The Book of Abraham and the Great Toilet Paper Test

In light of the recent controversy over whether and how the length of the original scroll of Hor (the source of Facsimile 1) can be determined, I thought I would have a little fun and try my hand at the problem. (For background see here, here, and here.) However, not having any papyri to measure, I decided to use toilet paper as a model. Below I describe my procedure and present my results.

Methods
I did not want to use a complete roll of toilet paper because that would be a lot of work, so I searched my house for a partially used one. Having located a suitable roll, I located the end of the roll and drew a line along the edge to the cardboard center. I knew that I would not be able to draw a precise line because I needed to use a marker, which bled a little. In order to have a better defined mark along the edge, I used a knife to cut along the line. I then gradually unrolled the roll and measured the length between marks (winding length), rounding to the nearest millimeter (0.1 cm). I found that this particular brand of toilet paper could stretch a few millimeters, so I tried to always measure it while it was relaxed but straight.

The last winding was not directly measured; the circumference of the cardboard tube was used instead. This alternate method was used because it was difficult to obtain an accurate measure of the last winding due to the way the toilet paper was glued to the tube. A small length of paper extended beyond the last mark, but was ignored for convenience. Having completed this series of measurements, I rewound the toilet paper. Anyone who has rewound toilet paper knows that it is never the same as before, with the windings being somewhat more loose. The purpose of this was to simulate thicker toilet paper of the same length. I cut and marked the opposite end of the roll, and measured the new winding lengths. This time I was able to measure the last (innermost) winding length directly.

Exterior windings can be used to calculate the maximum length of missing inner roll using the Hoffmann formula. John Gee has given the formula as Z=((E²-6.25)/2S)-E, where Z = the missing remainder of the roll, E = the length of the last existing winding, and S = the average change in winding length. Andrew Cook has given the Hoffmann formula as Z=(E²–6.25)/(2S)–E+S/2. Not having access to Hoffmann's original publication, I can't say which is correct. However, I found that the two formulas agreed to within a centimeter, so for my purposes the difference is irrelevant. Here I report the form given by Gee.

Cook and Smith expressed their formula as L_n = (W_n²-W_N²)/(4piT). Although it looks different, this formula is almost identical to the Hoffmann formula, which is illustrated to the right where I have colored identical terms. Hoffmann uses 6.25 (i.e. 2.5²) assuming that the missing innermost winding would be no less than 2.5 cm. For the toilet paper, I used the actual last winding measurement in its place. T represents the change in radius, while change in winding length (S) can be thought of as circumference. Thus S = 2(pi)T (i.e. circumference = 2(pi)r), and 2S = 4(pi)T. According to Cook,

Our centered convention for the winding numbers and definition of where the missing section begins removed the factor of –E+S/2 from the right-hand side.

I don't quite understand how that works, so I just went ahead and compared the results of the equations in a straightforward manner. I don't know which is most appropriate, but I found that using Cook's expression yielded a remainder of zero on the last winding, while Hoffmann's (via Gee) gave a negative number, which suggests to me that Cook's expression is better.

Results
The results of my two tests are shown below (click for large). Each graph represents the predicted maximum length of roll remaining at each winding based on the two formulas, compared to the actual remaining length. For each test, S (or T) was calculated from either a running average of change in winding length (i.e. average of all previous windings), or as a sliding window of the average of the three previous windings. (Note that the sliding window begins at winding 4; the running average begins at winding 2.)

The Hoffmann and Cook formulas gave nearly identical results, which is not surprising since the equations are essentially the same. Test 1 using a running average S (or T) tended to overestimate the maximum remaining length, but gave fairly consistent results. In contrast, the sliding window of S (or T) gave predictions that often varied from the true remainder, sometimes dramatically.

Results for the running average S (or T) for Test 2 tended to underestimate the maximum remaining length, while the running average gave more accurate predictions. I believe that this can be accounted for by the fact that I rerolled the toilet paper by hand, which resulted in the outer windings being looser than the inner ones. The initially large changes in winding length (S) of the outer windings--as the winding became tighter--had a large impact on the running average S (and T). The sliding window, in contrast, used only local changes in S (or T) and so calculations for inner windings were not affected by the loose outer windings.

In his rebuttal to Gee, Cook wrote:

Another way of determining a scroll’s original length, which involves less math, is to plot the lengths of the extant windings and fit a straight line to the results. The missing windings will reliably lie along the straight line.

I tested this by doing a linear regression (using the spreadsheet trend function) on either the first (outer) 3 or 10 windings. Using only 3 windings yielded a prediction that was reasonably accurate for Test 1, but clearly inaccurate for Test 2. However, in both cases using 10 windings yielded predictions that agreed well with the actual length.

Cook's rebuttal contains the first seven measured winding lengths of the Hor scroll, as well as 12 remaining interpolated winding lengths. I applied the Cook/Smith formula to the first seven windings using the T value reported in their original article. I then compared that to the length given by linear regression based on the same seven windings. Agreement between the two was quite good. Finally, I compared these results to a linear regression based on only the first three winding lengths, which gave the same result to within 3 cm of the other regression.

Note: Spreadsheet is available here

Conclusions

I undertook this toilet paper test primarily as a fun exercise, and its results are only as good as my competence and accuracy, both of which may be called into question. Nevertheless, based on my exercise I conclude the following:

1. The Hoffmann and Cook/Smith formulas are essentially the same and give very similar results. Why exactly Gee found otherwise will remain unknown until he is more specific about his method. Cook has suggested that Gee mistakenly applied Cook and Smith's derived value of T for the Hor scroll to the Royal Ontario Museum, not understanding that T is unique to each scroll and derived from the winding lengths.

2. Although they may be sound in theory, these formulas can give inaccurate results, which I think can partially be attributed to uneven tightness in winding. Whatever the source of error, I think it is wise to treat the results with caution, as my results in some ways mirror those of Gee's for the Royal Ontario Museum scroll.

3. Linear regression based on as few as 10 windings gave reasonably accurate results in both tests. This may represent a better approach and, at the very least, can serve as a cross-check.

4. In the particular case of the Hor scroll, linear regression based on Cook and Smith's measurements of extant winding lengths agrees well with results obtained from using their formula, even when the regression is done using as few as three windings. I therefore think that their conclusion that the missing inner portion of the scroll was 56 cm is correct to within 10 cm. (Cook has revised the length to 51 cm because of a calculation error. My linear regression using seven windings put the missing portion at 60 cm. However, if we assume that winding lengths could be as small as an unrealistic 0.2 cm, the linear regression puts the missing length at 65 cm.)

In summary, although I believe Gee has legitimate reason to be wary of the application of these formulas, Cook and Smith's estimate of the lost portion of the Hor scroll seems correct. What, if anything, this means for the Book of Abraham depends on other assumptions and considerations.

References:
1. John Gee, Some Puzzles from the Joseph Smith Papyri.

2. Andrew Cook and Chris Smith, The Original Length of the Scroll of Hôr (pdf).

3. John Gee, Formulas and Faith.

4. John Gee, Book of Abraham, I Presume.

5. Andrew Cook, Formulas and Facts: A Response to John Gee.

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