Secrets of Pi

This week’s conjoined Torah portions of Vayak’hel and Pekudei conclude the description of the Mishkan’s construction. The Haftarah for Pekudei is a passage from the seventh chapter of I Kings (the exact verses vary by community) describing King Solomon’s construction of the Jerusalem Temple. One of the most breathtaking structures standing in front of the Temple was the “Molten Sea”, a large bathtub for the kohanim to immerse in (as per Rashi and II Chronicles 4:6). The Tanakh describes that the bath was circular, sitting upon a base of twelve oxen statues, and had a total depth of five cubits, roughly ten feet. It held a volume of alpayim bat, “two thousand baths” of water (I Kings 7:26). In fact, the Hebrew bat (בת) is likely the etymology for the English word “bath”!

Illustration of the First Jerusalem Temple, or Solomon’s Temple, with the Molten Sea on the right.

What’s most perplexing in the description is that we are told the diameter of the circular tub was 10 cubits, yet its circumference was 30 cubits. Throughout history, many have pointed out that this seems to be an error! We all know that the ratio of a circle’s circumference to its diameter is π or Pi, which is 3.1415926 (and so on). So, the Tanakh should have said that the diameter was 10 cubits and the circumference was 31 or 31 and a half cubits. How do we solve this puzzle, and what deeper significance does Pi hold in the Torah?

Pi in the Molten Sea

One classic solution to the Molten Sea problem is to factor in the brim of the pool. What the Tanakh describes in saying the circumference was 30 cubits is the inner space of the pool. There would have been a brim measuring a cubit and a half going around the perimeter. This makes sense since the kohanim would need a platform to stand on to get into the pool. The issue with this is that the diameter should also have been given for the inner space, in which case it would be less than 10 cubits. We would expect the Tanakh to stay consistent.

Some say that ancient builders were limited by the tools they had: they definitely had a measuring stick for accurately measuring a straight line, but nothing to accurately measure circumference. So, the ten-cubit diameter could be exactly measured with a stick, but the circular circumference value was only an approximation. The builders did not need to know the circumference anyway, they only needed to work with diameter and flesh out a perfect circle, ten cubits across.

Another solution, often attributed to the Vilna Gaon but actually first appearing in a 1962 paper by Rabbi Max Munk, highlights the true beauty of Torah tradition. The verse in question (I Kings 7:23) has a famous kre u’khtiv—an instance where Scripture is written one way, but read another way. The word kav (קו), “circumference line”, is written strangely with an extra hei (קוה) in this verse. The values of these two words are 106 and 111, and their ratio (111/106) is 1.0471698. This value represents the adjustment that one should calculate to find the true circumference measure (kav) of the Molten Sea. When applying this adjustment (30 cubits x 1.0471698) one gets a Pi value of 3.1415! Thus, the Torah secretly encodes an accurate Pi value within its divine spelling and reading.

For those who don’t like either of the above, the simplest solution is just to conclude that the Molten Sea was not a perfect circle, and may have been slightly oval. Indeed, we know the ancient Babylonian builders used a simpler rounded Pi ratio of 3 instead of the more exact 3.14. The Midrash Tadshe (Ch. 2) implies the same, saying that God wanted a diameter of exactly ten to signify the Ten Sefirot, and a circumference of exactly thirty to represent the Ten Utterances of Creation, Ten Commandments, and Ten Sefirot together, as well as the length of a lunar month. Whatever the case, we find that Pi plays a role in other places in Judaism, and is even tied intricately to a couple of important names of God.

Names of God

One of the earliest names of God given in the Torah (and, according to Exodus 6:3, the only name which the Avot recognized) is Shaddai. This is traditionally understood as sh’dai, “The Sufficient One”, the one and only God necessary, and the One that gives us all that we need. The Talmud (Chagigah 12a) offers a deeper meaning, too, in line with modern scientific findings, that God began Creation with a command for the universe to expand rapidly, until He declared dai! and commanded the “cosmic inflation” to stop. In mystical texts, it is explained that Shaddai is a name representing blessings, fertility, and life cycles. This is why the name Shaddai is often invoked in the Torah specifically when God is giving a blessing (such as in Genesis 17 and 35). Note how all of these things deal with circles and cycles, whether it is the expansion of the circular universe, or a blessing for long life. Fittingly, the gematria of Shaddai (שדי), like the value of circular Pi, is 314!

Jewish mystical texts describe Creation within a process of concentric circles.

Another name of God of huge importance in Kabbalistic texts is actually Pi (פא״י) itself! One of the key places where this name is derived from is the verse in Ashrei, Psalm 145, that God “opens His hands” (פותח את ידך) and satisfies the needs of all living things. (It is a universal Sephardic custom, and in recent times adopted by some Ashkenazim as well, to literally open one’s hands and raise them upward when reciting this verse.) The initials of the phrase spell פא״י. This name is often found on Kabbalistic charms and artworks. The value of פא״י is 91, a number whose significance we’ve explored before, represents the very bridging of Heaven and Earth.

A classic Kabbalistic work, often used as a birkat habayit (blessing for the home) depicting the “Menorah Psalm” and various names of God, including פא״י.

Finally, in ancient times, when decimals were not used, the value of Pi was given as 22/7 (which works out to 3.1428). This is of great significance since the divine Hebrew alphabet has 22 letters, and seven of them are the kefulot, “doubles” that carry two possible sounds (for more, see here). These seven letters themselves correspond to the seven days of the week, as well as other sevens in nature like the seven visible colours of the rainbow and the seven musical notes. Today, some places in the world celebrate a “Pi Day” on the 22nd of July, ie. 22/7.* Most countries, however, celebrate Pi Day on the 14th of March (3/14), which the UN declared “International Day of Mathematics”. Indeed, one of the most amazing things about our universe—which has puzzled scientists for decades—is why it is so precisely mathematical. We live in a perfectly “fine-tuned” universe, and God’s Fingerprints are clearly all over it.

Happy Pi Day!

*The verse in Tanakh which begins the description of the Molten Sea comes at I Kings 7:23, directly following 22 verses in the 7th chapter of the book. Since the numbering of chapters and verses is not entirely Jewish in origin, I wonder if two of the shorter earlier verses should be counted as one, meaning the proper citation would really be I Kings 7:22!

For lots more on the incredible “fine-tuned” and encoded universe, and for a primer on gematria and Jewish numerology, please see the following: