Vanishing Point Update - September 2005

Following a recent discussion with Stephen Goodfellow, I've been taking another look at the Giza Vanishing Point, and have been checking my original calculations with the help of a CAD program. I have also been looking again at the anomalous boundary wall to the south of the Third Pyramid, and its connection with the two encompassing circles of the three pyramids.

In my original computation of the Vanishing Point, I used analytical geometry to determine the centres of the circles, based upon the coordinates of the pyramid-corners through which the circumferences of the two circles had to pass. Those coordinates were expressed in terms of Egyptian royal cubits, measured southwards and westwards from the north-east corner of the Great Pyramid, as determined by my analysis of the Giza Site Plan. Essentially, the coordinate points represented a conversion into cubits of Petrie's survey data in inches, although Petrie himself did not think that a connection between the three pyramids had been planned. The centres of the circles were obtained from the points of intersection of the perpendicular bisectors of the notional chords which can drawn between the respective corners of the three pyramids, as detailed in my original diagram below:

With the assistance of CAD, it is a relatively simple matter to determine the dimensions of circles passing through any three points, given the coordinates, and hence to confirm that the results of the calculations which I carried out back in April 1987 are correct. Note that the above diagram was sent to Stephen for his own information. If I had known that it would later appear on his web site (and now on my own), I might have taken more care with my handwriting....

The following diagram shows where the centres of the two circles are located in relation to the three pyramids. Only a portion of the larger circle has been drawn, since the dimensions are too vast for the full circumference to be represented effectively on the same scale as the pyramids themselves.

Now with regard to the dimensions of these circles, I had always taken the view that they were coincidental by-products of the Giza Site Plan, without any real significance of their own. After all, it seemed highly unlikely that the architect of the site plan could have calculated the radii and the centres, let alone have configured the layout to obtain specific results. In any case, I had already ascribed the exact dimensions and relative positions of the three pyramids to a highly logical and coherent design, in which every measurement had been explained and often with reference to more than one requirement. There was little reason to think that any further factors had to be taken account - least of all two circles of enormous size.

However, there was a nagging suspicion that my preconceptions were not entirely justified. From the outset, one or two of the dimensions were clearly significant, as I mentioned to Stephen in a letter many years ago. It was exceedingly strange that the centre of the large circle was just 11,000 cubits southwards from the vanishing point itself, with a computed discrepancy of only 0.13 cubit. Not only did the chances of randomly obtaining such a round number with such accuracy seem rather slight, but the number was significant in its own right. In the Giza plan, as we have seen, the modular design placed the south side of the Second Pyramid 1100 cubits southwards from the north side of the Great Pyramid, so that the north-south dimension encompassing these two pyramids was just 5/2 times the base of the Great Pyramid of 440 cubits. In addition, the coordinates of the vanishing point and the radius of the small circle corresponded to whole numbers of cubits to within 0.05 cubit, and although not particularly interesting, conformed to the cubit system.

It was only recently, however, that some further relationships came to light, when Stephen pointed out that the circumference of the large circle was just 20 times the diameter of the small circle. The exact factor of 20.04 was close enough to a round figure to suggest deliberate intent. Now for the reasons outlined above, I hadn't paid much attention to the dimensions of the circles, and cannot recall having calculated the circumferences. I knew that the radius of the large circle corresponded to a round 17,500 cubits, with a discrepancy of only 0.04%, but had declined to draw any further conclusions. The diameter of the large circle is, however, practically just 35,000 cubits, so that given the approximation to p of 22/7, the circumference will be 110,000 cubits. Not only is it inherently quite surprising to obtain these simple multiples of 10,000 cubits, but the circumference is also just 10 times the distance northwards from the centre of the circle to the Vanishing Point - through which, by definition, the circumference must pass.

Once again, therefore, it is not just the whole numbers of thousands of cubits which are significant, but the fact that these dimensions are mathematically meaningful. Being a multiple of 7000 cubits, the diameter of 35,000 cubits would have been an ideal choice - if indeed it was chosen - since the circumference would also contain a round number of thousands of cubits, according to the 'classic' value for p of 22/7. The same reasoning applies also to the Great Pyramid, which is thought to embody the p-proportion through its height of 280 cubits and side-length of 440 cubits. Furthermore, the circumference of the large circle is just 250 times the side-length of the Great Pyramid, and the dimensions of 440 and 250 cubits are consecutive in the Giza plan.

It must be noted that the diameter of the large circle is highly sensitive to the exact placing of the pyramid corners through which the circumference passes, owing to the flatness of the curve which connects them. Indeed, if the three corners had been placed in a straight line, then that line would belong to the circumference of a circle with infinite radius. Although the slight bend in the line reduced the diameter to a 'sensible' dimension, the dimension changes rapidly with slight adjustments of the pyramid corner positions. It turns out - and I still find this hard to believe - that the precise diameter of 35,000 cubits can be obtained for the large circle by shifting the south-east corner of the Second Pyramid a mere 0.006 of a cubit from the position as defined by the site-plan coordinates in whole cubits!

If it had ever been intended to define a circle with a diameter of 35,000 cubits by means of the three pyramid-corners, therefore, then it would have been virtually impossible to achieve a more accurate result than that actually obtained by the site-plan dimensions. It is true that the circumference of the circle will not be exactly 110,000 cubits if the exact value of p is used instead of 22/7, yet the relationship with the Great Pyramid still stands, since the dimensions of this pyramid arguably reflect p with greater accuracy than 22/7.

Now turning to the small circle, my computations had shown that the radius was 2742 cubits, while the centre was 2740 cubits eastwards from the Vanishing Point. This number was significant to me as being practically ten times the height of the Second Pyramid, which the survey-data and theoretical factors had shown to be 274 cubits. Whilst the dimensions of the large circle seemed to refer to the Great Pyramid, therefore, a comparable relationship existed between the small circle and the Second Pyramid. At the same time, the diameter of the small circle is a fair approximation to one-twentieth of the circumference of the large circle, as Stephen suggested - this requiring a radius of about 2748 cubits.

The Boundary Wall
We have already referred to the anomalous boundary wall to the south of the Third Pyramid, which seems to run straight over the Vanishing Point. Stephen wanted to know whether the wall described a large circle, and whether it was possible - given the fragment that exists - to accurately determine the size of the circle. It might be interesting at this point to quote from a letter I sent Stephen on 16th April 1987:

There are a few strange things about this wall, the first being that while all the other boundary walls at Giza are aligned north-south or east-west, this wall diverges by about 7 degrees from an east-west alignment - in such as way as to just encompass the vanishing point within the boundary. Secondly, this wall was not built in a straight line but in fact represents the arc of a very large circle, the radius of curvature of which is in the region of 11000 cubits - or comparable to the larger of your two ground circles. A chord joining the ends of this curved section of boundary wall actually falls precisely on the point of intersection of the two ground circles, which point would therefore have been covered over if the wall had been built in a straight line!
Although at that time I had answered Stephen's question, I now find that the details are not quite correct. The rough figure of 11,000 cubits which I gave for the radius of curvature of the wall, actually referred to the diameter! Apart from this simple mistake, much depends on the accuracy of Petrie's plan of the boundary wall, upon which I had based my calculations. In a letter dated 27th May 1987, I wrote:
These walls were all surveyed more than a hundred years ago by Flinders Petrie, who was also puzzled by the line of the southern wall and wrote: "it is impossible to suppose its skew and bowing line to have been laid out along with the very regular lines of the other parts." Yet this wall is of the same construction, and its curvature must have been produced by a deliberate effort of the builders - so I think it must definitely have had a special significance.
Looking again at Petrie's work, it is clear that he took great trouble to determine the exact lines of the walls around the Third Pyramid, as he says: "They were all fixed in the survey by triangulation, with more accuracy than the wall-surface can be defined." In my reassessment of the curvature of the south boundary wall, I took a scan of Petrie's plan, and determined the coordinates in pixels of three points along the wall - at either end and at the approximate mid-point. These coordinates were then related to the size and position of the Third Pyramid, scaled to the dimensions of the Giza plan in cubits, and entered into the CAD program. This gave a radius of curvature of around 4800 cubits.

As a check on this estimation, I imported the scan into the CAD program, and scaled and positioned it so that the base of the Third Pyramid matched the site-plan location. By this means, I could not only determine the centre and radius of the circle, but also superimpose the accurately-plotted circumference on Petrie's plan, in order to make a comparison with the curvature of the wall as it was actually built. As can be seen in the diagram below, in which the arc of the circle is shown in magenta, the agreement with the line of the wall is rather close, being in fact practically pixel-perfect:

Boundary Walls

This new evaluation also confirmed that the Vanishing Point is extremely close to the inner (north) side of the wall. It now appears that if the wall had been built in an exact straight line between the ends of its curvature, then the vanishing point would have been just outside the enclosure. Not only was the anomalous curve similar to that of the pyramid ground circles, therefore, but the position of the wall and the effect of the curve were significant in relation to the vanishing point. At this stage, however, the radius of curvature and the centre-point of the circle appeared to be arbitrary. Clearly, the circle of the wall did not make contact with the pyramids, or with any other structures as far as I could see, so the question arose as to whether the size and position had any significance of their own.

Again with the assistance of CAD, it was easy to plot the entire circle and relate the dimensions to the vanishing point circles. The result of this exercise is shown in the diagram below. Much to my surprise, the circle of the wall was bounded to the east and west by tangents to the small and large circles drawn parallel to the north-south axis of the plan. Consequently, the size and position of the wall circle were entirely a function of the pyramid ground circles. First, the diameter and east-west position of the wall circle are defined by the eastward extent of the small circle, and by the westward extent of the large circle, thus being equally dependent upon both circles. Second, the north-south position of the wall circle is such that the circumference intersects the vanishing point, which is itself defined by the intersection of the pyramid ground circles:

The circle of the wall

It will noted that there is a slight discrepancy between the circumference of the wall circle and the tangents as drawn to the small and large circles. Given, however, that the wall circle had to be extrapolated from a fairly short segment, the agreement seems remarkably good. It is, of course, possible to construct a theoretical circle which is defined exactly by the tangents to the pyramid circles; and when overlaid on the wall as built, the departure is no greater than the thickness of the wall itself.

Now as Stephen will confirm, I have always been extremely skeptical about the idea that the pyramid-builders had intended to lay out the Giza pyramids in such a way as to define anything resembling a vanishing point. However, when all the evidence is considered, it does seem to me that there is quite a strong argument to support the idea that the builders were aware of the fact that the three pyramids were bounded by circles which defined two points by their intersection - one being comparable to the artist's conception of a vanishing point. Perhaps after the three pyramids had been built, the architects turned their attention to the location of the boundary walls, which can be shown to have been laid out on a definite plan. It was then, perhaps, that the dimensions of the enclosing circles were considered, and found to be of sufficient interest to justify the enhancement of the unified plan by setting out the southern enclosure wall in a manner which was entirely contrary to their usual practice. For to construct a wall with such a continuous but slight curve can only have been the result of conscious effort...

John Legon, 17/09/05



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