Mathematics

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The Puzzle of Pi

March 14 is celebrated by nerds around the world as the Pi Day. When written in digits, 3/14 represents first three digits of the number traditionally represented by the Greek letter “π” (pronounced pi) – 3.14. Pi Day is celebrated by eating pie and discussing the significance of π. Some folks who have too much time on their hands compete at memorizing decimal digits representing the value of pi. The record presently stands at 67,890 digits! While eating pie is optional, discussing the significance of π is truly mandatory. So, what is π and why is to so important? π is arguably the most famous mathematical constant, expressing the ratio between the circumference of a circle and its diameter. If you take a circle with a diameter of 1, the circumference of this [...]

Vectors, Spins, and Gender

Vectors, Spins, and Gender Speak to the children of Israel, and have them take for Me an offering; from every person whose heart inspires him to generosity, you shall take My offering. (Ex. 25:2) Every physics professor teaching physics 101 to freshmen spends the first lecture teaching vector algebra. Why vectors are so important? Because most quantities in mechanics (a branch of physics dealing with motion and forces) are vectors. A vector is a mathematical object that has a magnitude and a direction. For example, velocity is a vector – it shows not only how fast the object is moving, but also in what direction it is moving (it is different from “speed” which is an average velocity and is just a number). Similarly, acceleration is a vector, force is a vector, angular [...]

Fractal Tu B’Shevat

When thou shalt besiege a city a long time, in making war against it to take it, thou shalt not destroy the trees thereof by wielding an axe against them; for thou mayest eat of them, but thou shalt not cut them down; for is the tree of the field man, that it should be besieged of thee? (Deut. 20:19) On Tu B’Shevat we celebrate the New Year of the trees. There is much to be said about the legal, ecological, homiletical, and mystical meaning of this day. And, indeed, much has been written about this minor holiday from these perspectives. In this post, however, I’d like to focus on the mathematical and biological aspects of this holiday that, to my knowledge, had not been discussed until now. When the Torah rhetorically asks, [...]

Jacob Meets Esau and his 400 men

And the messengers returned to Jacob, saying: “We came to thy brother Esau, and moreover he cometh to meet thee, and four hundred men with him.” (Gen. 32:7) Why does the Torah take pains to tells us the exact number of men accompanying Esau? Servants of Esau are of no particular significance. It is apparent that, by specifying the number of men in Esau's entourage, the Torah is trying to tell us something about Esau. It seems to me that Jacob understood the significance of the number 400. This number is a sum of the powers of 7 from 0 to 3: 70+71+72+73=400. This four powers of 7 hint at the lunar month that has, approximately, 28 days (4×7). The significance of this will become apparent later. 400 is a repdigit in base [...]

The Meaning of Life as Taught by Bayesian Angels

And he [Jacob] dreamed, and behold! a ladder set up on the ground and its top reached to heaven; and behold, angels of God were ascending and descending upon it. (Gen. 28:12) Rabbi Shneur Zalman of Liadi (a.k.a Baal HaTanya, or the Alter Rebbe) once remarked in Yiddish, “lieben mit di tzait” – live with the time, meaning one should live with Torah portion of the week.  Last week I was taught a lesson on how the current Torah portion affects my thinking, whether I know it or not. Last week I shared with my wife, who is a biophysicist, that while thinking about system biology, I realized that all organisms – from a single cell to multicellular organisms – are Bayesian systems. I knew that the brain is a Bayesian engine, but [...]

Tisha B’Av on Shabbat – A Relativistic Perspective

Yesterday was the 9th day of the month of Av or, in Hebrew, Tisha B’Av. Usually, Tisha B’Av is marked by mourning and fasting. Yesterday, however, we ate festive meals, drank wine and were prohibited from fasting or displaying any signs of mourning. Because yesterday was Shabbat. Shabbat pushes off the observances of Tisha B’Av by a day. Indeed, today, Sunday, we fast and mourn the destruction of the First and the Second Holy Temple  – Bet Hamikdash – in Jerusalem, we remember the Holocaust and many other tragedies that befell the Jewish people. Why couldn’t we observe Tisha B’Av on Shabbat? After all, that was the day when on the 9th of Av, both Temples were destroyed! The simple explanation, of course, is that on Shabbat there is no mourning. On Shabbat, [...]

Three-and-a-half Hakafot — Topology of Simchat Torah

Why is this night different from all other nights, asks a child on the Seder night. On this Simchat Torah I asked a different question—why is the day different from the night? Indeed, on the night of Simchat Torah, we dance seven hakafot-circuits. However, during the morning service of the next day, we only dance three-and-a-half hakafot. What is the meaning of this number—three and a half? There are a few instances the Torah, Talmud, and Rabbinic instances where this number is mentioned (e.g., during the Gaonic period, c. 590–1000 CE, in some communities in the Land of Israel, the Torah reading cycle was completed in three and a half years; Maimonides rules that only half of the tzitzit string should be dyed blue leaving three and a half strings white), but none of them [...]

Quantum Cheshire Cat and Resurrection

In memory of my father, Abraham Shamshin ben Reuven, ע"ה   For those of us who can't get enough of Schrödinger cat, comes a new feline—Quantum Cheshire Cat—the creation of an Israeli physicist, Yakir Aharonov. In Alice in Wonderland, Alice meets a grinning Cheshire cat. To her amazement, the cat disappears leaving only his grin behind: "All right', said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone. Well! I've often seen a cat without a grin, but a grin without a cat! It's the most curious thing I ever saw in my life!" According to Martin Gardner, the statement "a grin without a cat" is a reference to mathematics dissociating itself [...]

Symmetry and Love — Jewish Chromodynamics

Ye are standing this day all of you before the Lord your God: your heads, your tribes, your elders, and your officers, even all the men of Israel, your little ones, your wives, and thy stranger that is in the midst of thy camp, from the hewer of thy wood unto the drawer of thy water; that thou shouldest enter into the covenant of the Lord thy God—and into His oath—which the Lord thy God maketh with thee this day. (Deut. 29:9-11) The above verses at the beginning of the Torah portion Nitzavim that is always read in the week preceding the Jewish New Year, Rosh HaShanah, are usually interpreted in terms of the unity of Jewish people: You are standing this day all of you [read: standing together in perfect unity]. This is not [...]

Fractal Patterns in the Torah

In the Torah portion Emor (Leviticus 21:1–24:23), we are instructed to abstain from work every seventh day on Shabbat. Next week’s Torah portion, Behar, continues this theme and instructs us to abstain from agricultural work every seventh Sabbatical year, Shmita. And the Torah doesn’t stop there. It instructs us to count seven Shmitas and then observe a Jubilee, Yovel. Do you notice a pattern? Every seven days, every seven years, every seven Shmitas… Furthermore, the Midrash states the world will exist for seven thousand years with the seventh millennium being a thousand years of the kingdom of Mashiach (Messiah)—yom shekuloy Shabbat—one long Shabbat. A second-century sage, Rabbi Huniah ben HaKanah, interprets this Midrash to mean that the world will last seven Cosmic Shmitas, i.e., 49 thousand years (which, according to a prominent 13th–14th [...]

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