Math CentralQuandaries & Queries


Question from Vincent, a student:
hello, my name is Vince, I am wondering how the Golden ratio was used by early mathematicians.
What formula did they use to find it? open for anything... thank You!


The Ancient Greeks used it in geometry, and called it the "extreme and mean ratio" (The name "golden ratio" only goes back to Ohm, in the 19th century.) It turns up in Euclid's Elements in many places.

They were aware of its relationship to the regular pentagon and pentagram. They also knew that if you construct a rectangle with this ratio and cut off a square the remainder is in the same proportion but rotated 90 degrees. They knew it was an irrational number.

It is not clear whether the Ancient Greeks consciously used it in architecture; most examples that people give are much rougher approximations than Green architects and masons could have achieved if they had wanted to.

The relationship with Fibonacci numbers is comparatively modern; the Fibonacci sequence itself was not known until around 1200 CE.

New applications are still being discovered, Roger Penrose's aperiodic tiles being a famous example of a few decades back.

Good Hunting!

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