Symmetrically Designed Sifrei Torah: A Quantitative Analysis - Hakirah

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220 : Hạkirah, the Flatbush Journal of Jewish Law and Thought Appendix B A Mathematical Analysis of the 9 th Chapter of Hilchos Sefer Torah Thickness of Rambam’s Parchment Rashash, Baba Basra 14b, discusses the difference in the circumference of a one- and two-atzei-chaim S”T. This Appendix expands Rashash’s analysis to calculate the thickness of Rambam’s parchment. This is the one critical piece of information Rambam left out from his presentation in the 9 th chapter of Hilchos Sefer Torah on how he wrote a S”T that had equal circumference and height. The analysis is divided into several different cases, with each case building on the results of the previous one. Case 1—No Roller Assume n identical pieces of parchment, ‏,קלף of height h, width w, and thickness t (see diagram below) that are stitched together along the length of the pages and, without the use of an etz chaim, are tightly rolled from one end to the other with negligible space between successive layers of parchment. The surface area at the top of the roll is formed by the thickness and the width of the sheets (not length). t -------w ---------- If r is the radius of the cylinder formed by the rolled parchments, the surface area of the top of the rolled parchment is:
Symmetrically Designed Sifrei Torah: A Quantitative Analysis : 221 Surface Area = π r 2 = nwt Solving for the radius of the circular top of the roll we get: (1) r = √nwt/π and the Circumference of the roll and cylinder are given by (2) Circumference = 2√nwtπ. Case 2—One Roller 74 If one end of the parchment is wrapped around a central cylindrical Etz Chaim that has a radius r etz , then the surface area on top of the Etz is π r etz 2 , and this area must be added to the area of the parchment to arrive at the total surface area of the S”T, i.e.: Surface Area = π r 2 = nwt + π r etz 2 , and (1) and (2) become: 74 We assume in this analysis that when the Gemara and Rambam prefer that the Circumference of the S”T equal its height, it includes the contribution of the Atzei Chaim to the circumference. Rambam’s own language would seem to disagree, i.e. ט:ג כיצד יתכוון אדם,‏ עד שייעשה הספר שיהיה אורכו כהיקפו:‏ מתחיל ומרבע העורות בשווה,‏ ועושה רוחב כל עור מהן שישה טפחים,‏ קצב אחד לכול.‏ ואחר כך גולל העורות,‏ ועושה מהן כרך אחד,‏ מהודק יפה יפה;‏ ומוסיף בעורות ומהדק,‏ עד שייעשה היקף הכרך שישה טפחים,‏ שהוא רוחב העור.‏ ומודד בחוט של שני,‏ שמקיפו על הכרך.‏ with no mention of an etz chaim. However, Rambam here is offering a suggestion that is to be used only for approximation purposes. As we will show later, Rambam’s atzei chaim were considerably under an etzba in radius and contributed little to the actual surface area of the S”T. Thus, preliminary calculations could be done that completely ignored the atzei chaim. In the final analysis, however, it is the outside appearance of symmetry when the cover is on the S”T that gives the Sefer an esthetically pleasing look. If the circumference of the rollers was not meant to be included in this final measure, then what exactly is the significance of the symmetry that makes it so desirable?
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Page 3 and 4: Analysis Symmetrically Designed Sif
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Page 37 and 38: עח:‏ Symmetrically Designed Sif
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Symmetrically Designed Sifrei Torah: A Quantitative Analysis - Hakirah

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