pyramids were optically true and highly reflective.  Both a true surface and high reflectivity are requirements of a mirror.  Since each face is angled skyward, they'd allow a viewer looking at a face to see reflections of objects in the sky with superior clarity.

The courses of highly-polished limestone of optical precision were of a small range of sizes, but their courses varied at somewhat regular intervals.  Interestingly, in the dark, the 1/100th gap between casing stones would have provided a grid within which to assist in precisely locating object position relative to an observer surveying the surface.

Having a structure that is stable, oriented to true north, and is highly reflective, except for the thin lines of the gaps between stones comprising the shadowed grid, would appear to provide the means to view the reflections of objects in the sky and their position in the faces with superior precision.

Should an intrepid viewer have positioned themselves at the center of the base of the south face and simply looked up the apothem, the center line of the face, it would have been possible for them to have observed the passing of stars in the reflection of the face, simply by observing along the apothem:

Figure a. The Apothem

If our observer were to find themselves in possession of a pencil, papyrus roll, and time keeping device, they could not only have observed the location of the (primarily northern) stars passing
through the line marking the center of the face, but transferred that position detail to their papyrus.  The papyrus could conveniently roll along with the passing of time and be a collection of pencil dots
where bodies were recorded at the time they transited the meridian (the apothem is aligned with the meridian).  By collecting data in such a manner, it would seem possible that way back in the olden, olden, olden days, they could have made star maps with the help of the pyramid tombs.

Having one observer on one face represents an opportunity to collect good data, but it's limited.   Were a set of four observers positioned, each before a face of the pyramid, such that they were each observing the same celestial bodies in the reflections at the precise time the object reached the meridian it would appear to be possible to calculate the celestial longitude, latitude, and distance to bodies in the sky.

Figure b & c, show two views of the same observation in a somewhat iconic way.  Each observer would see in the reflection in their respective faces as S transits the meridian of the pyramid.  Both observers 2 and 4 note the time as S transits the meridian.  The time will be used to calculate the hour angle of S from a reference meridian.  All observers record position detail with the intention of calculating a set of angles to derive declination of the star relative to the celestial equator.

Figure b. Top view of a star to four observers on their respective faces.

Figure c. Side view of a star to three observers on their respective faces.

Declination is derived by:

1. Finding the position of the star by intersection of the vectors at a point S.
2. Projecting a vector from the center of the earth through the point S
3. Calculating the angle formed between the vector projecting from the center of the earth through the equator at hour angle T.

Because the structure was so stable, it would appear to have been possible to measure bodies in the sky with a great deal of precision over time.

Once it becomes possible to calculate distance to a point, it becomes possible to calculate distance to a myriad of points.  It would also be possible to record colors, though that's not the focus of this paper.

This paper will attempt to highlight how the pyramids of the Giza plateau were constructed such that they operated as a mathematical model of vision, complete with color mapping, and depth perception.

Constants and Structural Notes

The concept is initially explored using one pyramid.  First, some facts about the Great Pyramid:

1. At time of construction, the surfaces of both pyramids were complete and uniform.
2. The surfaces of both pyramids have "optical precision on a scale of acres." (Romer, 2007, p. 41)
3. The surfaces of both pyramids are highly reflective.
4. The faces are slightly concave, with a noticeable depression running down the center of each face.  The center of the four sides are indented with precision, thus forming an 8-sided pyramid.
5. The pyramid is oriented to true north.
6. The angle of the faces of the pyramid are each 51.8 degrees.

The math of a single pyramid as it relates to vision

In order to highlight the function of the pyramid as observatory, a series of experiments will be presented, from the basic to the complex.

Experiment 1 - Reflection of a vector on the face

An observer is positioned at the center of a face, orthogonal to the base, sighting the apothem on the horizontal.

Using Reflection

Having established the observer's position, attitude, and distance from the base, a point is marked on the viewers horizontal, coincident with the apothem.  By establishing a stable viewing position along the horizontal, it becomes possible to calculate the angle of reflection.  This is an essential starting point.  Because the angles of the face (A and B) are known, and the observation vector is horizontal, the angle of reflection is the special case, found by subtracting the angles:

R = 90.0 - 51.8 = 38.2

Figure 2. Finding the angle of reflection for degenerate case (horizontal observation.)

This case is meant to highlight how to begin using the system.  The ray from the observer's eye is projected to the surface and the incident angle is discovered.  In this base case, the incident angle of the reflected ray is calculated by virtue of the stable angles of the face (angle B is 38.2 degrees.)

Experiment 2 - The Intersection of Reflected Vectors

Before finding the intersection of a vector, the reflection vector needs to be calculated.  The following graphic highlights the sequence to solve for reflection vector P.

Figure 3.1. Sequence of calculations to find the projection vector P.

The sequence to solve for the projection vector P, as displayed in Figure 3.1, are repeated for clarity:

1. Measure distance CF, from point of observation to base.
2. Calculate triangle ABC
3. Calculate triangle BDE
4. Calculate triangle BEF
5. Calculate reflection angle R
6. Project vector P
7. Calculate the intersection of P with other vectors (not shown.)

The sequence above is repeated in the next example to find the point of intersection (I) for two observers surveying points at equal height e and e-prime along the apothem:

Figure 4.  Calculating the point of intersection from point of observation (f).

Figure 4 shows two vectors reflected at an equal altitude, along the apothem of opposing faces.  By virtue of their being projected along the apothem, they'll intersect somewhere, but in this special case, their being surveyed at equal altitudes means they'll intersect directly over the cap of the pyramid.

In order to observe how resolution changes with height, the viewers agree to sample a more points along the apothem at equal altitude.  As the reflection angle increases, the altitude of the point of intersection decreases.  As the reflection angle decreases, the altitude of the point of intersection increases.  The change in altitude is akin to depth perception in the eyes.

Figure 5. Change in altitude of vectors calculated at equal, but progressively greater heights.  An increase in altitude of the point of reflection results in a decrease in altitude at the point of intersection.

The following graphic shows the field of view available to a single observer at the center of the base of a face of the pyramid.  By combining the field of view available to observers at the center of all four faces, a single pyramid can be used to observe, via reflection, points that lie most anywhere within the celestial hemisphere.

figure 5 - Celestial Hemisphere

It's Three Dimensional

Figure B (above) shows the position of a star, and its reflection on four faces.  It shows iconically the reflection of the star in each of the faces.  By having each observer measure position on the face (height, distance from apothem) the angular measures can be discovered and the reflection vectors calculated.  In this case, the calculations are a bit more complex, especially finding the point of intersection of the vectors.  But the sequence is stable and consistent.


Stereoscopic Vision & Meridians

The following graphic shows how the star S from Figure B might appear to 8 observers at the faces of two pyramids.

Figure 7.  Stereoscopic view of star S in the reflections of the faces of two pyramids at time T(0).

Figure 8.  Stereoscopic view of star S in the reflections of the faces of two pyramids at time T(1).

Figure 9.  Stereoscopic view of star S in the reflections of the faces of two pyramids at time T(2).

In the series of figures (7-9), a star S is shown at three positions in time.  One of the nice features of having two pyramids aligned to true north is that each provide an arc time and angle at the meridian for each.  The earth's rotation provides a constant background to measure the arc angle and arc time against.  The arc time between the pyramids is a constant.  If something moves faster or slower than the earth's rotation, it would seem to indicate the direction of travel of the object.  Plus, if the declination or distance changed, especially as measured over time, the direction and velocity of the object would appear to be possible to determine.

While star calculations are interesting, scanning applications are pretty interesting too.  Calculating the distance to a point in space using one pyramid appears to provide high utility in that a number of vectors can be calculated, with accuracy increasing with the number of viewing positions.  The introduction of a second pyramid would seem to improve precision.

But, the ability to measure depth in the angled faces in the pyramids, combined with a relatively large distance between pyramids, means that each face can be used to construct patches of an object by collecting a range of data points from each face.  The patches constructed from the survey of the reflected points of the object in each face represent distinct views of the surface of an object, with depth, which when assembled (using reference points to align patches) would create a projection of the object in three dimensions.

It would seem that any object which could be recorded, could also be projected to reconstruct the object or location in space above a projection mechanism.  It'd be fun to see if this would work to create holographic projections.

Construction of Surfaces

Creating a scanner using the pyramids would involve utilizing a large number of observational positions to an object.  The number of observational positions and their location would vary, but the more positions records as points in space, the more surface detail to an object are recorded as a point cloud.  The ability to construct surfaces from point clouds has a set of well-defined solutions and products which facilitate surface construction.

While one pyramid is enough to get a sense of the shape of an object, using the two pyramids would seem to facilitate constructing stereoscopic images from a collection of manifolds, each created by virtue of collecting a large cross-section of points, converted to a manifold.

Conclusion

In conclusion, the math appears to support the distinct possibility that the pyramids at Giza were used to reckon stars accurately, and that the reckoning would be by arc angle, declination, and distance.  By virtue of a stable structure and a series of well-defined steps, it would be possible to calculate vector projections through a point on the face, extending into space.  The vector projections, when combined from observational positions near the original observational position, provide a stable mathematical base to calculate distance to an object.

It would seem, if the system is as stable as it appears, that with a little creativity and insight, this mechanism can provide hours of fun and entertainment to creative individuals as a surveying instrument, holographic projection system, medical scanner, surgical knife, medical 3D scanner.  Seems like whatever we would use our eyes to do could be modeled, from the highly focused to the wide angle.