Daf Ditty Eruvin 14: Pi, the Ratio of God

“Human beings, vegetables, or cosmic dust, we all dance to a mysterious tune intoned in the distance by an invisible player.” – Albert Einstein

“The world isn't just the way it is. It is how we understand it, no? And in understanding something, we bring something to it, no? Doesn't that make life a story?”

"Love is hard to believe, ask any lover. Life is hard to believe, ask any scientist. God is hard to believe, ask any believer... be excessively reasonable and you risk throwing out the universe with the bathwater."

Yann Martel, Life of Pi

1

The mishna continues: If the cross beam is round, one considers it as though it were square. The Gemara asks: Why do I need this clause as well? Similar cases were already taught in the mishna. The Gemara answers: It was necessary to teach the last clause of this section, i.e., the principle that any circle with a circumference of three handbreadths is a handbreadth in diameter.

2

The Gemara asks: From where are these matters, this ratio between circumference and diameter, derived? Rabbi Yoḥanan said that the verse said with regard to King Solomon:

And he made the molten sea of ten cubits from 23 גכ שַׂﬠַיַּו תֶא - ,םָיַּה :קָצוּמ ֶשֶׂﬠ ר אָ בּ מַּ הָ הָ מַּ אָ בּ ר brim to brim, round in compass, and the height thereof וֹתָפְשִּׂמ דַﬠ - וֹתָפְשׂ גָﬠ לֹ ,ביִבָס שֵׁמָחְו הָמַּאָבּ הָמַּאָבּ שֵׁמָחְו ,ביִבָס לֹ גָﬠ וֹתָפְשׂ was five cubits; and a line of thirty cubits did compass מוֹק ,וֹתָ הוקו ו( קְ ָ )ו םיִשְׁ שׁ אָ בּ מַּ ,הָ סָ י בֹ וֹתֹ א וֹתֹ א בֹ סָ י ,הָ מַּ אָ בּ םיִשְׁ שׁ )ו ָ קְ ו( הוקו ,וֹתָ מוֹק .it round about .ביִבָס I Kings 7:23

“And he made a molten sea, ten cubits from the one brim to the other: It was round all about, and its height was five cubits; and a line of thirty cubits did circle it roundabout”

The Gemara asks: But isn’t there its brim that must be taken into account? The diameter of the sea was measured from the inside, and if its circumference was measured from the outside, this ratio is no longer accurate.

3

Rav Pappa said: With regard to its brim, it is written that the brim is as the petals of a lily, as stated in the verse:

And it was a hand-breadth thick; and the brim thereof was 26 וכ וֹיְבָﬠְו ,חַפֶט וֹתָפְשׂוּ הֵשֲׂﬠַמְכּ תַפְשׂ - wrought like the brim of a cup, like the flower of a lily; it held כּ סוֹ רֶפּ חַ שׁוֹשׁ ָ ;ן לַא פְּ ַ םִ י ,תַבּ תבּם ְלא; וֹ ַרפּס .two thousand baths .ליִכָי }פ{ I Kings 7:26

“And it was a handbreadth thick; and its brim was wrought as the brim of a cup, as the petals of a lily; it contained two thousand bat” i.e. The brim was very thin.

The Gemara now calculates how many ritual baths should have been contained in Solomon’s Sea. The volume of the sea was five hundred cubic cubits, as it was ten cubits in length, ten cubits in width, and five cubits in height. The minimum volume of a ritual bath is three cubic cubits. Therefore, three hundred cubic cubits is the volume of a hundred ritual baths, and one hundred and fifty cubic cubits is the volume of another fifty ritual baths. Consequently, four hundred and fifty cubic cubits are enough to contain a hundred and fifty ritual baths; but the volume of the sea was five hundred.

The Gemara answers that there is an error in the calculation: These calculations with regard to the volume of the sea would apply to a square, but the sea fashioned by Solomon was round, and its volume was therefore smaller.

The Gemara continues to ask: Now, how much larger is a square of ten-by-ten cubits than a circle with a diameter of ten cubits? A quarter.

4 Consequently, four hundred cubic cubits of our original calculation must be reduced to three hundred, which is the volume of one hundred ritual baths; and the remaining hundred cubits must be reduced to seventy-five, which is the volume of twenty-five ritual baths. According to this calculation, Solomon’s Sea was the size of only one hundred and twenty- five ritual baths, not one hundred and fifty as stated above.

Tosafos:

תופסות ה"ד אכיאהו והשמ (Tosafos infers that the calculation is precise, but challenges this.)

משמ ע בשחהש ו ן צמ ו םצמ Inference: The calculation (the circumference is three times the diameter) is precise.

5 ןכו 'פב ק' 'בד ב' 'ד( ):די יבג ינש םיחפט ורייתשנש וראב ן םשש רפס הרות חנומ איהש איהש חנומ הרות רפס םשש ן וראב ורייתשנש םיחפט ינש יבג ):די 'ד( ב' 'בד ק' 'פב ןכו הפקיהב השש םיחפט ךירפו יכ ו ן ותיעצמאלד נ ללג שיפנ היל נשמ י םיחפט

Support: Also, in Bava Basra (14b) regarding two Tefachim that remained in the Aron, in which the Sefer Torah rested, for its circumference was six Tefachim, the Gemara asks "since it was rolled to its middle, it is more than two Tefachim!" (This implies that when all is on one roller, it is precisely two Tefachim.)

ןכו רתב יכה ינשמד רפסב הרזעד ותלחתל נ ללג ךירפו יתכא ירת ירתב יכיה ביתי עמשמ עמשמ ביתי יכיה ירתב ירת יתכא ךירפו ללג נ ותלחתל הרזעד רפסב ינשמד יכה רתב ןכו צמד ו םצמ ל ג רמ י

Also after this [there], it answers that Sefer Azarah is rolled to its beginning, and asks "still, how can two [Tefachim] fit in two [Tefachim]?!" (When taking it in or out, it scrapes the sides. Parchment can be compressed somewhat, but it is dishonorable to do so to a Sefer Torah. Also, perhaps letters will be rubbed out!)

אישקו יאד ן ובשחה ן מ קדקוד יפל ימכח :תודמה

Question: The calculation is not precise, according to mathematicians! (Tosfos HaRosh explained that the Gemara asked what is the source to rely on this approximation, even though it is not precise.) Tosafos points out that the ratio of the diameter to the circumference of a circle of 3:1 is not accurate. Tosafos leaves this unresolved.1

Rambam, in his Commentary to the Mishnah, states that the ratio of diameter to circumference is a number which— בוריקב אלא םלועל י גשו אלש —known to be “three and a seventh plus a bit more will never be precise, only approximated.”

[This is a clear indication that Rambam, and the Gemara, knew of pi. Even the approximation he gives of 3 1/7 is less than 13 ten thousandths more than pi.]

Rambam concludes that because the ratio of pi is an irrational number, and at some point, we have no choice other than to approximate, the Gemara regularly uses the round number of 3:1.

This is admittedly a wider approximation than 3 1/7, but the sages deliberately chose to be lenient and not burden the community with measurements of fractions.

The Rosh wonders why the Gemara probes to find the scriptural source for the ratio of 3:1, when this is an empirical fact which can be observed by everyone who does the measurement. He answers that we know that the ratio is not exactly three to one, and that it is slightly greater. The question of the Gemara, therefore, is how do we know that we are allowed to calculate halachic guidelines based upon a number which is approximate?

1 Daf Digest

6

In other words, we are technically allowed to use a round beam to fix a mavoi even if its circumference is exactly three tefachim, as the Mishnah states.

Yet, the measurement of three is actually too small to yield a diameter of a full tefach and using this round beam will lead to having a korah-beam which is too small. How do we know, nevertheless, that we are allowed to rely upon this ratio?

The Gemara answers that the halacha recognizes this approximation and considers measurements based upon it to be valid, based upon a verse found in Melachim 2.

It seems that according to the Rosh, we can rely upon this leniency not only in regard to rabbinic injunctions, but also in reference to Torah laws. 2

Circles and Hexagons

The Gemara relates that a circle that has a circumference of three will have a diameter of one. Based on this ratio, a koreh – crossbeam — that is three tefachim in circumference is valid since it will have a diameter of one tefach.

Tosafos notes that according to mathematicians this ratio is not accurate and the diameter of a circle with a circumference of three will be slightly less than one. Interestingly, Tosafos HaRosh2 explains that the reason the Gemara asks “from where we know these words,” is specifically because the ratio is not precise.

If the ratio was exactly three to one there would be no reason for the Gemara to ask for a source. The Gemara’s inquiry was for the source that indicates that even though the ratio is not precise it is still reliable in Halachah.

Later authorities question the extent to which one may rely upon this ratio. Mishnah Berura writes that one may certainly rely on this ratio when it comes to Rabbinic matters.

Regarding Biblical matters he writes that it may be that this ratio is Halachah L’Moshe M’Sinai and may be relied upon even for Biblical matters.

Chazon Ish writes definitively that this ratio is known from Halachah L’Moshe M’Sinai and may be relied upon even for Biblical matters.

He explains that this ratio is included in the Gemara’s statement that measurements are Halachah L’Moshe M’Sinai.

,who says that perhaps there was a tradition from Sinai to allow us to rely upon this approximation צה י ו ן רעש (See also (#18 372 2 although it is a leniency, even for Torah law. See also Chazon Ish 138:4

7

Sefer Eretz Chaim in the name of his father suggests the following explanation. The ratio of three to one happens to be precise when measuring a hexagon. He further proposes that in the past they did not make perfect circles due to the difficulty involved in forming a perfect circle. The best they could do was to make hexagon shaped objects which are easier to form. Therefore, one could assert that when the Gemara discusses a round koreh it was referring to one that was actually hexagon shaped which contains the exact ratio of three to one.

Rav Mordechai Kornfeld writes:3

THE VALUE OF "PI"

The Gemara quotes the Mishnah (end of 13b) which says that the circumference of a circle is three times greater than its diameter (if the circumference of the circle is 3, then the diameter is 1).

How do we reconcile this statement with the mathematical fact that the ratio of the circumference of a circle to its diameter is slightly more than three (Pi = 3.141592...)?

TOSFOS HA'ROSH explains that the Gemara itself addresses this issue.

The Gemara asks "from where do we learn" that the circumference of a circle is three times greater than the diameter.

Why does the Gemara need a source to teach the ratio of the circumference of a circle to its diameter?

We do not need a verse to teach us a mathematical, observable fact! It must be that the Gemara is asking for the source that teaches us that we may use a slightly inexact value to determine the circumference of a circle.

The Gemara answers that the verse that describes the circumference of the pool that Shlomo ha'Melech constructed (the "Yam Shel Shlomo") as three times its diameter, teaches that for all Halachic purposes we may use the approximate ratio of three to one.

RAMBAM (Perush ha'Mishnayos, Eruvin 1:7);

3 Daf Advancement Forum

8

RAMBAM: Hilchos Tum'as Mes 12:7

Rambam points out that Pi is an irrational number, and "the exact relationship of the diameter to its circumference cannot be known and it is not possible to speak of it... its actual value cannot be perceived." He writes that the value which is commonly used in calculations is 3 1/7 (3.142857...).

The Tanna’im of the Mishnah rounded this number and expressed it in terms of the nearest whole integer (3).

Vilna Ga’on provides a fascinating insight regarding the value of Pi. (Actually, there is no source to substantiate the claim that the Vilna Ga'on said it. The actual source for the insight may be credited to Matityahu ha'Kohen Munk (Frankfurt-London), who published the thought in the journals "Sinai," Tamuz 1962, and "ha'Darom," 1967).

9 In the verse that the Gemara cites as the source for the ratio of the circumference to the diameter:

And he made the molten sea of ten cubits from 23 גכ וַיַּ ַע שׂ ֶאת- ַהיָּ ,ם צוּמ ָ :ק ֶע ֶשׂ ר ָבּ ַא ָמּ ה ה brim to brim, round in compass, and the height ִמ פְשּׂ ָ וֹת ַעד- פְשׂ ָ וֹת ָע לֹג ָס ִב ,בי חְו ָ ֵמ שׁ ָבּ ַא ָמּ ה ה thereof was five cubits; and a line of thirty cubits did מוֹק ָ ,וֹת הוקו קְו( ָ )ו שְׁשׁ ִ םי ָבּ ַא ָמּ ,ה יָ בֹס וֹתֹא וֹתֹא בֹס .compass it round about ָס ִב .בי I Kings 7:23 there is a "Kri" and a "Kesiv" -- a word that is pronounced differently than it is spelled. The word in the verse is written "v'Kaveh" (with the letter "Heh" at the end), but it is pronounced "v'Kav" (with no "Heh" at the end).

The Gematria of the word "Kav" is 106, and the Gematria of the word "Kaveh" is 111. The ratio of the Kesiv (111) to the Kri (106), or 111/106, is approximately 1.0472. This value represents the ratio of the value for Pi to 3 (3.141592.../3, or approximately 1.0472).

Hence, the difference between the actual value of Pi and its practical, Halachic value is expressed by the difference between the Kesiv (the actual, but unread word) and the Kri (the word as we read it) of the verse that discusses the value of Pi!

Rabbi Moshe Taub writes:4

4 WIRED MAGAZINE -3/14/2013

10 It’s Pi Day! And in honor of this year’s celebration I decided to do a bit of historical research. While π—the ratio between a circle’s circumference and its diameter—has long been known and approximated, even since ancient times, only since the Eighteenth Century has it been proven to be an irrational number. Prior to that, various approximations were given, such as even the nice round number of 3.

I was recently reading the Wikipedia page on approximations of π and noticed that it included a note saying that Maimonides—the Jewish physician and scholar who lived nearly 1,000 years ago—seems to have alluded to its irrationality in his writing.

The source took me to the book The Ancient Tradition of Geometric Problems and using Amazon’s helpful Look Inside feature, I traced this putative statement to the commentary of Maimonides on the Mishnah, a collection of Jewish Law which forms part of the Talmud (... Eruvin 1:5)...

The ratio between a diameter of a circle and its distance around is not known. We cannot speak about it precisely…This ratio’s reality cannot be found, but it is known in approximation…we can place an approximation of this as one to three and a seventh…

Maimonides uses the approximation for π of 22/7 which is about 3.14 and is a good approximation. In addition, he does seem to imply that any value is necessarily an approximation and can’t be known precisely...

Nevertheless, these kinds of discussions are certainly intriguing, and it is exciting that scholars of the Middle Ages were perhaps already intuiting that π could never be calculated as a ratio.

Aruch HaShulchan, Orach Chaim 363:22

Rabbi Taub then speculates:

HOW FAR CAN WE TAKE THIS? CAN WE RULE LIKE THIS ROUNDED 'PI' EVEN WHEN IT WOULD LEAD TO KULAH IN BIBLICAL LAW?

11 THIS SEEMS TO BE A DEBATE BETWEEN THE CHOFETZ CHAIM (372:30 WITH #18 IN HIS SHAAR HATZION) AND RAV MOSHE FEINSTEIN

Of course, we cannot ignore our irreverent blogger at My Talmudology!5

The Talmud determined the that value of π is 3. How does it know this?

Because of this verse in the Book of Kings:

And he made the molten sea of ten cubits from brim 23 גכ שַׂﬠַיַּו תֶא - ,םָיַּה :קָצוּמ רֶשֶׂﬠ הָמַּאָבּ הָמַּאָבּ to brim, round in compass, and the height thereof was וֹתָפְשִּׂמ דַﬠ - וֹתָפְשׂ גָﬠ לֹ ,ביִבָס שֵׁמָחְו הָמַּאָבּ הָמַּאָבּ שֵׁמָחְו ,ביִבָס לֹ גָﬠ וֹתָפְשׂ five cubits; and a line of thirty cubits did compass it מוֹק ,וֹתָ הוקו ו( קְ ָ )ו םיִשְׁ שׁ אָ בּ מַּ ,הָ סָ י בֹ וֹתֹ א וֹתֹ א בֹ סָ י ,הָ מַּ אָ בּ םיִשְׁ שׁ )ו ָ קְ ו( הוקו ,וֹתָ מוֹק .round about .ביִבָס

I Kings 7:23

םיכלמ א קרפ ז קוספ גכ קוספ ז קרפ א םיכלמ

5 Mytalmudology.com

12 And he made a molten sea, ten amot from one brim to the other: it was round, and its height was five amot, and a circumference of thirty amot circled it.

So, one of the vessels in the Temple of Solomon was ten amot in diameter and 30 amot in circumference.

Since π is the ratio of the circumference of a circle to its diameter (π=C/d), π in the Book of Kings is 30/10=3. Three - no more and no less.

From this prooftext, today’s page of Talmud teaches a general rule:

אנמ נה י ?ילימ - רמא בר י י ו ח נ ן , רמא ארק : ו שעי תא םיה קצומ . לכ שיש ופקיהב השלש םיחפט שי וב בחר חפט בחר וב שי םיחפט השלש ופקיהב שיש לכ רשע המאב ותפשמ דע ותפש לגע ביבס שמחו המאב ותמוק וקו םישלש המאב בסי ותא ביבס ותא בסי המאב םישלש וקו ותמוק המאב שמחו ביבס לגע ותפש דע ותפשמ המאב רשע

"Whatever circle has a circumference of three tefachim must have a diameter of one tefach."

דומלת ילבב תכסמ ןיבוריע ףד די דומע א דומע די ףד ןיבוריע תכסמ ילבב דומלת

BUT PI IS MORE THAN THREE

However, this value of π =3 is not accurate. It deviates from the true value of π (3.1415...) by about 5%. Tosafot is bothered by this too.

אכיאהו .והשמ עמשמ ובשחהש ן םצמוצמ כו ן ק"פב ב"בד 'ד( ):די יבג ינש םיחפט ורייתשנש ןוראב םשש רפס הרות הרות רפס םשש ןוראב ורייתשנש םיחפט ינש יבג ):די 'ד( ב"בד ק"פב ן ומ חנ איהש הפקיהב השש םיחפט ךירפו יכ ו ן ותיעצמאלד ללגנ שיפנ היל ינשמ םיחפט ןכו רתב יכה ינשמד רפסב הרזעד הרזעד רפסב ינשמד יכה רתב ןכו םיחפט ינשמ היל ותלחתל ללגנ ךירפו יתכא ירת ירתב יכיה ביתי עמשמ םצמוצמד ירמגל אישקו יאד ן ובשחה ן קדקודמ יפל ח מכ י דמה ו תודה ימ

,תופסות יבוריע ן די א די ן יבוריע ,תופסות

Tosafot opens the objection with these words: “But [pi] is a little more [than 3]. Which means that the value [of pi] is rounded down” Tosafot can't find a good answer to this obvious problem, and concludes "this is difficult, because the result [that pi=3] is not precise, as demonstrated by those who understand geometry."

Another example of the approximation of π=3 can be found in Bava Basra 14b, where there is a detailed discussion about the space that was available inside the original Ark of Moses. According to Rebbi Meir, there was a space of was exactly two tefachim available for the Torah scroll written by Moses. That scroll, according to the tradition of Rebbi, had a circumference of six tefachim. Then comes the same general rule we found in today’s page of Talmud:

אבב ארתב י ד , ב ,די את ב

לכ שיש ופיקהב השלש םיחפט שי וב בחור חפט

13 Any circular object with a circumference of three tefachim must have a diameter of one tefach.

The two Torah scrolls, each of a circumference of six tefachim, could fit into a space only two tefachim wide. However, this only works if you round π down to three. But it is more than three, and so in reality, there cannot have been enough room in the Ark of Moses, unless of course, the scrolls had a smaller diameter, or the Ark was larger.

PI IN THE RAMBAM

In his commentary on the Mishnah on which today’s discussion is based, (Eruvin 1:5) Maimonides makes the following observation:

שוריפ הנשמה ם"במרל תכסמ יבוריע ן קרפ א הנשמ ה הנשמ א קרפ ן יבוריע תכסמ ם"במרל הנשמה שוריפ

ךירצ התא תעדל סחיש רטוק לוגיעה ופקהל יתלב ,עודי יאו רשפא רבדל ילע ו םלועל ידב ,קו ןיאו הז ורסח ן רחה יו ,ויבםול ל בלרפ יו ,וי יל וקללגע טקסי עלהאךר העידי נדצמ ו ומכ םיבשוחש ,םילכסה אלא רבדש הז דצמ ועבט יתלב נ עדו יאו ן ותואיצמב .עדויש לבא רשפא רשפא לבא .עדויש ותואיצמב ן יאו עדו נ יתלב ועבט דצמ הז רבדש אלא ,םילכסה םיבשוחש ומכ ו נדצמ העידי עשל ר ו ב ק י ר ,בו רבכו ושע יחמומ םיסדנהמה הזב ,םירובח רמולכ תעידיל סחי רטוקה ופיקהל בוריקב נפואו י פא ברק ויה טק ח תיי מל ,יוחהבםסנמ יממוערכ ב החכוהה ילע ו . בוריקהו םישמתשמש וב ישנא עדמה אוה סחי דחא השלשל ,תיעיבשו לכש יע ג לו ורטוקש המא המא ורטוקש לו ג יע לכש ,תיעיבשו השלשל דחא סחי אוה עדמה ישנא וב םישמתשמש בוריקהו . ו ילע החכוהה תחא ירה שי ופיקהב שלש תומא תיעיבשו המא .בוריקב יכו ו ן הזש אל י גשו ירמגל אלא בוריקב ושפת םה ה ופ ובשחב ן לודג ורמאו לכ שיש ופיקהב השלש םיחפט שי וב בחור ,חפט וקפתסהו הזב לכב תודידמה וכרצוהש ןהלןל כצה וימ כ ז ופסו ,פ חר בש םחטהל ויה י כ רא לד בח לכב ותה הר ת כ

...The ratio of the diameter to the circumference of a circle is not known and will never be known precisely. This is not due to a lack on our part (as some fools think), but this number [pi] cannot be known because of its nature, and it is not in our ability to ever know it precisely. But it may be approximated ...to three and one-seventh. So, any circle with a diameter of one has a circumference of approximately three and one-seventh. But because this ratio is not precise and is only an approximation, they [the rabbis of the Mishnah and Talmud] used a more general value and said that any circle with a circumference of three has a diameter of one, and they used this value in all their Torah calculations.

IS THE VALUE OF PI HIDDEN IN THE BIBLE?

There are lots of papers on the value of pi in the Bible. Many of them mention an observation that כאלמ י ם א ׳ seems to have been incorrectly attributed to the Vilna Gaon. The verse we cited from רבד י ה םימי ב ׳ In) . וק but it is pronounced as though it were written , הוק spells the word for line as The ratio of the numerical (. וק II Chronicles 4:2) the identical verse spells the word for line as) .is 111/106 ( רק י ) to the pronounced word ( תכ י ב ) value (gematria) of the written word

Let's have the French mathematician Shlomo Belga pick up the story - in his paper (first published in the 1991 Proceedings of the 17th Canadian Congress of History and Philosophy of Mathematics, and recently updated), he gets rather excited about the whole gematria thing:

14 כאלמ י ם A mathematician called Andrew Simoson also addresses this large tub that is described in and is often called Solomon's Sea. He doesn't buy the gematria, and wrote about it in The College ׳א Mathematics Journal.

A natural question with respect to this method is, why add, divide, and multiply the letters of the words? Perhaps an even more basic question is, why all the mystery in the first place?

Furthermore, H. W. Guggenheimer, in his Mathematical Reviews...seriously doubts that the use of letters as numerals predates Alexandrian times; or if such is the case, the chronicler did not know the key. Moreover, even if this remarkable approximation to pi is more than coincidence, this explanation does not resolve the obvious measurement discrepancy - the 30-cubit circumference and the 10-cubit diameter.

Finally, Deakin points out that if the deity truly is at work in this phenomenon of scripture revealing an accurate approximation of pi... God would most surely have selected 355/113...as representative of pi...

Still, what stuck Simoson was that "...the chroniclers somehow decided that the diameter and girth measurements of Solomon's Sea were sufficiently striking to include in their narrative." 6

DID THE RABBIS OF THE TALMUD GET Π WRONG?

So, what are we to make of all this? Did the rabbis of the Talmud get π wrong, or were they just approximating π for ease of use? After considering evidence from elsewhere in the Mishnah (Ohalot 12:6 ) Judah Landa, in his book Torah and Science, has this to say:

We can only conclude that the rabbis of the Mishnah and Talmud, who lived about 2,000 years ago, believed that the value of pi was truly three. They did not use three merely for simplicity’s sake, nor did they think of three as an approximation for pi. On the other hand, rabbis who lived much later, such as the Rambam and Tosafot (who lived about 900 years ago), seem to be acutely aware of the gross inaccuracies that results from using three for pi.

Mathematicians have known that pi is greater than three for thousands of years. Archimedes, who lived about 2,200 years ago, narrowed the value of pi down to between 3 10/70 and 3 10/71 ! 7

Still, don't be too hard on the rabbis of the Talmud. The rule that the circumference of an object is three times its diameter is pretty close to being correct and is usually a good enough approximation. But it is not accurate, and never will be.

6 See also B'Or Ha'Torah - the journal of "Science, Art & Modern Life in the Light of the Torah." 7 Judah Landa. Torah and Science. Ktav Publishing House 1991. p.23.

15

The Pool of Shlomo Ha’Melech and the Value of π

MORRIS ENGELSON writes:8

The value of pi based on Solomon’s Pool (Yam Shel Shlomo) in I Malachim, 7:23 has a rich literature in Torah and general secular and mathematical sources. This article is not intended as a survey of this literature, and certainly not as a commentary on the extensive Torah-based analyses. My intent is to acquaint the Torah-focused reader with some of the many approaches to a resolution of the puzzle posed by the stated dimensions of the Yam Shel Shlomo. In particular, we will show how and why these approaches are deficient when these ignore information derived from Torah. Only an approach based on Torah, and especially one that considers the volume discussed in Eruvin, yields consistent results.

8 http://www.hakirah.org/Vol22Engelson.pdf. Morris Engelson is a retired scientist; formerly a Chief Engineer at Tektronix Inc. and Adjunct Professor at Oregon State University, with publication of scientific papers and reference books in his specialty of spectrum analysis. He has also published on the intersection of science and Torah in publications such as Inyan and Kolmus, and is the author of the book, The Heavenly Time Machine: Essays on Science and Torah.

16 It would appear from a simple reading of the text that neither the people Yisroel nor their leaders at the time of Shlomo Ha’Melech nor Chazal over a thousand years later were aware that the ratio of the circumference to diameter of the circle, currently designated by the symbol π (pi), is greater than 3. Thus, we are provided the dimensions of Solomon’s circular pool as being 10 amos (cubits) in diameter, 30 amos in circumference, 5 amos in height and with a wall 1 tefach (hand breadth) thick. The volume is given as 2000 bas. The same dimensions are also found in Divrei HaYamim (4:2). Our Daf provides information about circles:

“Whatever has a circumference of three tefachim has a width of a tefach.”

The text in both places clearly designates the value π=3.

This creates a puzzle because it is virtually impossible that these people were not aware that the result is more than.9 The Bais HaMikdash was built with precision and skill by experienced craftsmen. Surely Hiram (I Malachim 7:13) and his master builders knew of approximations to pi that had long been established by measurement and observation.

We know that the Egyptian and Babylonian approximations for pi (3.16 and 3.125 respectively) were established some five centuries before the time of Shlomo Ha’Melech. Ordinary people might have been ignorant on this matter, but how is it that Yirmiyah, the author of Malachim, knew nothing about this?

Furthermore, by the time of the Talmud over a millennium later, knowledge of pi was much improved due to mathematical analysis. Archimedes (ca. 225 BCE) had centuries previously established the relationship of 223/71 < π < 22/7. Moreover, numerous people were involved in Talmudic discourse over a period of over 200 years. It is impossible that some of these people did not know that π is not equal to exactly 3. So why did they retain the statement that a circle with circumference equal to 3 has a diameter equal to 1?

A Matter of Volume

We have proofs in Masechet Eruvin page 14b that the volume of the pool was 450 cubic standard amah (CSA), of 6 tefachim to the amah. This is the same as 150 mikvah volumes at their minimal volume of 40 seah each [at 3 CSA per 40 seah].

9 “On the Rabbinical Approximation of π,” Historia Mathematica 25 (1998) http://u.cs.biu.ac.il/~tsaban/Pdf/latexpi.pdf.

17 The key point for us is that the volume should be 450 CSA. The Talmud quickly concludes that the volume of a cylinder of the given dimensions (assuming the diameter is on the inside) will yield only 375 CSA (using π =3).

Thus, Chazal conclude that the bottom 3 amah of the pool were in the shape of a square whose volume is 300 CSA, while the upper 2 amah is a cylinder whose volume is 150 CSA. The total volume is the required 450 CSA.

The above calculation goes further than simply stating that we use π=3 for ordinary usage. The statement, “π=3” is simply an approximation that most people can easily understand and accept. But claiming that a cylinder with a diameter of 10 amos and height of 2 amos has a volume of 150 CSA implies that π actually equals 3. Thus, there are those who argue that Chazal were ignorant of basic mathematics.

This is obviously nonsense to anybody who learns Eruvin or other areas of Torah and sees the sophisticated mathematical analyses involved. Chazal are focused on Halachah; mathematics is invoked only as an aid and not in its own right.

No one could possibly claim, for example, that Chazal were not aware that a gap, which is halachically treated via lavud as if it did not exist, does in fact exist physically. The same applies here. This article, however, is not focused on Halachah but rather on the mathematical implications resulting from the dimensions of the Yam Shel Shlomo.

Hence, we are obliged to not only show a value of pi greater than 3, but we need to do so with a shape whose volume is 450 CSA.

The simplest resolution is along the lines proposed by Graf, as previously discussed, where the diameter of the outer part of the cylinder is equal to 10, while the inner diameter of the cylinder is less than 10.

This can result in a value for pi precise to as many decimal places as one might wish. Unfortunately, the result violates various restrictions imposed by Chazal. Simoson discusses several possibilities. Here are some examples in addition to the many ingenious solutions discussed by Simoson.

18

The hexagonal pool solution.

The verse in Malachim states that the pool’s top lip was in the shape of a lily flower. This flower has six petals, which some take to mean a hexagon.

A regular hexagon whose side is s=5 amos will have a circumference of 30 amos and a maximal diameter of 10 amos.

This fits perfectly the ratio of 3:1. The area of a regular hexagon with side length s=5 is given by [(3√3)/2]s2 ≈64.95, and we do not get the required volume of 450 CSA for a height of 5 amos. This solution does not work based on the position of the Talmud that the volume of the pool is 450 CSA.

19

Russian icon of King Solomon. He is depicted holding a model of the Temple (18th century, iconostasis of Kizhi monastery, Russia).

The flared lip solution.

This solution is discussed in a general fashion by Simoson; here is a more detailed analysis. Malachim states that the brim was “like the lip of a cup.” This analysis is based on the idea that top of the pool was flared and the bottom cylindrical, making the diameter on top larger than the rest of the pool. Peter Aleff argues on his web site that “the rim was flared,” and elaborates in his book Ancient Creation Stories told by the Numbers in a chapter on “The old myth of King Solomon’s wrong pi.” The essentials of this analysis are available on the internet, where Aleff quotes the simplified pi formula from Eruvin: “that which in circumference is three hands broad is one hand broad.” He comments that “scholars of the Enlightenment era were glad to concur with that interpretation because it allowed them to wield this blatant falsehood in the Bible as an irresistible battering ram…” But Aleff disagrees with Enlightenment conclusions.

Here the 10 amah diameter is measured across the flared top, while the circumference is measured on the outside lower cylindrical body. Referencing various works on archeology and ancient science/mathematics (e.g., van der Waerden: Science Awakening, Egyptian, Babylonian and Greek Mathematics and Leen Ritmeyer: The Temple and the Rock), Aleff provides a well annotated argument for a fairly accurate value of π. He uses 7 tefachim to the amah based on

20 archeological analysis and Egyptian units. The bottom of the pool is taken as one tefach thick, and this is deducted from the 5 amah height. Unlike other calculations that ignore this feature, the required volume of 2000 liquid measure bas is accounted for in the calculations. Aleff demonstrates that it all works out perfectly. His calculation of the volume after, accounting for the thickness of the bottom and the flared lip is 304.04 cubic amah, which he equates to 2000 bas. This appears to be contrary to the position of Masechet Eruvin where 2000 bas is equated to 450 CSA. But it is not that simple, because Aleff uses a super large amah of 7 tefachim. The volumetric ratio to the Talmudic amah of 6 tefachim is (7/6)3 and the two volumes can be made to agree by a judicious choice of the volume lost to the flared rim. The only issue is the choice of 7 rather than 6 tefachim to the amah.

Solution by the Ralbag.

The Ralbag proposes a solution that on the surface appears identical to that proposed by Graf; the diameter is for the outside surface of the cylinder while the circumference is on the inside. But Ralbag, who in addition to his greatness in Torah was also a world-class scientist and mathematician, does not make the error that Graff makes.

His amah has 6 tefachim and he is mindful of the required volume at 450 CSA. He proceeds with the shape proposed by Chazal with the 3 bottom amos in a square and the top 2 in a cylinder. There are several ways to calculate the volume of the shape chosen by Ralbag.

The best result, noted in the paper below, shows a volume of 446.8 CSA, and Ralbag states that his result is approximate; that is, he does not ignore the need for 450 CSA10.

Ralbag is not the first Torah authority to suggest a good approximation for pi embedded within the shape of this pool. We have the value π = 31/7 in a book on mathematics, Mishnat ha-Middot, attributed to Rabbi Nehemiah (ca 150 CE).

I previously noted an objective to show a value of pi greater than 3, but also with a shape whose volume is 450 CSA. None of the examples discussed in this article or its references meet this objective. Even Ralbag, with his great erudition in Torah and mathematical skill, falls short on the volume.

I propose two possibilities for pi which fulfill the volume requirement. The two approximations for pi which come immediately to mind are the ancient Mesopotamian approximation at 3.125 (3+1/8 on clay tablet from Susa) and the ancient Egyptian approximation of near 3.16 (per Rhind Papyrus). He then goes on to suggest solutions which work well mathematically but introduce numerous questions. Neither is a credible candidate for the pool’s actual shape but do obtain the called-for volume and have pi set at a known ancient approximation.

In summary: none of the proposed solutions meets all of the called-for parameters.

10 An analysis of the position taken by Ralbag is provided by Shai Simonson, of the Department of Mathematics and Computer Science in Stonehill College, in The Mathematics of Levi ben Gershon, the Ralbag: http://u.cs.biu.ac.il/~tsaban/Pdf/MathofLevi.pdf.

21

Thomas Degeorge (1786–1854), The Death of Archimedes (detail), 1815. Collection of the Musée d’Art Roger-Quilliot Museum [MARQ], City of Clermont-Ferrand, France.

David Wilson writes:11

Pi (π) has been known for almost 4000 years—but even if we calculated the number of seconds in those 4000 years and calculated π to that number of places, we would still only be approximating its actual value. Here’s a brief history of finding π.12

11 https://sites.math.rutgers.edu/~cherlin/History/Papers2000/wilson.html

12 https://www.exploratorium.edu/pi/history-of pi#:~:text=The%20first%20calculation%20of%20%CF%80,mathematicians%20of%20the%20ancient%20world.&text=Mathem aticians%20began%20using%20the%20Greek,who%20adopted%20it%20in%201737.

22 The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for π, which is a closer approximation.

The Rhind Papyrus (ca.1650 BC) gives us insight into the mathematics of ancient Egypt. The Egyptians calculated the area of a circle by a formula that gave the approximate value of 3.1605 for π.

The rabbinic discussion above in our daf the ratio of pi to 3 is very close to the ratio of 111 to 106. In other words, pi/3 = 111/106 approximately; solving for pi, we find pi = 3.1415094... This figure is far more accurate than any other value that had been calculated up to that point and would hold the record for the greatest number of correct digits for several hundred years afterwards. Unfortunately, this little mathematical gem is practically a secret, as compared to the better-known pi = 3 approximation.13

When the Greeks took up the problem, they took two revolutionary steps to find pi. Antiphon and Bryson of Heraclea came up with the innovative idea of inscribing a polygon inside a circle, finding its area, and doubling the sides over and over.

"Sooner or later (they figured), ...[there would be] so many sides that the polygon ...[would] be a circle"14.

Later, Bryson also calculated the area of polygons circumscribing the circle. This was most likely the first time that a mathematical result was determined through the use of upper and lower bounds. Unfortunately, the work boiled down to finding the areas of hundreds of tiny triangles, which was very complicated, so their work only resulted in a few digits. At approximately the same time, Anaxagoras of Clazomenae started working on a problem that would not be conclusively solved for over 2000 years. After imprisonment for unlawful preaching, Anaxagoras passed his time attempting to square the circle. Cajori writes:

"This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, the rock that upon which so many reputations have been destroyed.... Anaxagoras did not offer any solution of it and seems to have luckily escaped paralogisms"15.

13 Tsaban, Boaz and David Garber. "On the Rabbinical Approximation of pi." Historia Mathematica 25, Article HM972185. Academic Press, 1998.

14 Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.

15 Cajori, Florian. A History of Mathematics. MacMillan and Co. London, 1926.

23 Since that time, dozens of mathematicians would rack their brains trying to find a way to draw a square with equal area to a given circle; some would maintain that they had found methods to solve the problem, while others would argue that it was impossible. The problem was finally laid to rest in the nineteenth century.

The first man to really make an impact in the calculation of pi was the Greek, Archimedes of Syracuse. Where Antiphon and Bryson left off with their inscribed and circumscribed polygons, Archimedes took up the challenge. However, he used a slightly different method than they used. Archimedes focused on the polygons' perimeters as opposed to their areas, so that he approximated the circle's circumference instead of the area. He started with an inscribed and a circumscribed hexagon, then doubled the sides four times to finish with two 96-sided polygons.16 (Archimedes, 92) His method was as follows...

Given a circle with radius, r = 1, circumscribe a regular polygon A with K = 3(2n-1 sides and semiperimeter an and inscribe a regular polygon B with K = 3(2n-1 sides and semiperimeter bn. This results in a decreasing sequence a1, a2, a3... and an increasing sequence b1, b2, b3... with each sequence approaching pi. We can use trigonometric notation (which Archimedes did not have) to find the two semiperimeters, which are: an = K tan ((/K) and bn = K sin ((/K).

Also: an+1 = 2K tan ((/2K) and bn+1 = 2K si n ((/2K). Archimedes began with a1 = 3 tan ((/3) = 3(3 and b1 = 3 sin ((/3) = 3(3/2 and used 265/153 < (3 < 1351/780. He calculated up to a6 and b6 and finally reached the conclusion that 3 10/71 < b6 < pi < a6 < 3 1/7. Archimedes ended with a 96-sided polygon, and numerous delicate calculations.

16 Archimedes. "Measurement of a Circle." From Pi: A Source Book.

24 (Archimedes, 95). The fact that he was able to go that far and derive such a good estimation of pi is a "stupendous feat both of imagination and calculation"17

Archimedes approximated the area of a circle by using the Pythagorean Theorem to find the areas of two regular polygons: the polygon inscribed within the circle and the polygon within which the circle was circumscribed. Since the actual area of the circle lies between the areas of the inscribed and circumscribed polygons, the areas of the polygons gave upper and lower bounds for the area of the circle. Archimedes knew that he had not found the value of π but only an approximation within those limits. In this way, Archimedes showed that π is between 3 1/7 and 3 10/71.18

A similar approach was used by Zu Chongzhi (429–501), a brilliant Chinese mathematician and astronomer. Zu Chongzhi would not have been familiar with Archimedes’ method—but because his book has been lost, little is known of his work. He calculated the value of the ratio of the circumference of a circle to its diameter to be 355/113. To compute this accuracy for π, he must have started with an inscribed regular 24,576-gon and performed lengthy calculations involving hundreds of square roots carried out to 9 decimal places.

Mathematicians began using the Greek letter π in the 1700s. Introduced by William Jones in 1706, use of the symbol was popularized by Leonhard Euler, who adopted it in 1737.

An eighteenth-century French mathematician named Georges Buffon devised a way to calculate π based on probability. You can try it yourself at the Exploratorium's Pi Toss exhibit.

17 O'Connor, J.J., and E.F. Robertson. The MacTutor History of Mathematics Archive. World Wide Web. 1996.

18 https://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html

25

The picture is a public domain U.S. Army photo of the ENIAC. The wires, switches, and components are all part of the ENIAC with two of the team of operators helping run the machine.

20th Century and Pi

With the twentieth century, computers took over the reigns of calculation, and this allowed mathematicians to exceed their previous records to get to previously incomprehensible results.

In 1945, D. F. Ferguson discovered the error in William Shanks' calculation from the 528th digit onward. Two years later, Ferguson presented his results after an entire year of calculations, which resulted in 808 digits of pi.19 (Berggren, 406) One and a half years later, Levi Smith and John Wrench hit the 1000-digit-mark. (Berggren, 685) Finally, in 1949, another breakthrough emerged, but it was not mathematical in nature; it was the speed with which the calculations could be done.

The ENIAC (Electronic Numerical Integrator and Computer) was finally completed and functional, and a group of mathematicians fed in punch cards and let the gigantic machine calculate 2037 digits in just seventy hours. (Beckmann, 180) Whereas it took Shanks several years to come up with his 707 digits, and Ferguson needed about one year to g et 808 digits, the ENIAC computed over 2000 digits in less than three days!

19 Berggren, Lennart, and Jonathon and Peter Borwein. Pi: A Source Book. Springer-Verlag. New York, 1997.

26

"With the advent of the electronic computer, there was no stopping the pi busters" (Blatner, 51).

John Wrench and Daniel Shanks found 100,000 digits in 1961, and the one-million-mark was surpassed in 1973. In 1976, Eugene Salamin discovered an algorithm m that doubles the number of accurate digits with each iteration, as opposed to previous formulas that only added a handful of digits per calculation. (Blatner, 52)

Since the discovery of that algorithm, the digits of pi have been rolling in with no end in sight. Over the past twenty years, six men in particular, including two sets of brothers, have led the race: Yoshiaki Tamura, Dr. Yasumasa Kanada, Jonathan and Peter Borwein, and David and Gregory Chudnovsky. Kanada and Tamura worked together on many pi projects, and led the way throughout the 1980s, until the Chudnovskys broke the one-billion-barrier in August 1989.

In 1997, Kanada and Takahashi calculated 51.5 billion (3(234) digits in just over 29 hours, at an average rate of nearly 500,000 digits per second! The current record, set in 1999 by Kanada and Takahashi, is 68,719,470,000 digits. (Blatner, 59)

There is no knowing where or when the search for pi will end. Certainly, the continued calculations are unnecessary. Just thirty-nine decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the radius of a hydrogen atom. (Berggren, 656) Surely, there is no conceivable need for billions of digits.

At the present time, the only tangible application for all those digits is to test computers and computer chips for bugs. But digits aren't really what mathematicians are looking for anymore. As the Chudnovsky brothers once said: "We are looking for the appearance of some rules that will distinguish the digits of pi from other numbers. If you see a Russian sentence that extends for a whole page, with hardly a comma, it is definitely Tolstoy. If someone gave you a million digits from somewhere in pi, could you tell it was from pi? We don't really look for patterns; we look for rules" (Blatner, 68).

Unfortunately, the Chudnovskys have also said that no other calculated number comes closer to a random sequence of digits. Who knows what the future will hold for the almost magical number pi?

27

Tanya Lewis writes:20

Every year, math enthusiasts celebrate Pi Day on March 14, because the date spells the first three digits (3.14) of pi, or π, the mathematical constant that represents the ratio of a circle's circumference to its diameter. This year, the event is even more special because, for the first time in a century, the date will represent the first five digits of pi: 3.14.15.

Pi is an irrational number, meaning it cannot be expressed as a fraction, and its decimal representation never ends and never repeats.

There are many ways to celebrate Pi Day, including consuming large amounts of its delicious homophone, pie. But a handful of people take their admiration further, by reciting tens of thousands of digits of pi from memory. [The 9 Most Massive Numbers in Existence]

20 https://www.livescience.com/50134-pi-day-memory- experts.html#:~:text=In%201981%2C%20an%20Indian%20man,recited%2067%2C890%20digits%20of%20pi.

28

In 1981, an Indian man named Rajan Mahadevan accurately recited 31,811 digits of pi from memory. In 1989, Japan's Hideaki Tomoyori recited 40,000 digits. The current Guinness World Record is held by Lu Chao of China, who, in 2005, recited 67,890 digits of pi.

Despite their impressive achievements, most of these people weren't born with extraordinary memories, studies suggest. They have simply learned techniques for associating strings of digits with imaginary places or scenes in their minds.

For many of these memory champions, the ability "to remember huge numbers of random digits, such as pi, is something they train themselves to do over a long period of time," said Eric Legge, a cognitive psychologist at the University of Alberta in Edmonton, Canada.

Expert pi memorizers often use a strategy known as the method of loci, also called the "memory palace" or the "mind palace" technique (like the one used by Benedict Cumberbatch's character in the BBC TV Series "Sherlock").

Applied since the time of the ancient Greeks and Romans, the method involves using spatial visualization to remember information, such as digits, faces or lists of words. "It's one of the more effective, yet complex, memory strategies out there for remembering large sets of information," Legge told Live Science.

Here's how it works: You place yourself in a familiar environment, such as a house, and walk through that environment placing chunks of the information you wish to remember in various places. For example, you might put the number "717" in the corner by the front door, the number "919" in the kitchen sink, and so on, Legge said.

"In order to recall [the digits] in order, all you simply have to do is walk in the same path as you did when you were storing that information," Legge said. "By doing this, people can remember huge sets of information."

29

Nurture, not nature

Anders Ericsson, a professor of psychology at Florida State University in Tallahassee, has studied Lu and others who have set records for reciting digits of pi, to find out how they achieved these stunning feats of memorization.

Like most other pi reciters, Lu used visualization techniques to help him remember. He assigned images such as a chair, a king or a horse to two-digit combinations of numbers ranging from "00"

30 to "99." Then he made up a story using these images, which was linked to a physical location, Ericsson said.

A few years ago, Ericsson and his colleagues gave Lu, as well as a group of people of the same age and education level, a test that measured their "digit span" — in other words, how well they could remember a sequence of random digits presented at a rate of one digit per second.

Lu's digit span was 8.83, compared with an average of 9.27 for the rest of the group, according to the study, which was published in 2009 in the Journal of Experimental Psychology. The results suggest that, unlike some other memory experts who have been studied, Lu's skill in memorizing long lists of digits was not the result of an innate skill in encoding information.

Rather, it was the result of years of practice, Ericsson said.

So does this mean anyone can learn to remember tens of thousands of digits of pi? "There have been a lot of demonstrations showing that regular people, given training, can dramatically improve their performance" in memorizing long lists, Ericsson said. "But I have to be honest," he said. "When you make that commitment to memorize pi …we're talking years before you can actually reach record performances."

31

A slice of π: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist:21

Subject PI demonstrated superior memory using a variant of a Method of Loci (MOL) technique to recite the first digits of the mathematical constant π to more than 216 decimal places. We report preliminary behavioral, functional magnetic resonance imaging (fMRI), and brain volumetric data from PI. fMRI data collected while PI recited the first 540 digits of π (i.e., during retrieval) revealed increased activity in medial frontal gyrus and dorsolateral prefrontal cortex. Encoding of a novel string of 100 random digits activated motor association areas, midline frontal regions, and visual association areas. Volumetric analyses indicated an increased volume of the right subgenual cingulate, a brain region implicated in emotion, mentalizing, and autonomic arousal. Wechsler Abbreviated Scale of Intelligence (WASI) testing indicated that PI is of average intelligence, and performance on mirror tracing, rotor pursuit, and the Silverman and Eals Location Memory Task revealed normal procedural and implicit memory. PI’s performance on the Wechsler Memory Scale (WMS-III) revealed average general memory abilities (50th percentile), but superior working memory abilities (99th percentile). Surprisingly, PI’s visual memory (WMS-III) for neutral faces and common events was remarkably poor (3rd percentile). PI’s self-report indicates that imagining affective situations and high emotional content is critical for successful recall. We

21 https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4323087/

32 speculate that PI’s reduced memory for neutral/non-emotional faces and common events, and the observed increase in volume of the right subgenual cingulate, may be related to extensive practice with memorizing highly emotional material.

Behavioral data

The behavioral data show that PI’s most obvious strength was an exceptional working memory: score for working memory fell above the 99th percentile. PI’s general memory, in contrast, tested at the 50th percentile. This result is perhaps not surprising because most tests of working memory use digits wherein PI may have applied a variation of the MOL. PI’s performance on the test of implicit memory (Silverman and Eals task) and procedural learning (mirror tracing and rotor pursuit tasks) was comparable to that of an age-matched group. This result is perhaps not unexpected because other mnemonists were not good at all types of memory, but PI’s superior performance on working memory subtests was consistent with previous reports of the working memory of superior memorists.

Anatomical data Comparing the volumetric measures of the neuroanatomical structures of PI with those of the sample shows that the only statistically significant (p<.05) difference, by both moderate and conservative estimates, occurs for the right subgenual region of the cingulate gyrus,2 an anatomical area located below the genu of the corpus callosum. Findings suggest that the anterior cingulate cortex and subgenual region plays a role in mentalizing (Frith & Frith, 2003), emotional processing (Bush, Luu, & Posner, 2000), and autonomic arousal (Critchley et al., 2003).

Conclusions The greatest contribution of this preliminary case report is its suggestions and directions for future research. We intentionally did not include in this study a control group of non-superior memorists: such non-experts would be MOL amateurs and would undoubtedly use the method in a way different from PI. In addition, their MOL proficiency may rely on an alternative, less-established strategy that would likely not be comparable to PI’s, thus nullifying the comparison. Instead, we opted to elucidate the neural substrates of PI’s unusual mnemonic capability by using PI as PI’s own control. Although we did collect fMRI data on tests wherein both PI served as a self-control and PI’s memory was not superior, those data were inadvertently lost through a computer malfunction. While PI performed at 100% accuracy when tested outside of the magnet immediately after the scan, we can only assume that PI’s performance during the scan was comparably accurate. Nevertheless, that learning occurred during the scan is evident in PI’s accuracy immediately following the scan. Finally, we studied PI after having mastered MOL, so our findings cannot distinguish exceptional memory from the learning of effective memory strategies. Still, the bulk

33 of the evidence suggests that an important component of a superior memory is the acquired skill in and diligent practice of mnemonic strategies (Ericsson 2003; Wilding & Valentine, 1997).22

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34 The Spirituality of Pi

Michael Piper writes:23

What is Pi? A math teacher will tell you it is the ratio of the circumference of a circle to the radius of that circle. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and you always get exactly the same number; 3.14159… . It doesn’t matter what size the circle is, Pi remains the same. To the Ancients, Pi was revered, even, a Holy ratio. Pi is often written using the symbol and is pronounced “pie”, just like the dessert. In reality it is the basis of Everything because everything is composed of cycles that spin, orbit or rotate. It is a number that goes on forever, never repeating itself, ad infinitum. Pi goes to infinity and has never been calculated with finality because it is impossible for material men, infinity cannot be calculated within a finite Universe. Within its never repeating stream of digits is the definition of everything, a foundation of information locked in the most elegant geometry, what you could term, a spiritual geometry of God. It defines all the cycles of the spinning and orbiting of atoms, planets, solar systems and Galaxies. Even the DNA molecule, as the basis of all life has its basis in the entangled double helix with many internal ‘connections’, and within its never- ending factory of numerical simplicity lies the foundation of the Universe itself.

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23 https://pipermichael.wordpress.com/pi-the-ratio-of-god/

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For within this infinite stream of never repeating numbers lies the Big Bang and the meaning of life, the programming code for DNA and the Fibonacci sequence, all the prime numbers, and even your telephone number and all the texts or messages you ever sent, all the books and novels ever written, all the knowledge of man and God, as One. Enoch said the Creation was started with the begotten creation of the Logos, which was the manifestation of all of time from the beginning to the end of time itself. If you could but calculate Pi to infinity, and somehow convert those numbers into the appropriate selection and representations of the data, you could create a Universe. But those numbers wouldn’t be seen in the familiar decimal or even binary digits you are familiar with, they would be a stream of fuzzy vibrations, harmonic and dissonant frequencies of energy.

Analog Waves are the language of Pi, waves and cycles in a never repeating glory of infinite harmonic variety. This is the basis of the Entanglement of the Infinities, known in our reality as quantum entanglement, and to eastern philosophy as the Oneness, and to some as the collective consciousness. To see this concept in reality, consider the simple prism, and what happens when sunlight hits its surface. All the colors of the rainbow are disconnected from each other in a properly constructed facet. The colors are entangled, until their true beauty is revealed by the power of a magnetic density shift at an angle identical with their entanglement phase. The sudden shift of the alignment of the magnetic bubbles within the prism face, are at the proper angle to the incident light waves, and the entangled colors are harmonically separated by the power of God within the spin of the magnetic bubbles. This is the same functional principle of why and how the primal energies, with the spin, and the Knowledge, are harmonically entangled in the Magnetic Aether to form the Firmament and Life itself.

To the ancients, Pi was sacred, a spiritual number, the basis of pyramids and massive stone edifices, that to this day, so called modern man, cannot fathom. But it is easy to fathom these things if you but read the clues written in the spiritual geometry of the Universe. If you can see the Primal Energy matrix, and the Source of life and death, the Giver and the Taker, the Producer and the

36 Consumer, the many names of Sources of the Light and the Dark energy, given down through time in the Perennial Philosophy of the ancients. To them they were not simply metaphors of some preachers sermon, but the basis of the reality in which they lived. They built these massive stone structures, to reach upwards to The Light, even their glyphs of pyramids have the Light as the Goal, but modern scholars interpret this as the Sun, and they consider them Sun worshipers. How simple minded, but it is understandable, when your reality itself is not understood. These edifices concentrated the primal energies, by the spiritual geometry of Pi, they were trying to create an invisible tunnel to the Light of God.

The ancients understood the mechanisms of the Hypostasis, the Hyper-Reality, even better than modern man, who is the product of the Mystery School. A school that can tell you the How of everything, but not the Why. A school that sees the surface layer of reality, but not the interior structures that infinitely repeat in chaotic beauty. It is a school of ‘Mysterious Phenomena’, whether you are of religious bent or scientific mind, their reality is composed of mysterious particles and mysterious spiritual entities.

As the power of Pi forces all things into the spiritual geometry of the cycles and circles, the intentions of higher powers are made manifest in the simple alignments of the magnetic bubbles according to their place on the wheel, as the power of higher intention merely has to synchronize the attraction and repulsion of bubbles, by the harmonic power of the entanglement of the infinities, the wave entanglements, exactly like the white light is composed of all the other colors, and this is revealed in the prism. The surface of the prism, is the surface of the magnetic waters, as the hyper-light energy is focused and the entanglement of the helical waves is strengthened and weakened at their peaks and troughs, to come into alignment in 3d reality, aligning and manifesting, according to the master database(what the Gnostic texts referred to as the Treasury of the Light), and the programmers of the delegated powers of the Tree of the Knowledge of Good and Evil, and the aetheric prism of the medicine wheel of DNA is the natural and mathematically provable result. The Universe is thus an intelligently designed life manufacturing machine, and its product is ever increasing complexity, otherwise known as; Evolution.

Science has recently shown that reality itself is like a hologram of energy, a hologram of chaos. They have also proven that magnetic bubbles or currents are manifested at different orientations outside the solar system, but they have no explanation, and refuse to accept the evidence of their eyes and instruments concerning the existence of the magnetic ‘waters’ of the Ancient texts, which is the Aether of Einstein’s General Relativity. They are blinded by their own arrogance and refusal to accept that the Creation they study, could not have been a random event of something from nothing. That only something can give something, and that something is the gift of the knowledge of the spiritual geometry, through the mind of the master creator, the Intelligent Designer and the gift of Pi.

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Perhaps the most profound mystical property of π is that it has the inexplicable power to make people believe it has various mystical properties. This great power of π is due in large part to π being both irrational and transcendental — both of which can suggest mystical connotations. Of course, mathematicians can provide perfectly rational definitions of these properties, but these definitions do little to break the mystical spell of π.24

24 Tom McFarlane https://www.quora.com/profile/Tom-McFarlane

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The fact that π is an irrational number suggests it cannot be rationally comprehended. Indeed, its decimal representation is non-terminating and non-repeating, which means that it has infinitely many digits. And infinity, of course, has mystical connotations as well. So, this only adds further to the mystical power of π.

Of course, mathematicians will point out that almost all real numbers are irrational, and π is not at all unique in having these properties, but such facts also do little to break the mystical spell of π. Nor does it seem to help that mathematicians have no trouble at all dealing with the infinite in a consistent way.

The fact that π is a transcendental number also suggests that it transcends something or other, and transcendence is, of course, a mystical notion.

Again, mathematicians can provide a clear and rigorous definition of what it means to be a transcendental number, and they will also point out that almost all real numbers are transcendental, so π is not at all special in that way. But, again, this does little to break the mystical spell of π.

The mystical power of π to make people believe it has special mystical properties is indeed mystical.

How good is good enough? Even the 1.2 trillion-digit approximation of π made by Professor Yasumasa Kanada of Tokyo University in 2002 is still only an approximation. It is humbling to realize that there is something that we can never really know, and π provides us with this experience.

Elishakoff and Pines25

25 B’Or HaTorah 17 (5767/2007) http://u.cs.biu.ac.il/~tsaban/Pdf/Elishakoff Pines.pdf.

39 Ivy Roberts26writes:

A Hero’s Journey

Many myths and legends tell of the hero's journey out into the wilderness where he gains a respect for the natural world, the social order, and learns to locate the strength within himself. It's a common moral in many stories: One must experience the hardships of life in order to appreciate its wonders.

In Life of Pi, author Yann Martel draws on many of these quintessential mythical storytelling techniques. In exploring the relationships between humans and animals, humans and Gods, Martel imbues his characters and their world with mythic, religious, and symbolic importance with the use of allusions, calling something to mind with indirect or passing reference.

Part II (The Pacific Ocean) opens with the sinking of the Japanese freighter ship Tsimtsum. Pi, his family, and many of their zoo animals board the Tsimtsum in India for the trans-Pacific passage to their new home in Canada. But due to mechanical failure and an undisciplined crew, the ship sinks during a storm. That leaves Pi and Richard Parker, along with a hyena, zebra, and orangutan, drifting on a life raft.

What is the significance of the Tsimtsum in relation to larger themes of religion in the novel Life of Pi.

What's in a Name?

Many of the names in Life of Pi are highly symbolic. Pi's name recalls the importance of circles and reason, as well as his given name Piscine, which refers to the French words for both 'fish' and 'swimming pool'. Also, by giving the tiger a formal name (Richard Parker), Martel encourages readers to question animal-human relations, respect, and friendship.

The ship, Tsimtsum, is no exception to this rule. If you don't initially recognize the reference, it's easy to jump to the conclusion that Tsimtsum might be a Japanese word. After all, it's a Japanese ship. But Martel leaves the door open for readers to wonder about the ship's unusual name.

Tsimtsum is actually a Hebrew word. It refers to the reduction or contraction of God from the universe at the moment of creation. This is an important concept because it raises the question of free will vs. predetermination. In retreating from the world, God leaves room for human beings to express their faith and independence. But His departure also opens the space for human beings to sin and to give in to temptation.

Jewish Mysticism From the beginning, the first page of the novel, in fact, Yann Martel infers to his readers the important role that religion will play throughout the book. Adult Pi mentions that he studied at the

26 https://study.com/academy/lesson/life-of-pi-tsimtsum-allusions.html: is an adjunct instructor in English, film/media studies and interdisciplinary studies

40 University of Toronto. His capstone project 'concerned certain aspects of the cosmogony theory of Isaac Luria, the great sixteenth-century Kabbalist from Safed.' By referencing the Kabbalah in the first paragraph of the novel, Martel hints that Jewish mysticism will be important for readers to keep an eye on. Isaac Luria was an important Jewish Rabbi and mystic recognized for authoring and popularizing the modern Kabbalah text.

The name Pi had many relevant associations. It is a letter in the Greek alphabet that also contains alpha and omega, terms used in the book for dominant and submissive creatures.

Pi (uppercase Π, lowercase π) is the sixteenth letter of the Greek alphabet and it has a value of 80 in the Greek numeral system. The letter is pronounced as "p". It came from the Phoenician letter pē, which meant mouth. The Phoenician letter also gave rise to the Latin P and Cyrillic П. The capital letter pi is used in mathematics and chemistry. The lowercase letter pi (π) has been used as a symbol for mathematical constant since the mid-18th century. This symbol is also called Archimedes' constant or Ludolph's number.

The mathematical Pi has so many decimal places that the human mind can’t accurately comprehend it, just as, the book argues, some realities are too difficult or troubling to face. These associations establish the character Pi as more than just a realistic protagonist; he also is an allegorical figure with multiple layers of meaning.

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