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An Ancient Egyptian Catenary Construction Curve

Refer to companion paper: The Saqqara Ostracon

When I first examined the curve displayed by the Saqqara Ostracon I recognized that it was not circular in form. My first attempt to define the curve also showed it was not elliptical. Refer to remarks by Gunn, who confirms both assessments. I also could not believe it was a hyperbolic.

The following solution should be regarded as incorrect, but curious. I offer it as an example of how solutions can reach far afield in attempts to understand the ancient Egyptian geometricians.

I was haunted by the possibility that it might be a catenary. This form is well known to engineers, who see it displayed in suspension bridges and other architectural structures. They favor it in conscious design because it is a natural curve, created, for example, by gravity acting on free-hanging chains. The necklace to the left hangs in the form of a catenary. The catenary can also be created in the form of an arch, and offers a very rigid structure. The famous Gateway Arch in St. Louis, Missouri, was able to be constructed with thin cross-section at extraordinary height because of its natural rigidity.

 

 

 

College instructors have a neat demonstration of this unique property. They will arrange a group of toy wooden blocks in the form of a catenary on a flat bed. They then raise the bed until the blocks are standing upright and remove the bed. The blocks will continue to stand and will even undulate if lightly touched, without falling down. A circular arch cannot achieve this stability.

The mathematical form of a catenary is:

Y = a cosh (x/a) = 1/2 a (ex/a + e-x/a).

This equation is so well known in engineering designs that it is given a special name as the hyperbolic cosine of "ax," where "a" is a constant that defines the general shape of the curve. Larger "a" values open the curve. The set of curves to the right show a graphical plot of this function with "a" values from 0.05 to 1.00 in steps of 0.05. Note that the vertex of the curve moves downward vertically as "a" increases. (I turned the curves upside down in order to follow our discussion.

We can easily see that the bottom-most curve is similar in shape to that plotted by Gunn in the companion paper.

In order to test the validity of the assumption that the Saqqara structure fit this curve I calculated a catenary to different values. The following table shows the results for "a" = 4.

Calculation of Ostracon Curve based on assumption of catenary form.
Calculation in Cubits, converted to Fingers for graphical plotting
Catenary Curve given by Y = a(ex/a + e-x/a)

x
in
cubits
x/a
a = 4
ex e-x y chart y
(100 - 28y)
chart y + vert.
offset = 110
Specified
Value
Difference from
Specified
Value
0 0 1.00 1.00 4.00 -12.00 98.00 98.00 0.00
1 0.25 1.28 0.78 4.13 -15.52 94.48 95.00 0.52
2 0.5 1.65 0.61 4.51 -26.29 83.71 84.00 0.29
3 0.75 2.12 0.47 5.18 -45.00 65.00 68.00 3.00
4 1 2.72 0.37 6.17 -72.83 37.18 41.00 3.83
5 1.25 3.49 0.29 7.55 -111.50 -1.50 0.00 1.50

 

I then plotted the graphical values, as shown here.

I used cubits, not fingers to calculate the values, and then converted back to fingers to plot the graph. I did this for more convenient calculation numbers. I could have calculated directly with fingers.

I also adjusted the curves to a common vertex location in order to compare with the curve specified by the Saqqara ostracon. This is seen in the vertical offset. This was necessary because increasing "a" values cause the curved to move downward, as I described above. The amount of vertical offset is determined by the value of "a". For "a" = 4 this was four cubits). (I inverted the curve by subtracting from 100 fingers, and then normalized to a common vertex. Both operations are mathematically sound to obtain true comparisons between the theoretical and the structure "as-built.")

The difference between the coordinate values specified by the ancient architect and the theoretical catenary is four fingers for one point, three for another, and the rest less than two. Several of the points are almost directly on the theoretical, within the ability of the architect to specify by the number of fingers.

Before going on to discuss the knowledge of the ancient architect and the implications of this solution, it is helpful to examine the intellectual environment in which Gunn made his assessment.

He naturally assumed that the saddle-back had been completed, and then dismantled with the passing centuries. According to this view, as the stones were removed, rubble piled up inside the saddle-back. Why the builders would have created a saddle-back merely as an object without perceivable usefulness, merely as decoration, is unexplained. Of course, those ancient people engaged in many activities we do not fully understand today; we should not bias an objective assessment. On the other hand, why the architect's instructional flake would have been left lying around was also not addressed by Gunn. If I were a builder I would have cleaned up the neighborhood when I had completed my project.

Another scenario is possible. Perhaps the project was never completed. Then the project, with the ostracon still useful because the job was incomplete, would have been forgotten when everyone walked away. Perhaps limited manpower was demanded for more urgent building elsewhere. We should remember that the Zoser complex was the first, truly innovative, and truly immense, stone building project in ancient Egypt. Many new architectural ideas were explored. These included the rapid evolution of an original stone mastaba into a step pyramid within a few years. That evolution broke the tradition of burial in rectangular block structures.

The possibility exists that the architect suddenly realized uncertainty in the stability of the structure and decided to abandon the project.

Discussion continues in the Zoser Complex. Click here.

 

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