What is the Golden Ratio or Fibonacci's Golden Section?

Hello, I would like to ask about the concept of the Golden Ratio, also known as the Golden Section, which appears in nature in various subjects and is closely echoed in our sacred Torah in the Ark of the Covenant and Noah's Ark (1.66 in both). My question is, what is the Torah's view on such a "proof," and are there other examples like this? Another question related to mathematics: Why is the pi number rounded to 3 in the Talmud?

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Hello and blessings,

Indeed, the Golden Ratio is a fascinating fact that many have pointed out, and some call it the "fingerprint" of Hashem in creation, or "Hashem's signature" in the universe. Unfortunately, there aren't enough articles in Hebrew on this important topic, from which we can learn about the divine planning in the entire creation.

See the Wikipedia entry on the Golden Ratio

"It is hypothesized that this ratio was discovered by one of Pythagoras's students; it was described in Euclid's book, "Elements," about 2,300 years ago. Back then, the ratio was called "the extreme and mean ratio." Luca Pacioli, an Italian mathematician from the Renaissance period, dedicated a whole book to the Golden Ratio and called it "The Divine Proportion." The ratio represents many measurements and sizes in nature and has been used in art and architecture since the time of classical Greece."

"The Golden Ratio appears or is claimed to appear in the following areas:
In geometry: in pentagrams and pentagons, in golden rectangles and golden triangles.
In botany: in the arrangement of leaves around the stem, the placement of seeds in an apple, in the pine cone, in the arrangement of sunflower seeds in the flower head, in the petals of a rose.
In crystal formations: Professor Dan Shechtman won the Nobel Prize for Chemistry in 2011 for discovering a pentagonal quasi-crystal that maintains the Golden Ratio.
In animals: in the horns of deer, the ratio among bees in a hive.
In art, architecture, and certain psychological processes."

Regarding the topic of pi in the Talmud, it is mentioned by the sages in Tractate Eruvin (page 14a):

'Every [circle] that has a circumference of three handbreadths, has a width [diameter] of one handbreadth. From where do we know these words? Rabbi Yochanan said: The scripture states: "And he made [Solomon] the Sea [a large basin] cast, ten cubits from brim to brim, round in compass, and its height was five cubits, and a line [circumference] of thirty cubits did circle it" (1 Kings 7:23).'

According to the Talmud, the ratio between the circumference of a circle and its diameter, known as "pi," is 3.

Rabbi Menashe Israel writes: On the contrary, precisely because of the inaccuracy in this matter, the sages needed to learn the issue from the verse to show that we should not be overly precise about what exceeds pi 3. If it were halacha to be precise about the calculation, there would have been no need to bring a verse on how much a circle's circumference is to its diameter. It's recommended that you read the following article from the Talmudic Encyclopedia on the subject, also called "the Golden Section," "the Golden Ratio." It is indeed a fascinating subject that proves divine creation in the universe. 

In blessings,

Daniel Bales

For more on the Golden Ratio/Golden Section, click here.