Winding Measurements

[Chris Smith]

"...after his two hands have been clasped to his heart. The Document of Breathing which <Isis> made shall (also) be buried, which is written on both the inside and outside of it, (and wrapped) in royal linen, being placed <under> his left arm near his heart..."

This is a ritual formulation and likely does not reflect the actual disposition of this particular scroll. (Can't remember where I read this. Maybe Ritner?)

[Mortal Man]

You never made any mention of the tube and the idea of the scroll exhibiting the properties of a rolled-up poster until AFTER I explained the implications of your measurements.

I discussed all this back in early April.

http://www.mormonapologetics.org/index.php...mp;p=1208632358

http://www.mormonapologetics.org/index.php...mp;p=1208633588

http://www.mormonapologetics.org/index.php...mp;p=1208633611

This is a ritual formulation and likely does not reflect the actual disposition of this particular scroll. (Can't remember where I read this. Maybe Ritner?)

Do you think you could dig up the reference for this?

[Chris Smith]

I seem to recall that the "royal" references are left over from the Book of the Dead, but I can't track down where I read that at the moment. Do note that the Louvre papyrus uses exactly the same formula you quoted from the Hor document. In the Aug 1968 Dialogue (p. 120) Klaus Baer wrote, "One hopes that the description of the scroll as being inscribed on both sides is not literally true in this case. It would imply that whoever mounted the papyrus on its present backing sacrificed about a third of the text to do so." (As an aside, you'll also notice that on p. 113 Baer shows the size of the "gap" as being closer to what I suggested than to Nibley's or Rhodes' reconstructions.)

[William Schryver]

Mortal Man,

I've engaged in some analysis of your proposed measurements over the course of the past few days. I even had "mormon fool" (whose expertise in terms of the mathematical issues greatly exceeds my own) graphically plot your measurements. Here is the result:

It is interesting to me how precisely your measurements seem to correspond to the assumption with which you commenced (and yes, after re-reading your posts from April, I see that you started with this assumption. Although it wasn't clear to me and others at the time, it can be seen in retrospect). In any case, your numbers seem well-designed (too well designed?) to achieve the desired result. Are you certain that confirmation bias has not played a role in the measurement points you ultimately settled on?

Either way, it all comes down (as I have made clear above) to whose measurements are correct. There is a HUGE discrepancy between your measurements performed on the grafted photos and those performed by Professor Gee on the original papyri themselves. Gee's measurements indicate a long scroll of thin papyrus wound around an inner core; your measurements indicate a short length of thick papyrus loosely rolled up and placed in a tube. It remains to be seen whose measurements will triumph in the end.

[Chris Smith]

It is interesting to me how precisely your measurements seem to correspond to the assumption with which you commenced (and yes, after re-reading your posts from April, I see that you started with this assumption. Although it wasn't clear to me and others at the time, it can be seen in retrospect). In any case, your numbers seem well-designed (too well designed?) to achieve the desired result. Are you certain that confirmation bias has not played a role in the measurement points you ultimately settled on?

The very first analysis I posted on MDB of this issue was based on measurements I took from the papyri before I had a full understanding of how the Hoffmann equation worked or how it would respond to my particular figures. My result was similar to MM's in the sense that the S-value increased significantly as I moved toward the center of the roll. I am thus reasonably confident that confirmation bias is not really the problem. MM's measurements are more refined than my initial ones, and I think his anchor points are very well-chosen. If there is a major error here, we can say with some confidence that it resides in the photographs used rather than in anything MM did. (Although I'm sure he would welcome alternate anchor-point proposals if you think you can find better ones.)

Best,

-Chris

[Mortal Man]

Thanks mormon fool for your nice polar plot.

It is interesting to me how precisely your measurements seem to correspond to the assumption with which you commenced

I didn't commence with any assumptions.

In any case, your numbers seem well-designed (too well designed?) to achieve the desired result.

The only desired result here is to determine the upper bound on the scroll's length from accurate measurements of the windings.

Are you certain that confirmation bias has not played a role in the measurement points you ultimately settled on?

This is a legitimate question and one that should be asked of anyone who reports winding measurements. There is no way to read an analyst's mind and the level of trust that can be established on a message board is paper thin (papyrus thin?) at best. That is why anyone who reports measurements must show the details of his methodology. I have laid out all the details of my measurements so you don't have to read my mind. You can examine what I've done and decide for yourself.

All that being said, there is a better way to do this that would eliminate all subjectivity in the placement of winding end-points. The procedure is as follows:

1. Use an edge-detection algorithm (e.g., wavelets) to map out the top edge of the papyrus. You could probably use Photoshop for this or other digital imaging software.

2. Draw a horizontal baseline through the papyrus and, at each pixel location, plot the distance from this line to the top edge. This will yield y (edge distance) as a function of x (horizontal location).

3. Let y' = y(x+L) be the distance obtained by shifting the edge curve by a horizontal distance L.

4. Plot the correlation coefficient between y & y' as a function of L (shifting distance). The correlation coefficient is defined as r(L) = < yy'> - <y><y'>/[sqrt(<y²> - <y>²)sqrt(<y'²> - <y'>²)], where the angle brackets (<>) denote expected (average) value.

5. The correlation coefficient will be 1.0 when the curves are perfectly aligned (L=0). It will drop down into the range ~-0.2<r<~0.2 as the y' curve is displaced w.r.t. the y curve. As winding N comes into alignment with winding N+1, r will spike up, probably peaking above 0.9. Then it will drop down again until the N+2 winding comes into alignment etc.

6. Once this process is complete, the winding lengths will correspond to the L values where the spikes occur.

This has been on my "To Do" list for a while now but I haven't yet found the time to do it. Perhaps mormon fool would care to take a stab at it. This purely mechanical procedure has the merit of being entirely objective; i.e., other than setting threshold parameters in the edge-detection algorithm, the analyst has no direct control over the process. I doubt however, that this procedure would change the results by much, other than shifting the third digit up or down a bit. The layered images I showed give a visual representation of where the spikes would occur and the eye is pretty good at identifying patterns.

[DonBradley]

The way to settle the question of scroll length would be for the discussants on both sides to agree in advance what would constitute an objective test. Without agreement in advance either side may find an ad hoc way to wiggle around the new evidence, and may be accused of doing so even if they don't. To agree on an objective test--or one that is objective enough--would iron out the wrinkles of post hoc rationalization.

Mortal Man has put his position on the line with what would he would consider an objective and acceptable test.

Will , Chris, and Mormon Fool, what research or analysis that has not already been done would constitute, for you personally, an acceptable test of scroll length?

Don

[Chris Smith]

Will , Chris, and Mormon Fool, what research or analysis that has not already been done would constitute, for you personally, an acceptable test of scroll length?

Hi Don,

I don't think there's a perfect test. We'll always be working with limits, approximations, inferences and the like. But I think that the test we're working with here is very good, and Mortal Man's suggestions for increasing its objectivity are also very good, though it would probably be quite a time-consuming endeavor. (I certainly wouldn't have the know-how to pull it off.) I think the obstacle presently standing in the way of results all can agree on is not the test itself, but the unavailability of mutually-accepted measurements.

-Chris

[Mortal Man]

S factor

There is another possible contributor to the difference in S between X and I-XI. If the tears in the layers were not radially aligned; i.e., if they zig-zagged inwards, then the change in winding length would vary as a function of radius, even if the scroll were a perfect archimedean spiral.

This effect would be very small for the outer layers though; since, for windings with a large radius of curvature, S is sensitive only to the curvature (second derivative) of the tear pathway and not the angle (radial misalignment) of the tear.

Tube

It seems to me that the lack of periodic lacunae along the bottom edge of the papyrus is further evidence that the scroll was stored in a tube and that the lacunae pattern along the top edge was produced suddenly when the scroll was grasped at the bottom and pulled out.

Sanity check

A good way to verify the location of winding end-points is to shift the sequence and see if it lines up with another set of cracks/tears. This is what I did for W1.4-W3.4 to get W1-W3.

Notice how well the crack in Duamutef matches up with the crack in the instructions column. This adds confidence to the end-point identifications.

[DonBradley]

Hi Don,

I don't think there's a perfect test. We'll always be working with limits, approximations, inferences and the like. But I think that the test we're working with here is very good, and Mortal Man's suggestions for increasing its objectivity are also very good, though it would probably be quite a time-consuming endeavor. (I certainly wouldn't have the know-how to pull it off.) I think the obstacle presently standing in the way of results all can agree on is not the test itself, but the unavailability of mutually-accepted measurements.

-Chris

Hi Chris,

Thanks for your answer. I didn't ask what you would see as a perfect test, and I agree this is much too high a standard. Rather, I was curious what you each thought would constitute an acceptable test. It sounds like obtaining measurements everyone could agree were acceptably reliable would be Step One for such a test.

Don

[DonBradley]

No one besides Mortal Man is willing to stake their position by spelling out what would constitute an adequate test?

Don

[Chris Smith]

Hi everyone,

An update. Brent Metcalfe helpfully provided two more sets of photographs with rulers in the margins: one from George D. Smith and one from Wesley P. Walters. Walters' photographs materially agree with the scale of the Improvement Era photos, appearing to verify the accuracy of Mortal Man's analysis. The George D. Smith photos, however, portray the papyrus as dramatically smaller. They show PJS X, for example, as being a full two inches shorter than the Walters photos do. Brent checked his own exact-size photographs for us, and they add further confirmation for the Walters and Improvement Era measurements. The reason for the anomaly of the George D. Smith photos has not been definitively determined, but we suspect upon examining them that they may actually be photographs of photographs. Brent is trying to contact Smith to ask for clarification.

In any case, the Improvement Era scale appears to be more or less doubly confirmed, and the George D. Smith photos certainly provide no support for the apologists' case since they would actually result in a considerably smaller estimate of the length of the missing papyrus.

It is likely that I or Mortal Man will post measurements from the Walters photos in the near future.

-Chris

[Mortal Man]

It appears that

[William Schryver]

It appears that

[William Schryver]

Incidentally, Andrew, I'm not quite sure what you mean to imply by "the thickness method." Perhaps you could elaborate. I view the measurements of the papyrus thickness as serving two distinct purposes:

1) to easily calculate the "upper bound" length of the scroll;

2) as a "control" on the results of calculating the scroll length on the basis of the available winding measurements.

[Mortal Man]

I'll have to submit my winding length measurements to him and see what results his formula returns.

The winding lengths are what his formula returns when you apply the integral over 2pi segments; e.g., if t_o=2pi then L is the length of the innermost winding. The inverse problem; i.e., solving for r, epsilon and b_o given the winding lengths, is ill posed.

The key (and the challenge), as you probably well know, is to produce reliably accurate winding length measurements.

Yes, we are in complete agreement on this. It's all about the winding lengths.

I view the measurements of the papyrus thickness as serving two distinct purposes:

1) to easily calculate the "upper bound" length of the scroll;

2) as a "control" on the results of calculating the scroll length on the basis of the available winding measurements.

The winding lengths lead directly to an upper bound on the scroll length. The thickness measurements set a loose upper bound on the upper bound.

[Mortal Man]

Our paper is now available at the Dialogue website.

During the course of our investigations, we discovered that the rulers in the Improvement Era sepias were not photographed alongside the papyri, as shown in the Feb. 1968 article, but were repositioned later in the production process; i.e., the images are photographs of photographs. As a result, measurements based on the IE rulers do not precisely match the originals. This and other sources of error caused the estimate in the OP to be a bit off.

Although the basic approach outlined in the OP is still sound, our analysis of the originals, described in the paper, is far more objective and robust. We believe it is accurate to within 5 cm.

I now consider the length issue a solved problem.

[Calm]

I now consider the length issue a solved problem.

No doubts at all?

[Mortal Man]

No doubts at all?

The math is exact, except for equation (9) which is a very good approximation. The windings lengths obtained from the correlation analysis exhibit a consistent progression from right to left for all three fragments. Additionally, the top windings are consistent with the bottom windings. For there to be more than 60 cm missing from the scroll, you would have to assume that the papyrus sheet suddenly became extremely thin and was wound impossibly tight.

Suppose you were asked to estimate the original height of the Parthenon's pediment.

You could simply take the tangent of the angle or you could assume that the angle of rise suddenly increased to generate a spike the size of the Empire State Building.

Which approach is more realistic? I suppose you could postulate a really tall statue at the apex but then the analogy breaks down because, unlike the Parthenon, there is not an infinite amount of space inside the Hor scroll to add material; i.e., the available space is circumscribed by the innermost extant winding.

The innermost extant winding of the Hor scroll is 8 cm. This corresponds to a radius of 1.27 cm. Try drawing a circle of radius 1.27 cm and asking yourself, how much papyrus could fit inside?

[Mortal Man]

As promised, our data are now freely available for download from Dialogue's website.

The paper is now "fully published and released for citation."