Parshat Matot describes the war of retribution with the Midianites, whose women seduced Jewish men and caused the terrible calamity of the plague described at the end of parshatBalak. One of the highlights of the war was the killing of Balam, the sorcerer who had encouraged Balak, the king of Mo'ab, to send the Midianite seductresses in the first place.
The Torah relates that: "And Balam the son of Be'or they killed by the sword."1 Yonatan ben Uziel's translation of the Torah into Aramaic relates the following about this verse:
It happened that when Balam saw Pinchas the priest chasing him he spoke a magic word and took off into the air. Pinchas then spoke the great and holy Name and flew after him. Holding him by his head, Pinchas drew his sword to kill Balam.
Balam opened his mouth pleading for mercy. He said: "I swear to you that as long as I live, I will not curse your people."
But, Pinchas replied: "You are Laban the Aramite who wished to destroy our patriarch Jacob. It was you who followed him to Egypt to ensure that his offspring perish, and it was you who followed us out of Egypt and encouraged the wicked Amalekites to make war upon us. And then you tried to curse us. But, when you saw that your actions were not successful and your words were not effective, you had a wicked king appointed, Balak, whose daughters you sent to seduce us and because of whom 24,000 Jews perished. For all of these reasons we cannot let you live."
Pinchas took his sword from its sheath and killed him.
One of the most important tools for learning Torah is to search for other instances where a particular event or person is mentioned. Balam's death by a sword is mentioned once more in the Book of Joshua. There it says, "And Balam the son of Be'or the sorcerer was killed by the Children of Israel by a sword."2 Note how similar these two verses are, but with two additions: "the sorcerer" and "by the Children of Israel."
Let us write out these two verses in Hebrew:
וְאֵת בִּלְעָם בֶּן בְּעוֹר הָרְגוּ בֶּחָרֶב
וְאֶת בִּלְעָם בֶּן בְּעוֹר הַקּוֹסֵם הָרְגוּ בְנֵי יִשְׂרָאֵל בַּחֶרֶב
The verse in parshat Matot contains 21 letters, whereas the verse in Joshua contains 34 letters. Both of these numbers are part of the famous Fibonacci series, or as we call it, the Love series of numbers.
Let's write out the first ten numbers of this very special series:
1 1 2 3 5 8 13 21 34 55
The Love series is an additive series, meaning that every number is the sum of the two numbers that precede it. One of the most important characteristics of this series is that the ratio between any two sequential numbers is an approximation of the golden number 1.618 (or its reciprocal 0.618). For this reason, as a matter of definition, when discussing this series we say that the golden section of each number is the two numbers that precede it.
If we look at the text of the verse in Matot, we will see that the cantillation marks divide it in two. The first part comprises the first 4 words and contains 13 letters while the second part, which comprises the final 2 words contains 8 letters. Thus, the 21 letters of the verse are divided internally into their golden section: 13 and 8 letters.
Looking at the text of the verse in Joshua, we find the same phenomenon. The verse in Joshua has 13 more letters than the verse in parshat Matot. But, these letters are divided into 3 words (1 word and 2 words, the golden section of 3). The first additional word, "the sorcerer," in the first half of the verse, has 5 letters and the other two additional words, "the Children of Israel," in the second half of the verse, have 8 letters. The 8 letters of the two words of the phrase, "the Children of Israel" divide into 3 letters and 5 letters.
Now, let us focus on Balam's name in each verse. In parshat Matot, he is called "Balam the son of Be'or" and in Joshua he is called "Balam the son of Be'or, the sorcerer." The sum of the numerical value of these two phrases is 1155 or 21 times 55, the two love numbers that precede and follow 34 in the love series.
… By Numbers
One of the most important and ubiquitous uses of love numbers and the golden section is found in art. The greatest artists in history were aware of the importance of the golden section in the form of the human body. Artists such as Leonardo da Vinci and Piet Mondrian were careful to make precise measurements before drawing in order to get beautiful results. Indeed, as discussed elsewhere in length, the true source for the art that is the golden section is found in the Torah in the chapter describing the ancient kings of Edom.3 The Arizal adopted many of the mathematical structures found in that chapter into his Kabbalah. In fact, Balam is considered to be equivalent to the first of those ancient kings, Bela the son of Be'or.
All this is to say, that since Balam exemplifies the golden section, when Pinchas killed him with his sword, it would be safe to say that he did so using the golden section! In other words, even when destroying one of the greatest enemies of the Jewish people, Pinchas included the special quality of the Love numbers.
Another property of the Love numbers
Returning to our mathematical discussion, note that 1155 is just one less than 1156, which is the square of 34! This is not by chance. There is a beautiful mathematical property of the love series that the square of any number in the series is equal to the product of the number that precedes it and the number that follows it, plus or minus 1 (alternately).
22 = 1 ∙ 3 ┴ 1
32 = 2 ∙ 5 – 1
52 = 3 ∙ 8 ┴ 1
82 = 5 ∙ 13 – 1
132 = 8 ∙ 21 ┴ 1
212 = 13 ∙ 34 – 1
342 = 21 ∙ 55 ┴ 1
Equidistance in Love Numbers
We can take this rule and extrapolate it to find an even more astounding result regarding Love numbers. We begin by asking, why is it that we have to add or subtract 1 in the previous rule? The answer is that 1 is the first Love number and that 1 is how far away in the series the numbers we are multiplying from the number we have squared. [Indeed, as we shall see, the divergence of plus or minus 1 should be understood to mean plus or minus 12 (= 1).] For example, if we start with 8—the 6th Love number—by multiplying the 5th Love number, 5, by the 7th Love number, 13, we get within 1 difference of the square of 8.
We might now ask, what would happen if we would take the Love numbers that are equidistant from 8, but whose distance is greater than 1? Let's define the distance d as the distance between the number we are squaring and the two numbers we are multiplying. In our previous example, d was set equal to 1.
Now let's look at what happens when we define the distance as 2, meaning d = 2. Let's use 8 once more. Since 8 is the 6th love number we need to multiply the 4th and 8th Love numbers, which are 3 and 21.
1 1 2 3 5 8 13 21 34 55
Their product is 63 which is 1 off from 64, the square of 8. So again, it seems that we are simply 1 off from the square. But, actually, this 1 is different from the first difference of 1 (when d was equal to 1). This 1 is actually the 2nd Love number, which is also 1, hence the similarity. This will become clearer when we set d = 3. Using 8, we need to multiply the 3rd Love number by the 9th Love number:
1 1 2 3 5 8 13 21 34 55
These are 2 and 34, whose product is 68, which is 4 more than 82 = 64. What is this 4? It is the square of the 3rd Love number (remember that we have selected d = 3), 2.
Let's check this once more for d = 4. We need the 2nd Love number and the 10th, which are 1 and 55:
1 1 2 3 5 8 13 21 34 55
Their product is 55, which is 9 less then 82 = 64. But, 9 is simply the square of the 4th Love number, 3.
If we express our finding in mathematical language we can write:
(Li)2 = Li–d ∙ Li┴d ± (Ld)2
Where Li designates the ith Love number and d is the distance as defined above.