Background

Adam Young and Ben van Kerkwyk made interesting claims that certain ancient Egyptian stone vases are astonishingly precise. This motivated me to conduct an independent metrological study of similar artifacts from Matt Beall’s collection. The objective of this study was to evaluate the precision of the stone vessels using a rigorous scientific method.

Importance of Statistics

Because all scientific analysis is comparative, examining a large dataset is the best way to arrive at a meaningful conclusion. In science one rarely relies on a thesis that hinges on a single data point because such a thesis is difficult to defend. Randomness and chance play tricks on us, forcing us to rely on statistics when we wish to know anything with certainty. As such, a thesis supported by statistics is much more sound since it tends to exclude biases and errors arising from insufficient sampling.

Stone Vessels

To obtain the necessary volume of data, I decided to study a sample of 25 stone vessels: 22 purportedly of ancient Egyptian origin from Matt Beall’s collection (Fig. 1), and 3 modern (Fig. 2).

The reason I say ‘purportedly’ is that in opinion of the academic archeologists, objects in private collections are not considered genuine and therefore are generally excluded from scientific research because they often lack provenance and archeological context. Indeed, when there are no proper records, how can we tell if these objects were indeed found in Egypt? Or how can we attribute them to the predynastic period without knowing how and where they were found? This is a legitimate objection, which can be overcome by examining objects from a museum collection. Museum objects from the Flinders Petrie collection were recently scanned by Karoly Poka and Brothers of the Serpent, and I am eagerly awaiting their results.

The three modern objects (two made of onyx and one of marble) were chosen because they are broadly similar to the vessels in Matt Beall’s collection in shape, size, and visual quality.

Dataset

The source dataset consisted of 25 CAT scans processed into 3D models that were saved as STL files. The scanning was performed at EMS, Inc. using a Nikon 4M-RTSS scanner.

An EMS technician manually aligned each scanned model to ensure maximum axial symmetry. Each model was centered on the origin in the XY plane with the principle axis of symmetry being the Z axis – Fig. 3.

Model Slices

To analyze the models I wrote code in MatLab (using the MatGeom 1.2.8 library) to slice each STL model into 101 slices evenly spaced along the Z-axis – Fig. 4.

Because most objects exhibited wear along the top and bottom surfaces, the first slice and the last slices were offset 0.040” from the bottom and the top of each object.

Note that the vast majority of the model slices (except those that cut through the handles) are circular. This is expected since the vases are axially symmetric and we are slicing the model across the axis of symmetry (which is the z-axis).

Quality Metric

First, we must clarify what we mean by ‘precision’. Precision is not an absolute but rather a relative measure characterizing how close an object is to its ideal. For example, when machining a part we use tolerances to specify the maximum allowed deviation of the actual shape of the machined part from its ideal given by a dimensional drawing or a CAD model.

However, in the case of ancient Egyptian vases, we do not have such a priori design documents, which we can use for comparison. Therefore we must abandon the idea of tolerances and define another quality metric.

Since the model slices are approximately circular, I decided to evaluate the quality of each slice’s fit to a perfect circle in the least squares sense. The result of the fit is the best-fit radius R, the root mean square error (RMSE), and the best-fit center (x, y), from which I compute the centering error dR =√ (x²+y²) – Fig. 5.

Small values of RMSE and dR mean that the slice is ‘very circular’ and ‘well centered’ on the origin, while large values of RMSE and dR mean that the slice has ‘poor circularity’ and is ‘poorly centered’ on the origin.

Combining the results for all slices, we can compute the average RMSE (<RMSE>) and the average dR (<dR>) for the inner and outer surfaces of a model. Then we can define the quality metric M as follows: M = <RMSE_outer> + <RMSE_inner> + <dR_outer> + <dR_inner>.

Slice Filtering

Because slices cutting through the handles are non-circular we ought to exclude them from the analysis. Other slices to be excluded are the slices through badly worn top and bottom surfaces and the slices that are too oblique. For example, slices through the lower portions of the rounded-bottom vases and top and bottom surfaces of the lip of a vase are ‘noisy’ because the oblique slicing angle disproportionally magnifies surface imperfections (this effect is similar to the elongation of shadows that occurs when objects are illuminated from low angles).

Through trial and error I implemented the following slice filtering algorithm: all slices with RMSE greater than twice the <RMSE> are eliminated; this procedure is repeated 3 times to eliminate the noisiest outliers.

An alternative approach is to include only the slices from the bottom portion of a vase up until the handle. Such a procedure requires an operator input to specify the value of z where a handle begins. This is not ideal since manual intervention introduces an operator-dependent degree of subjectivity into the analysis. For the rounded-bottom vessels, this approach requires even more operator input to specify the value of z, below which the slices must be excluded due to the excessively oblique slicing angle.

As such, I settled on the first approach because it allows for complete automation and does not require human participation, which could lead to undesirable selection effects.

Calibration

To ensure the validity of measurements, EMS, Inc. has calibrated the CT scanner by scanning a ruby T-stylus sphericity set (a NIST traceable metrological standard), which contained a small bead with the radius R = 1.99820 mm – Fig. 6.

I analyzed the resulting STL file using the same MatLab code I used to analyze the stone vases and obtained the following results: <RMSE> = 0.0001”, <dR> = 0.0003”. The measured radius of the standard was R_max = 2.0026 mm, which corresponds to the error of 0.0044 mm or 0.00017”.

Thus, I was able to establish that my analysis of the CT scans is accurate to within 0.2 thousandths of an inch.

Sample Results

Below are the examples of the analysis of the two stone vases from Matt Beall’s collection characterized by the vastly different quality metrics: the vase ‘V18’ on the left is a lot more ‘circular’ and ‘well centered’ than the vase ‘V8’ on the right – Fig. 7-16.

For convenience, the results of the analysis of the two vases are summarized in Table 1. The vase ‘V18’ is 20 to 30 times more precise than the vase ‘V8’ in terms of <RMSE> and <dR>.

The ‘Total Error’ in the last column of the table, which is a sum of all metrics listed in the table, paints an even starker picture since it amplifies the gap between the ‘precise’ and the ‘imprecise’ artifacts.

Two Classes of Objects

A scatter plot of <RMSE> vs <dR> for all objects in Matt Beall’s collection is shown in Fig. 17 and a bar blot of the quality metric M is shown in Fig. 18.

Figures 17 and 18 show that objects in Matt Beall’s collection fall into two distinct categories. This classification is particularly well evident in the clumping of data points in Fig. 17.

As such we can use this dichotomy to classify the objects as follows:

• Precise class: <RMSE_inner> less than 0.005” and <dR_inner> less than 0.005”;
• Imprecise class: <RMSE_inner> greater than 0.005” or <dR_inner>  greater than 0.005”.

Similar classification can be accomplished using the quality metric:

• Precise class: M < 25 thousandth of an inch;
• Imprecise class: M > 25 thousandth of an inch.

Parabolic Fit

The outer and the inner surfaces of the ‘precise’ vases exhibit excellent fit to a parabola with RMS error on the order of 0.005”  – Fig. 19.

The parabolic fit is evaluated for the slice points beginning from the narrowest slice at the bottom of the vase and ending with the widest slice.

For comparison, parabolic fits of the outer surfaces of the ‘imprecise’ artifacts are on the order of 0.030”.

Modern Artifacts

At the moment of publication of this blog post, the data on the modern artifacts was not yet available. I am going to update this post as soon as I receive the data from EMS, Inc.

Conclusion

It is nothing short of astonishing to find that the most ‘precise’ vessel in Matt Beall’s collection is characterized by the circularity error <RMSE> = 0.6 thousands of an inch (15 microns) and the centering error <dR> = 0.3 thousandths of an inch (7 microns). These errors approach the limitations of my analysis algorithm (which is 0.2 thousandths of an inch).

The ‘precise’ class average errors are as follows:

• <<RMSE>> = 1.3 thousandths of an inch (0.03 mm);
• <<dR>> = 1.3 thousandths of an inch (0.03 mm).

 Such surprising precision indicates a highly advanced manufacturing technique.

On the other hand, the ‘imprecise’ vessels in Matt Beall’s collection are characterized by the following class averages:

• <RMSE> = 12 thousandths of an inch (0.3 mm);
• <<dR>> = 23 thousandths of an inch (0.5 mm).

This manufacturing quality is indicative of a much less advanced manufacturing technique.

In other words, there is a huge gap in quality between the two classes of artifacts, which spans more than an order of magnitude in precision.

Given these results, I speculate that the ‘precise’ vases were machined using advanced tools since the lathe marks are clearly visible on the inner surfaces of the vases where they were not polished away completely – Fig. 20.

The ‘precise’ vases are strikingly symmetric and beautiful – Fig. 21-22. 

The 3D models of  CAT scans of the ‘precise’ vases look like CAD models: their symmetry is so perfect that it is difficult to spot any surface profile variability when these models are rotated – Fig. 23-24.

At the same time, the ‘imprecise’ vases appear noticeably imperfect to the naked eye, their lopsidedness made particularly clear by the CAT scan  – Fig. 25.

Modern Recreation

An effort to recreate a manual stone vase making process using only the primitive tools that were used by the ancient Egyptians was undertaken by Olga Vdovina in collaboration with the antropogenez.ru. The optical scan of the two resulting stone vases are shown on Fig. 26 and 27.

I must point out that Olga Vdovina made the vase ‘O1’ using a plastic rotary table supported by a ball bearing to control the outer surface accuracy through rotation by painting the elevated spots with a sharpie marker – Fig. 28.

The use of modern technology in making of the ‘O2’ vase represented a significant deviation from the initial objective of antropogenez.ru to use only the tools available to the ancient Egyptians. Nevertheless, I have included the 3D optical scans of both ‘O1’ and ‘O2’ into my analysis and plotted the resulting data points in red on Figures 17 and 18. As one can see, the both vases ‘O1’ and ‘O2’ are consistent in quality with the ‘imprecise’ class although the vase ‘O1’ lies somewhat close to the ‘precise’ class because the RMSE of the outer surface circularity was only 5 thousandths of an inch. This impressive circularity was achieved due to the use of the ball-bearing based rotary table, which is a contemporary piece of technology that was not available to the ancient Egyptians.

Nevertheless, what Olga Vdovina’s work proves is that one can achieve a remarkable degree of precision when working the stone manually. However, such precision is impossible without ‘high technology’ such as precision support and precision rotation.

Next Steps

An effort is underway to study these artifacts using electron microscopy (EDS) and nuclear science, e.g. using high-resolution germanium gamma spectroscopy, x-ray luminescence (XRF), and hopefully neutron activation.