Scroll Length?

[William Schryver]

Wade:

Excellent. Are the next two by chance:

9.96

9.78

As I have stated several times in the past, most recently in Post 17 of this thread: “… in the case of the scroll of Hor, I argue that it is only possible to obtain a reliable measurement for the first three windings …”. I am personally convinced that the irregularity of the lacunae beyond the third winding precludes any possibility of confidently obtaining further measurements, regardless of the methodology employed.

I noted this a couple years ago, but I proposed, in conversation with John Gee, and completely independent of any knowledge of what Cook/Smith would do, a methodology remarkably similar to what Cook/Smith chose to attempt to determine the winding lengths. My proposal entailed processing a converted version of the digital scan files of the papyri. As a computer programmer, I thought it would be possible to write a program to analyze the data from the lacunae in order to determine winding lengths. But ultimately, I became convinced that the irregularity of the lacunae, even in the area of the first three windings, was such that only human reasoning and judgment could be relied upon to make the necessary decisions. Although I still feel that way, I intend to explore the possibilities of getting the digital scan file processed in order to produce a data file susceptible to analysis.

The discrepancy may be accounted for naturally by considering the scroll as being rolled up asymmetrically, into a slight conical shape.

There is no evidence that the scroll of Hor was rolled into a slight conical shape. To the contrary, the evidence shows it was orginally wound quite roundly (or in a constant elliptical shape). The Egyptians were quite adept at rolling papyrus scrolls into extremely compact rolls, as evidenced by the scroll John Gee saw unrolled in Toronto (the one Mortal Man doesn’t believe in). The measurements from the scroll of Hor indicate that it was likely quite similar to this scroll from the British Museum:

ETA: Added phrase concerning the scroll being originally wound in a elliptical shape. The process used to roll scrolls would have likely produced a somewhat elliptical shape, as seen in the photos above.

Edited November 10, 2011 by William Schryver

[wenglund]

Wade: As I have stated several times in the past, most recently in Post 17 of this thread: “… in the case of the scroll of Hor, I argue that it is only possible to obtain a reliable measurement for the first three windings …”. I am personally convinced that the irregularity of the lacunae beyond the third winding precludes any possibility of confidently obtaining further measurements, regardless of the methodology employed. I noted this a couple years ago, but I proposed, in conversation with John Gee, and completely independent of any knowledge of what Cook/Smith would do, a methodology remarkably similar to what Cook/Smith chose to attempt to determine the winding lengths. My proposal entailed processing a converted version of the digital scan files of the papyri. As a computer programmer, I thought it would be possible to write a program to analyze the data from the lacunae in order to determine winding lengths. But ultimately, I became convinced that the irregularity of the lacunae, even in the area of the first three windings, was such that only human reasoning and judgment could be relied upon to make the necessary decisions. Although I still feel that way, I intend to explore the possibilities of getting the digital scan file processed in order to produce a data file susceptible to analysis.

I wonder if a CAD program could be used to produce a 3D visual representation of the respective theories/calculations?

I can't be sure, but from what I mentally visualize from Andrews calculations (what little I understand of them), given the lengths of the several known outer windings, there seems to be too much of a gap between the windings--to me they don't represent what would be expected given the elastic tension of the papyri (the outward force that causes a rolled up document to move towards its flattened state, causing the outer windings of the rolled document to be compressed together against a fixed outer circumference). To illustrate what I mean, role up a sheet of typing paper and slide it into a 2 inch diameter PVC pipe and notice what happens. To me, the notion that a rolled up papyri would form an Archimedian spiral with "white space" (gaps between windings) that is uniformly or on average more than three times the average thickness of the papyrus, seems questionable to me. But, what do I know. I had to take "bonehead" math courses to complete my undergrad degree in Public Policy, and so I don't have a stellar background in mathematics.

There is no evidence that the scroll of Hor was rolled into a slight conical shape. To the contrary, the evidence shows it was orginally wound quite roundly (or in a constant elliptical shape). The Egyptians were quite adept at rolling papyrus scrolls into extremely compact rolls, as evidenced by the scroll John Gee saw unrolled in Toronto (the one Mortal Man doesn’t believe in).

What I am talking about amounts to a diameter difference of only ~222 micron (the thickness of typing paper) between the lower and upper edges of the scroll. Given all the hypothesized windings of the roll, I am not sure it is unreasonable to except such nominal (conical) asymmetry at the outer edge of the scroll, even given the most talented of human hands rolling it. Again, what do I know.

Thanks, -Wade Englund-

Edited November 10, 2011 by wenglund

[William Schryver]

What I am talking about amounts to a diameter difference of only ~222 micron (the thickness of typing paper) ...

Normal printer/copier paper is actually only ~100 microns thick.

Paper in a fine set of scriptures is about ~50 microns thick.

[wenglund]

Normal printer/copier paper is actually only ~100 microns thick.

I got the information from an online source so it may not be reliable. But even still, I hope you get my point. I am talking about a difference in diameters equivent to two sheets of printer paper.

Thanks, -Wade Englund-

Edited November 10, 2011 by wenglund

[Mortal Man]

I’ll respond to Zeta’s questions first and the rest as I find the time.

2. How was equation (7) derived?

For simplicity, let n_b=0 for the time being. Also recognize that n is a continuous variable and W_n is a continuous function.

The correlation function, equation (5), returns the average winding length for the segment of the papyrus that you feed into it. From the mean value theorem, the average winding length of the segment is the same as the local winding length at the angular center of the segment; this is called W_n.

Now suppose that you feed exactly two windings into the correlation function (e.g., the blue and red windings in the spiral figure). Then n goes from 0 to 2 for the segment and the angular center (the winding number we are after) is at n=1. The best fit for the segment will occur when the curve is shifted by the average length of the two windings, such that the windings are completely overlapped. For completely overlapped windings, ξ=W_n; i.e., the length of the overlapping region is the mean winding length for the segment. Setting ξ=W_n in equation (7) yields n=1. So equation (7) tells us that the winding length coming out of equation (5) is W₁.

Now suppose that only slightly more than one winding is fed into equation (5). Then n for the segment goes from 0 to slightly greater than 1. The best fit will occur when only a tiny segment of the curve overlaps, such that ξ will be nearly 0. Setting ξ=0 in equation (7) yields n=0.5, so equation (7) tells us that the winding length coming out of equation (5) corresponds to W_0.5.

Since W_n is a linear function of n, equation (7) returns the correct winding number for any segment between one and two windings.

Also, some of the terminology is hard to follow. For example, the sentence after equation (7) has n=0. But then in the next sentence you derive n=0.8983. So where is it you use n=0 in equation (7) to get n=0.8983? (If only in the derivation of \zeta, then that was an unfortunate use of n twice.)

Look at Figure 9 on page 17, which summarizes the results. We define the rightmost (outer) edge of the papyrus scroll as n=0 (e.g., the outer end of the blue winding). We call the rightmost edge of each segment that we feed into the correlation, “location b”. The value of n at location b is n_b. So, for the segment containing the outer edge of the scroll, we set n_b=0 in equation (5). The purpose of equation (5) is to solve for n; i.e., n is an output not an input.

As a side note, bear in mind that x is the linear distance going left to right along the scroll; whereas, n is the angular distance going right to left.

3. Do we know that the scroll was wound from the right edge to the left?

The scroll was wound left to right, such that Facsimile 1 was on the very outer winding facing inward; e.g., the vignette would correspond to the blue winding.

4. I'm not sure I followed the discussion at the bottom of page 15. Are you assuming that JS 1.2 follows directly after JS 1.1?

Yes, we know they were originally joined because the pieces fit together.

(As a mathematician, it is hard for me to follow some of the formulas because the variables are being using in multiple different contexts. In the future, for example, it might be better to write n_{b1} and n_{b2}, where b1 and b2 refer to the ends of JS 1.1 and JS 1.2 respectively.)

We use “a” and “b” to denote the left and right ends, respectively, of each segment that we feed into the correlation function. Since each segment is analyzed independently, the notation is unambiguous and nontechnical readers are not spammed with extra notation.

In particular, I don't understand how you get n_{b2} by taking 18.72/10.42.

For the first segment, n_b is trivially obtained because we know the right edge is at n=0. The next segment to the left begins where the first segment ends; therefore, in order to obtain its starting winding number (n_b) we have to know the winding number where the first segment ended (n_a). The number of windings in a given segment is the length of the segment divided by the average winding length in the segment. The total length of the first analyzed segment is 18.72. The average winding length in this segment is 10.42; hence, the first segment contains 18.72/10.42=1.797 windings. So n_a=1.797 for the left edge of the first segment and n_b=1.797 for the right edge of the second segment.

5. Using the top edge and bottom edge, the derived lengths of the first winding are (respectively) 10.42cm and 10.49cm. There is a discrepancy of 0.05cm. I imagine this is not unusual, but can anyone confirm that this sort of error is natural?

This is not an error. The winding number, n, is defined independently for the top and bottom; i.e., if you draw a vertical line down from n=0 on the top edge, you don’t hit n=0 on the bottom edge. IOW, we define one set of winding numbers for the top and another set for the bottom. There is no reason to use the same set for both, since the top and bottom are independent curves.

6. Page 22, why is the winding length on the top of JS 1.2 anomalously large, as opposed to the winding length on the bottom being anomalously small (especially since the bottom edge admittedly has more damage)?

The winding length at the top of pJS 1.2 is anomalously large because of secondary damage to the major lacuna. This results in the autocorrelation curve shifting an extra amount to the right in order to “fill in” the damaged area. The bottom edge does not have this problem and is not anomalously small; i.e., it exhibits consistency with the other windings.

7. Using JS 1.3, we have two consecutive windings. Isn't that sufficient to tell us how much material could appear later? (Modulo possibilities like a tight inner wind, followed by a looser outer wind.)

Yes it is, which is why we analyzed it both top and bottom.

The best answer is obtained by considering all of the data, which is why we analyzed all of the extant papyrus. There is enough papyrus available in pJS 1.1, 1.2 & 1.3 to obtain four independent estimates of the missing length. Three of our four estimates agree with each other. The fourth is an outlier because of the extra damage to the top edge of pJS 1.2. Since the damage is to the right edge of the lacuna, which causes an extra shift to the right, we’d expect the correlation function to return a high value, and indeed it does (causing the scroll to appear longer). If the secondary damage had been to the left side of the lacuna, there would have been an undershift and the correlation function would have returned a low value (causing the scroll to appear shorter).

Okay, it is bed time. My brain has stopped trying to parse the math so I have to quit. If I missed something or skipped something while reading, please take mercy.

Thanks for your questions Zeta. I’m glad you’re taking the time to go through it carefully. If I can get your questions cleared up then perhaps you can help explain the method to others.

[Mortal Man]

My real question was: have you tried your methodology on a papyrus of known length to see if it actually works?

The fundamental principle of what you are calling "your methodology" has, in fact, been applied by Egyptologists to numerous ancient scrolls. For example, Dr. Stegemann states that "one can measure the arithmetic progression [of winding lengths] with exactitude in all of the larger Qumran scrolls," (p. 195) and he gives several examples. Thus, our assumption of a regular arithmetic progression has been validated on actual scrolls of known length. We are using essentially the same formula everybody else is using. This formula predicts the arithmetic progression of inner winding lengths, given two or more measured outer windings. So, for example, we can use it on pJS 1.1 & 1.2 to predict the windings in pJS 1.3. We have done this and it works extremely well.

The only difference in our approach is that we've introduced a more accurate and objective method for measuring the winding lengths.

On a completely irrelevant side note, Stegemann indicates on p. 196 that larger scrolls tend to be wound more tightly, whereas "shorter scrolls, which had a length of only about 1.5 or 2 m ... were not as tightly wrapped as the larger scrolls." Thus even if physical thickness were relevant to the length estimate (which it is not and I'll soon spell out why) it would tend to overestimate the length of shorter scrolls.

I was really boring

Freudian slip?

[William Schryver]

We are using essentially the same formula everybody else is using. This formula predicts the arithmetic progression of inner winding lengths, given two or more measured outer windings.

I doubt very much that there is anything inherently wrong with your formula that calculates the length of the scroll based on the outer winding lengths. The issue is how you go about obtaining the values for those outer winding lengths, and whether or not that methodology is reliable. Based solely on what you report as the winding lengths, I am personally convinced that your methodology is flawed. It is returning winding length values that do not accord with what can be observed, with the human eye and the ruler tool in Photoshop.

That you even attempt to report any winding length values beyond the third one is, in my judgment, indicative of the fact that your methodology cannot be reliable. It is my considered opinion that JSP X is so damaged and irregular as to make it virtually impossible to reliably obtain the length of any winding, one of whose anchor points lies within its boundaries.

Edited November 10, 2011 by William Schryver

[William Schryver]

/snip comment about Bill Hamblin being boring /

It is interesting that you criticize Hamblin for finding your paper "boring." It is also interesting that you seem to be using Bill Smith's name in vain, as one who (so you say) has affirmed its findings. Bill's (Smith) response to me concerning your paper was, and I quote, "I found their argument about scroll length obtuse and said so."

I have now had two other people report the same thing to me in the past week, after reading your paper. I'm beginning to wonder if you hoped to hide the potential flaws of your methodology in the thick fog of the paper's impenetrability.

Edited November 10, 2011 by William Schryver

[Chris Smith]

William,

Our methodology doesn't use "anchor points" or "measurement points". It shifts the papyrus's edge function along a horizontal axis, and finds the shifting distance that gives the highest mathematical correlation between the shapes of the shifted and unshifted functions. You don't even have to read our paper to understand this; just look at the pictures. See Figures 6, 8, 11, 13, 15, 18, 21, and 23. What our method does, in effect, is simultaneously considers all possible measurement points along both the top and bottom edges of the papyrus.

Furthermore, it's fairly easy to visually confirm the accuracy of our results by manually measuring the distance between obvious anchor points and comparing them to our findings, as we illustrate in Figures 25 and 26.

Peace,

-Chris

Edited November 10, 2011 by Chris Smith

[William Schryver]

William,

Our methodology doesn't use "anchor points" or "measurement points". It shifts the papyrus's edge function along a horizontal axis ...

I understand what your methodology does, Chris. Don't you understand that, at any given point on this "shifting horizontal axis", there are two points presumed that constitute anchor points for a winding length value?

My point (forgive the pun) is that whenever the "shifting horizontal axis" moves into JSP X, it cannot possibly have accuracy, since that fragment of the scroll is so heavily damaged and the lacunae are so irregular.

Gratefully (from my perspective), there are now several people with the expertise sufficient to describe technically the flaws that I am only able to describe conceptually. I look forward to their critiques of the Cook/Smith paper.

[Chris Smith]

The lacunae on pJS X aren't that irregular. The homologies are actually fairly obvious when you look closely. Furthermore, the results agree for the top and bottom edge functions and can be confirmed by visual examination of obvious measurement points. In short, there's no indication that the irregularity of pJS X compromised our analysis.

[Mortal Man]

I think you may be surprised at how much the questions of thickness/tightness can tell us.

I'll try to make this as plain as I possibly can.

Setting the innermost winding to zero and plugging equation (4) into equation (3) yields the following formula for the missing length:

L_n=[(m-n)W_n²]/[2(W_n-W_m]

where:

n and m are outer (extant) winding numbers (with m>n),

W_n and W_m are the winding lengths associated with these numbers,

L_n is the length of the scroll interior to the n-location.

This formula expresses a fundamental property of an Archimedean spiral. It represents an absolute ontological truth of the Universe. This equation was written by the fiery finger of God into the heart of Creation. It is immutable. Its veracity exists independent of humanity or anyone's opinion of it. Burn it into your psyche. Inscribe it into your phylactery. Paint it in blood on the upper door post of your house and reverence it when ye go in and when ye come out.

Note that the formula contains only two inputs: W_n and W_m. The missing length depends on these two windings and nothing else. The thickness of the papyrus appears nowhere in this formula. There is no place to plug it in. Thickness has nothing to do with the missing length.

By forcing your thickness into the formula, you have effectively altered the winding lengths. You cannot introduce a third input without destroying the formula itself. This is a crime against nature and all existence. You have blasphemed Archimedes by twisting and bending his elegant spiral into a horrific labyrinth. Where once there was beauty and harmony, there is now only corruption and pain. Your altered formula is an atrocity, the monstrous spawn of an unholy man/beast union. You must shun it. Renounce it. Cast it into Mt. Doom and flee for your life before it devours your soul.

I hope that helps clear things up.

Edited November 12, 2011 by Mortal Man

[Mortal Man]

Will appears to be drawing conclusions based on his own measurements rather than on any previous discussion. He takes around a 100 measurements of actual thickness with the various fragments and observes the nature of the separation. He reports less separation as you progress toward the inner portion of the scroll as well as a decrease in the actual thickness. Average thickness for JSP I (1.1) = 185, JSP XI (1.2) = 165, and JSP X (1.3) = 152 (all in microns). If we get less separation and less thickness as we progress inward, doesn't that suggest a stronger tensile strength exerting pressure on the inner windings? And if the actual thickness is around 160 microns but the effective thickness is 700, doesn't that seem like a scenario that isn't exerting much pressure on the windings? I.e., if you have a coiled spring and relax the tension over 300%, does the spring exert that much pressure any more?

This all makes for a fun activity but it has nothing to do with determining the missing length. As Prof. W. V. Smith found out when "someone" (probably Gee) asked him to look into it, determining scroll length from thickness measurements is an ill-posed problem. (Unfortunately, he took his paper describing this approach down from his blog.)

[Mortal Man]

I would like to plot the measurement points you propose on the backdrop of the images of the papyri. I think that might be a rather illustrative thing to do.

I'm not sure what you mean by this. Please elucidate.

You seem to believe that you have removed the “human element” from the act of determining the measurement points for the winding lengths you propose. I am very doubtful of that.

Once the sections to analyze are chosen (actually they were chosen for us by whoever cut up the scroll), it is a purely mechanical process, devoid of all human emotion. There is no love, no hate, no anger, no joy. It is a sterile ascetic procedure.

So it would require a steamroller, huh?

Or a hefty chef with a meat tenderizer and rolling pin.

I am puzzled that you so confidently dismiss Professor Gee’s report of the 3.3 cm diameter (while still rolled/sealed) Ptolemaic-era scroll that unrolled to a length over 6 meters.

I am puzzled that, after several inquiries, you still won't say if 3.3 cm corresponds to the major or minor axis of this (presumably) elliptical scroll.

I am also puzzled that you so confidently dismiss (as you have on several occasions) my report that the actual physical thickness of JSP I, JSP XI, and JSP X averages 167 microns.

Even though this argument is entirely separate from the length dispute, I nevertheless disagree that the sum of the recto and verso layers of the papyrus laminate is less than the typical diameter a single black human hair.

After all, (assuming John Gee and I are not simply lying) these are presumably demonstrable facts.

The actual thickness of the papyrus is a fact which could be obtained by careful measurements by qualified people. However, even if an army of metrologists came to unanimous agreement regarding the thickness, it still wouldn't help in determining the original length of the scroll or improving the accuracy of the formula. The formula does not call for the thickness.

You do agree with me and Professor Gee that the scroll of Hor, when rolled, measured more or less 3.3 cm in diameter. I’m glad we agree on at least that point.

Yes, that is a pleasant point of agreement. Like a feather pillow in an iron maiden.

That said, I agree 100% that the difference in our results is related to the difference in our winding length measurements.

Yup, given that we agree on the outermost winding, this whole argument boils down to W_m. (Of course, I'm talking about a W_m for each section.)

Edited November 11, 2011 by Mortal Man

[Mortal Man]

I am a little confused. In your article you said: "In an Archimedean spiral, the length and radius of each winding (proceeding inward) decreases by a constant amount per revolution. Note that the first (blue) winding is slightly longer than the second (red) winding and that there are twelve windings in total. We could compute the length of each black winding if we knew the lengths of the blue and red windings. Equivalently, we could compute the radius of each black winding if we knew the radii of the blue and red windings, since the distances across the white gaps (differences in radii between successive windings) are all the same.

I bolded where you used integers to refer to the windings, and your measurement reference seems to be the beginning point and end of the winding rather than the angular.ly centered point. Be that as it may, I don't really care what you call the windings or where you place your measurement points. I am just trying to find out what you calculate as the length for what is equivolent in the Hor scroll to the blue and red and first black windings in your Archimedien example--or, if you prefer, the distances between each of the 1st through 4th angularly centered points. Thus far you have given me 10.49cm and 9.74cm, so all I need is the next measurement (I am trying to figure out if there is some consistency or pattern to the decline in length for each successive wrap, which may provide a formula for predicting the lengths of missing wraps and thus the length of the scroll.)

See figures 14 and 24 for a summary of the mean winding lengths and winding numbers for both bottom sections. Note that <W> is the average winding length for the indicated section not the distance marked by the double arrow.

[Mortal Man]

Excellent. Are the next two by chance:

9.96

9.78

Remember, there is a gap of unknown length (which has been estimated but not with sufficient accuracy to be useful) between pJS 1.2 & 1.3. In working around this gap, the progression of windings is interrupted.

[Mortal Man]

I can't be sure, but from what I mentally visualize from Andrews calculations (what little I understand of them), given the lengths of the several known outer windings, there seems to be too much of a gap between the windings--to me they don't represent what would be expected given the elastic tension of the papyri (the outward force that causes a rolled up document to move towards its flattened state, causing the outer windings of the rolled document to be compressed together against a fixed outer circumference). To illustrate what I mean, role up a sheet of typing paper and slide it into a 2 inch diameter PVC pipe and notice what happens.

We can imagine what the papyrus ought to be like or we can measure it directly.

To me, the notion that a rolled up papyri would form an Archimedian spiral with "white space" (gaps between windings) that is uniformly or on average more than three times the average thickness of the papyrus, seems questionable to me.

Don't get too hung up on the white space in the spiral figure; it's just an artifact of the drawing program I used. We only included this figure as a visual aid in discussing certain concepts. It's not meant to actually represent the Hor scroll.

[Zeta-Flux]

Mortal Man,

Thank you for your clarifications and corrections. I think I understand better now what your equations mean. (I might have preferred a different notation, but that is a bit nit-picky.)

If I'm understanding you correctly, using the lacunae on the top and bottom of JSP 1.1, the measurements suggest a first winding length of approximately 10.48cm (or close enough, that you could agree with William on that point). Is that correct?

However, it is the second winding length that you and Chris differ from William on. He suggests 10.30cm and 10.13cm for the next two windings. Your correlation function using the top edge suggests something similar, but your correlation function on the bottom edge yields something significantly smaller. Is that correct?

If so, would you say that this single difference is one of two root causes in the difference in the total length computations, the other being calculations involving JSP X? If not, please clarify. (It seems to me that your equation (7) relies heavily on the computation of T, and this computation relies heavily on the winding lengths. If we agree on winding lengths, we are likely to agree on what happens afterwards.)

[wenglund]

We can imagine what the papyrus ought to be like or we can measure it directly.

Measurements in theory may prove problematic in practice.

Don't get too hung up on the white space in the spiral figure; it's just an artifact of the drawing program I used. We only included this figure as a visual aid in discussing certain concepts. It's not meant to actually represent the Hor scroll.

Really? LOL However the principle may still apply. Have you used your drawing program to plot out an end view of your Hor scroll calculations in spiral form so as to see what I mean?

Thanks, -Wade Englund-

Edited November 11, 2011 by wenglund

[wenglund]

How about this formula for calculating the scroll length, which shows why the distance between each spiral branch (i.e. the "white space" plus the average thickness of the papyri) explains the disparity in scroll length calculations:

Thanks, -Wade Englund-

[Chris Smith]

Wade,

Despite Andrew's reservations, I think it's fair to conceptualize the effective thickness parameter as equivalent to papyrus thickness plus "white space" thickness. But what you have to keep in mind that the scroll may have been an elliptical spiral rather than a circular one, in which case the "white space" would be "thicker" at the ends of the ellipsis than at the sides. In that case, the parameter represents a sort of "average" effective thickness for one revolution.

On a different note, I look forward to seeing how William came up with his numbers. The winding lengths in our paper aren't really comparable to Will's, since our winding numbers aren't integers. But we can get a fairly good idea of the first, second, and third winding lengths from a manual examination of the papyrus, as follows:

Edited November 11, 2011 by Chris Smith

[shalamabobbi]

How about this formula for calculating the scroll length, which shows why the distance between each spiral branch (i.e. the "white space" plus the average thickness of the papyri) explains the disparity in scroll length calculations:

Thanks, -Wade Englund-

http://apps.fasson.c...b/rollLength.do

5 meters Will's data,

1 meter Andrew's data.

Edited November 11, 2011 by shalamabobbi

[Bill Hamblin]

I'm probably missing something here because I don't understand the math. (My choice was calculus or Hebrew.) But it seems obvious to me that a variable in these formulas would be how tightly wound the papyrus was. A loosely wound scroll with a 3cm diameter would have less papyrus than a tightly wound scroll with a 3cm diameter. If the measurements were taken only from the outermost three windings of the scroll--which seems to be the case here with Chris and Andrew's experiment--then it couldn't tell us about the length of the scroll unless we knew how tightly or loosely wound the inner part of the scroll was. Wouldn't a 3cm diameter scroll produce the same measurements on the outermost three rings of the scroll regardless of how tightly wound it was in the interior?

Also, it seems to me that this could be tested empirically, but for some reason Chris and Andrew haven't done it. I guess Gee is trying it. Roll up a bunch of papyrus or paper tightly and run the formulas. Then roll it up loosely and run the formulas with the same diameter of the tight roll. Then see if the formulas accurately predict the length of the papyri.

Edited November 11, 2011 by Bill Hamblin

[shalamabobbi]

A loosely wound scroll with a 3cm diameter would have less papyrus than a tightly wound scroll with a 3cm diameter.

It just calculates the upper bound.

[wenglund]

Wade,

Despite Andrew's reservations, I think it's fair to conceptualize the effective thickness parameter as equivalent to papyrus thickness plus "white space" thickness. But what you have to keep in mind that the scroll may have been an elliptical spiral rather than a circular one, in which case the "white space" would be "thicker" at the ends of the ellipsis than at the sides. In that case, the parameter represents a sort of "average" effective thickness for one revolution.

My calculations assume an original round shape, with the diameter being calculated based on the length or circumference of the outermost ring.

On a different note, I look forward to seeing how William came up with his numbers. The winding lengths in our paper aren't really comparable to Will's, since our winding numbers aren't integers. But we can get a fairly good idea of the first, second, and third winding lengths from a manual examination of the papyrus, as follows:

Like Will, I question your measurement of the second and third winding lengths because it seems way too loose a winding for a scroll given the elastic tension I mentioned earlier.

Thanks, -Wade Englund-