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Solving the Problem of pi and Solomon’s Sea

Solomon Dedicates the Temple at Jerusalem, James Tissot Date: c.1896 - c.1902

Also titled “Appreciating the True Flavor of Solomon’s Pi”

Why does the biblical text describe a mathematical fallacy equating pi to be 3? Why have so many solutions been offered but none adequately resolving the issue?

TLDR; Without manipulating the text, the true value of pi used by the engineers was 3.125 which is equivalent to their most accurate contemporaneous knowledge. What is fascinating is how that value of pi derives naturally from the text as well as why so many others failed to understand it. What follows is a detailed explanation.

The book of Kings begins with a sprawling description of the extravagant contents of the Solomonic Temple. Meticulous detail is provided for some minute cosmetic components of the temple yet the text is cryptically vague about other more critical elements. Despite this dichotomy, there is a perplexing structure that is both meticulously detailed yet objectively irrational. The notorious inaccuracy of Solomon’s Sea persists as a vexing contradiction to both reason and biblical inerrancy, but a solution exists.

The Kiyyor was a basin that provided water for ritual cleaning to the priests in the Mishkan (Tabernacle, Shmot 30:18-21). Despite its pragmatism, it lacked the majestic grandeur appropriate for the temple Solomon imagined so he designed a grandiose alternative that would hold the equivalence of numerous Kiyyor volumes. This impressive reservoir rested atop 12 copper bulls but its dimensions are superficially paradoxical. Designed as a hemisphere, its diameter was 10 cubits and its circumference was 30 cubits (Kings I 7:23). Given pi as the absolute ratio between a circle’s circumference and diameter, even a rudimentary understanding of geometry would immediately recognize the significant error in the dimensions. Either the circumference must have been around 31.4 cubits or the diameter had to be around 9.5 cubits. An error of ~5% is significant whether one considers the text to be inerrant or simply an accounting of construction. Furthermore, if this was the mathematical understanding of the text for pi then it places the text embarrassingly less sophisticated than contemporaneous societies, even those preceding the Temple’s construction.

Many have tried to rationalize the flaw as merely an approximation. The Talmud itself is content with an approximation of pi equal to 3 (Eruvin 14a). However, in this case, this explanation is intellectually dishonest. Excusing the issue as an approximation is feasible for a pragmatic text such as the Talmud, but not when considering meticulous historical documentation, all the more so biblical inerrancy. For the religiously devout every word of the bible has infinite value and therefore must be intentional. After all, the text could simply have left out one of the dimensions and the reader could have calculated for themselves the remaining dimension using their contemporaneous appreciation for pi. With this simple change the text would have been accurate to every reader at every stage of societal math evolution, as well as a more succinct text. Yet the text included both dimensions, cornering itself and demanding an explanation from the intellectually honest reader. Even a secular reader feels pangs of curiosity when considering all the effort employed to describe this lavish construction only to have it overshadowed by a mathematical error known by any credible engineer of that day. Many have proposed solutions to this quandary of varying complexity and veracity. Ultimately none of them are intellectually satisfying and more often require a significant amount of imagination to sustain them [1].

The solutions deriving from the religious community are contrived but suffer an additional conundrum arising from the religious assumption of biblical inerrancy. Inaccurate mathematical constants within the text, such as pi, appear archaic to later readers requiring devotees to rationalize how their omniscient deity is less sophisticated than mere mortals. Further complicating the issue, many religious personalities will argue their divine texts are repositories of all worldly knowledge therein obligating themselves to contain accurate depictions of all mathematical and scientific knowledge (e.g. Avot 5:22). Endorsing this position necessarily invokes a contradiction with the text. Attempts to resolve these conflicts result in awkward solutions replete with logical fallacies and intellectual dishonesty.

Most rabbinic solutions explain the inaccuracy as an approximation. Approximations are a norm in biblical descriptions of large numbers: census numbers, quantity of spoils, etc. While superficially compelling, the comparison is incorrect. First, the text generally reserves approximations for large numbers only, not relevant in this case of 2 digits. Furthermore, when approximating the large quantities of circumstantial items, e.g. census, the reader cannot verify the data approximated and its accuracy is of no consequence to universal truth. In contradistinction to an objective mathematical fact that is easily contradicted by a reader and necessarily complicates the material reality of the design in question. Furthermore, if the goal of approximating as per the Talmud is to err on the side of caution, this argument would be backwards for a construction project, as it is better to assume greater in such cases. Approximating pi as 3 would result in structural failure and inadequacy of materials. Additionally, the entire problem could have been avoided by simply providing only one dimension as explained above.

A modern solution celebrated by many, likely dubiously credited to the Vilna Gaon [1], argues that a textual nuance within the verse can be used to provide a ratio of pi far more advanced than any used by a contemporaneous society, although only to a few more decimal places. While this post hoc rationalization is certainly a creative solution it fails to resolve the philosophical dilemma of a divine being that doesn’t seem to know the true value of pi and must still resort to an approximation.  If the divine went to the extent of encoding the text with a value for pi, then why stop short of the actual value?

Secular solutions are equally unsatisfactory. Some of the more radical solutions involve altering the simple geometry of the Sea into a sophisticated shape of scalloping angles in order to explain away the mathematical shortcomings [2]. Surely such a detail would have deserved mention if an author intended to paint a believable tale, especially when specifically providing all the dimensions.

A plausible solution I find compelling derives from a combination of two important assumptions. First, the text is conveying architectural and design details, not attempting to teach pi. The values provided would therefore derive the contemporaneous pragmatic value of pi not true (idealized) pi. The hypothesis that derives from this assumption would be that the value of pi used should only be contemporaneously accurate. Second, the diameter and circumference do not specify internal or external faces of the rim and therefore need not be the same. This nuance is noted by the Talmud (Eruvin 14a) but strangely ignored as negligible. If the diameter was measured from the outside face but the circumference was internal, then the measurements could be feasible depending on the thickness of the brim.  The thickness of the Sea was a tefach (Kings I 7:26), a measure analogous to a closed fist. The hypothesis derived from this assumption is that a plausible value for a tefach should result in a pragmatic value of pi.

Although independently conceived, I am not the first to entertain the possibility of a solution derived from the differences of inner and outer measures. In particular, the mathematician and religious scholar Ralbag proposed a similar solution to the problem. Ultimately he could not result in a value of pi comparable to even the value known in his day due to his erroneous assumptions for the ratio of the tefach [3].

The majority opinion in the Talmud endorsed an amah to tefach ratio of 1:6. The alternative of 1:5 is provided legitimacy by the rabbis, and considering how uncertain they were of the correct ratio, it is reasonable to conclude 1:5 a feasible option (Eruvin 3b-4a, Bava Batra 14a). Furthermore, Talmudic discussion supports the idea that the 1:6 ratio was used for the dimensions of the physical structure, but the items within the temple used a 1:5 ratio (Menahot 97a). Based on these considerations, it is remarkable that serious consideration by most religious thinkers in regards to resolving the problem with Solomon’s Sea, including the Ralbag, never considered the 1:5 ratio.

Using the modern value of pi and assuming the 10 amah diameter extends from one exterior wall to the opposite wall, the outside circumference would be ~31.416 amah. If the circumference of 30 amot was the interior rim then the diameter of the interior rim was ~9.549 amot. The difference between the two diameters being twice the thickness of the rim, ~0.451 amot, establishing the thickness of one wall to be 0.225 amah. If a tefach is assumed to be 1:5 an amah, then a tefach is equal to 0.2 amah, an error of 0.025 amot, 11% from our idealized calculation. If assumed to be 1:6 an amah then the error is even greater. Therefore, if a solution exists it likely uses 1:5 as the ratio.

To support my argument, the only assumptions will be the dimensions provided by the text and the relationship between an amah and tefach from the Talmud of 1:5. There are two circumferences and diameters, inner and outer. The nature of pi necessitates that the outer diameter must be the 10 amah value and the inner circumference 30. If the inner diameter is the 10 amah measure, then both the outer and inner circumference will be greater than 30. Similarly, if the outer circumference is 30 it would make both the inner and outer diameters both smaller than 10. Therefore, the outer diameter must be 10 and the inner circumference must be 30, which is what was assumed by the Talmud (Eruvin 14a). The value of pi for the unknown dimensions will be determined as per pi = C/D, therefore pi will be dependent on the derived second circumference or diameter. Instead of assuming a value of pi and attempting to post hoc rationalize a process to approximate it, this method assumes only the known dimensions provided by the text and Talmudic historical account and then derives the presumed value of pi employed by the engineers.

All these relationships can now be expressed algebraically. Starting with the amah:tefach ratio of 1:5 determines the outer diameter as 50 tefach. This makes the inner diameter 48 tefach. A tefach can now be expressed as equal to the inner circumference divided by 48 pi. Since we know that the thickness of the wall was a tefach this will be half the difference of both diameters. Setting both these tefach equations equal to each other cancels out pi and results in the outer circumference equal to 31.25 amot. This derives pi to be exactly 3.125 for both the inner and outer walls. If theoretically applying the same method using the 1:6 ratio the resulting value of pi approximates 3.103.

The use of pi as 3.125 has an error of 0.5% compared to true pi but has far more important historical value. Contemporaneous knowledge of pi during the construction of the Solomonian Temple was arguable between 3.125 [4] at best and 3.160 (Rhind Mathematical Papyrus). The Babylonians held 3.125 [5] and were a dominant cultural force at that time, exemplified by their use of the most sophisticated value for pi for the era. Had the ancient engineers used 3.125 as their practical value for pi, the interior diameter would have been calculated to be 9.6 rather than an idealized value of 9.549, a difference of 0.051 amot or a fourth of a tefach, in modern terms a little smaller than 1 inch, equivalent to an error of 0.53% for the entire structure. While certainly not perfect, the error was likely manageable and within acceptable ranges even by current construction standards [6,7]

The prior argument satisfies my dilemma with Solomon’s Sea as per the criteria detailed above. In summary, the text details the measurements used by the engineers to design the Sea using 3.125 as the value of pi. This places the engineers at the highest standard of mathematics for that era and within tolerance for competent construction. Using this value of pi and the provided dimensions for the Sea also supports the conclusion that the correct amah to tefach ratio was 1:5. This should clarify that the conundrum of Solomon’s Sea is not in the text but due to erroneous assumptions of the reader.

[1] Elishakoff, Isaac, and Elliot M. Pines. “Do Scripture and Mathematics Agree on the Number π?.” B’Or Ha’Torah 17 (2007): 5767. pg 142

[2] Simoson, Andrew J. “Solomon’s Sea and π.” The College Mathematics Journal 40.1 (2009): 22-32.

[3] Simonson, Shai. “The Mathematics of Levi ben Gershon, the Ralbag.” BDD–Bekhol Derakhekha Daehu: Journal of Torah and Scholarship 10 (2000): 56.).

[4] George M. Hollenback, “Another Example of an Implied Pi Value of 3.125 in Babylonian Mathematics,” Historia Scientiarum, vol. 13, no. 2 (2003).

[5] The Mathematics Teacher, vol. 50 no.2, February 1957, pp 164

[6] Residential Construction Performance Guidelines for Professional Builders & Remodelers, Fourth Edition Builder Books, a Service of the National Association of Home Builders

[7] ANSI/AISC 303-22 An American National Standard Code of Standard Practice for Steel Buildings and Bridges May 9, 2022