We learned to look at infinity from different angles: mathematics, philosophy, artistic. We also discuss how sound and music relate to it. While doing research for this extension, I was so excited to learn that for the Pythagoreans, numbers, music and harmony were considered among the first principles guiding the universe.
Relating to mathematics, the Greeks first understood harmonics that vibrating strings and columns of air produced overtones. And, Pythagoras, specifically, “described the arithmetic ratios of the harmonic intervals between notes, for examples: Octaves, two-to-one ... fifths, three-to-two ... and fourths, four-to- three” (Rockmore) Pythagoras also noted that if two commensurate strings were strummed to vibrate, then the tones that they produced would be pleasing in harmony. Thus, the Greeks believed that all the harmonious things in the world must be based on whole number ratios. And all measurements must be rational.
Diagonal of a square
The Pythagoreans’ belief now changed because they discovered that the diagonal of a square is never commensurable with its side. No matter how many units they divided up each side of the square into, there was always a small amount leftover when they tried to measure the diagonal with this basic unit.
In another words, “in any measuring system that gave a whole number of units to the side, the diagonal would have a length that has to be expressed as an infinite decimal expansion” (Rockmore). So, square root of 2 is irrational and therefore cannot be expressed as a ratio of two integers.
I'm glad you posted this to the blog. I was doing some research and read about this too. Very cool! Harmony plays a vital role in so many things.
ReplyDeleteInteresting. I remember learning about how much the Greeks used math to figure out proportions for art and architecture. Everywhere around us we are surrounded by the Greeks' influence!
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