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Mathematics

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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 Time

In the current Torah portion Emor, we are instructed to abstain from work on Shabbat—every seventh day. Next week’s Torah portion, Behar, continues this theme and instructs us to abstain from agricultural work every 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 c. kabbalist Rb. Isaac [...]

Joseph and His Brothers

I grew up In Russia and was raised on the metric system based on the decimal arithmetic. When we immigrated to the U.S., I had to learn a new to me (but, actually, very old) Imperial system of measurements having 12 inches in a foot or 12 ounces in a pound. I still struggle with it (decimal system, apparently, was hardwired in my brain). There are many number systems or positional notation systems. Ancient Babylonians used base-60, Hindu-Arabic system uses familiar decimal base-10 system, Mayans used base-20 system. There are base-2 binary, base-12 duodecimal, and base-16 hexadecimal systems, to name a few. Aside from the binary system used by computers, we humans mostly use decimal and, sometimes, duodecimal systems. The all-familiar decimal system is used in the metric system of measurements. In base-10 [...]