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Mathematics for Vayigash: Fruitfulness, Love, and Lucas Numbers

Parashat Vayigash, the 11th and second-to-last portion of the Book of Genesis, marks a turning point. In it, we read about the shift from the lives of the patriarchs in the land of Canaan to their descent into a semi-permanent exile in Egypt. Yet in spite of the conditions in Egypt—conditions that would turn harsh and inhuman over the years—the Children of Israel were fruitful and multiplied there many times over their original numbers. Jacob and his children and their families consisted of 70 individuals. The number that left Egypt is quoted by the Torah at 600,000, more than 8571 times more!

As we shall now see, the beginning of this increase happened with the clash between Judah and Joseph in the first verses of parashat Vayigash. Let us begin our exploration of this clash between Judah and Joseph by noting that the word chosen for “multiplication” by the Medieval Jewish philosophers literally means a “clash” (הַכָּאָה). This meeting began the process of adding to or multiplying the numbers of Jacob’s offspring.

Additive Series: Fibonacci and Lucas

Being fruitful and multiplying is intrinsically related to one of the most important mathematical topics, additive series, which are more readily recognized by the name of their most famous member, the Fibonacci series:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, …

The Fibonacci numbers relate to fertility from generation to generation and are reportedly the original problem that Fibonacci himself tried to solve and in the process “discovered” this series. Suppose a newly-born pair of rabbits, one male, one female, are put in a field. Rabbits are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits. Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month on. The puzzle that Fibonacci posed was How many pairs will there be in one year? The answer is 1 in the first month, 1 in the second month (the pair had to reach maturity), 3 in the third, 5 in the fourth, and so on, until at the end of a year there will be 233 pairs. All these numbers are Fibonacci numbers.

For a number of reasons, which we will not go into here, we designate this series, the “Love” (אַהֲבָה) series of numbers. As their name implies, additive series are derived not through a function, as are the quadratic series we have seen in previous chapters, but through a mechanism of addition. The sum of every two consecutive numbers becomes the next number in the series and the addition is repeated again. It is not difficult to realize that additive series can therefore be exhaustively distinguished from one another by merely stating the first two numbers in the series.

Apart from the Fibonacci series, the additive series that is most studied is the Lucas series of numbers, which begins with 1, 3. The Lucas series begins:

1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …

Additive series such as the Fibonacci and the Lucas series are an important interface between the Torah’s mathematics, which is mostly dicrete (i.e., it uses integer, or whole numbers), and the mathematics used to model nature, which is mostly continuous. As such, they create a medium between the mathematics of human experience and the mathematics of nature. Elsewhere, we treat this subject in length.

Lucas and Vayigash

The numerical value of the name of our parashah, “Vayigash” (וַיִּגַּשׁ) is 319, which connects it very strongly with the Lucas numbers because the sum of the first 10 Lucas numbers is:

1 ┴ 3 ┴ 4 ┴ 7 ┴ 11 ┴ 18 ┴ 29 ┴ 47 ┴ 76 ┴ 123 = 319

Vayigash is linked to Lucas numbers in another way: the product of 5th and 7th Lucas numbers, which are 11 and 29 is also 319:

11 ž 29 = 319

Since there seems to be a connection between our parashah and the Lucas numbers let us see some more properties of this series. First, note that the first four Lucas numbers are 1, 3, 4, and 7. Their sum is 15, the value of the first two letters, yud-hei (י-ה) of God’s essential Name, Havayah (י-הוה). But, the next number in the Lucas series is 11, which is the sum of the final two letters, vav-hei (וה), of Havayah, thus the first 5 Lucas numbers sum to 26, the value of God’s essential Name, Havayah (י-הוה) and even divide into 15 and 11, neatly.

The next Lucas number is 18 and it is the filling part[1] of the smallest possible filling of Havayahוד הא וו הא).

The sum of the first 6 numbers is thus 44, which is the value of the entire smallest possible filling of Havayah (יוד הא וו הא). Alternately, the sum of the first 6 Lucas numbers, 44 is 4 times the value of the fifth Lucas number, 11.

Adding the next Lucas number, 29, the sum of the first seven numbers is 73, the value of “wisdom” (חָכְמָה). The next Lucas number is 47, the value of “self-nullification” (בִּטּוּל), the way that wisdom is experienced inwardly in the soul. The sum of the first eight Lucas numbers is thus 120, Moses’ lifespan.[2]

The First Lucas Number

The pair of integers that start of the Fibonacci-Love additive series is actually 1, 2. However, by convention, this series is usually said to start with 1, 1, 2. The 1 that precedes 1, 2, is indeed the previous number in the series, since additive series can be extended backward. Doing so shows that the reverse extension of the series is very similar to its continuation:

…, -21, 13, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

If we do the same for the Lucas series and extend it backward we get:

…, 18, -11, 7, -4, 3, -1, 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, …

The number preceding 1, 3 (the start of the Lucas series) is 2, thus if were to follow the same convention used regarding the Fibonacci-Love series, we would say that the Lucas series begins, 2, 1. Doing so, we would see that the first 4 numbers are 2, 1, 3, 4, which are the number of sefirot as divided by categories (with the first and second categories in reverse order):

  • 1 super-rational (crown)
  • 2 intellectual (wisdom and understanding)
  • 3 emotional (loving-kindness, might, and beauty)
  • 4 habitual (victory, acknowledgment, foundation, and kingdom)

Since there can be no other additive series with these 4 numbers in sequence, just as we gave the Fibonacci series a name that has meaning in Torah (the Love series), so we will designate the Torah name of the Lucas series, the Sefirah series of numbers. The word sefirah in Hebrew (סְפִירָה), literally means “to count.” The next two numbers in the Sefirah series are 7 and 11—the two numbers that are inherent to the counting of the female menstrual cycle in Torah: 7 days and 11 days.[3] The next Sefirah number is 18, which is the length in days of the menstrual cycle for Torah calculations. The next Sefirah number, 29, represents the length in days of the natural menstrual cycle, known as the “average cycle” (עוֹנָה בֵּינוֹנִית).[4]

Properties of Lucas-Sefirah Numbers

Further analyzing this way of starting the Lucas numbers, we find that the sum of the first four numbers, 10, is a product of 5: 2 times 5. The sum of the next four numbers is 65, also a product of 5: 13 times 5. The sum of the next four numbers is 445, which is 5 times 89. In each sum, the product is the fifth Love number (5) and another Love number (3, 7, and 11, in our short example) skipping 4 numbers each time, ad infinitum.

Another important relationship between these two additive series is discovered when we notice that every number in the Lucas-Sefirah series is the sum of two Fibonacci-Love numbers with a gap in between them. For example, 1 is the sum of 0 and 1 (skipping over 1). 3 is the sum of 1 and 2 (skipping over 1). 4 is the sum of 1 and 3 (skipping over 2). 7 is the sum of 2 and 5 (skipping over 3), and so on.

In conclusion, a phenomenon that is a bit more complex to understand. The sum of the products of the first 11 pairs of numbers, one from each series, is:

(2 ž 1) ┴ (1 ž 1) ┴ (3 ž 2) ┴ (4 ž 3) ┴ (7 ž 5) ┴ (11 ž 8) ┴ (18 ž 13) ┴ (29 ž 21) ┴ (47 ž 34) ┴ (76 ž 55) ┴ (123 ž 89) = 17712

17712 is itself the product of the 12th Love number, 144, and the 11th Sefirah number, 123. This result can be generalized algebraically.

[1]. For more on fillings, see the article on parashat Mikeitz.

[2]. As is well known, in the verse that describes how mankind’s lifespan was decreased to 120 years, Moses is alluded to, “Havayah said, ‘My breath shall not dwell in humankind forever, since it too is flesh; let the days allowed them be one hundred and twenty years’” (Genesis 6:3). The value of the word translated as, “since [it] too” (בְּשַׁגַּם) is 345, the same as “Moses” (מֹשֶׁה).

[3]. Hilchot Issurei Bi’ah 6:6.

[4]. Meaning, 29 days after the inception of the previous cycle. Shulchan Aruch Yoreh Dei’ah 199:1.

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