I saw this Numberphile video and thought it was interesting.

https://www.youtube.com/watch?v=sj8Sg8qnjOg

Of course, I already knew that φ equals the continued fraction [1; 1, 1, 1, 1, ...], but didn't realize what this had to do with "optimizing" the seed arrangement of a sunflower.

If φ is "the most irrational number", it raises the question of how to quantify what makes one real number "more irrational" than another. Any ideas for how to define "relative irrationality"?

One practical use of "almost rational" irrational numbers is musical tuning. For example, the equal-tempered major third has a frequency ratio of the cube root of two (1.259921 decimal, 1.315188 dozenal), approximating the just intonation ratio 5/4. So, would a tuning based on setting the minor sixth to φ (or equivalently, setting the major fifth to (8/φ)^(1/4), or 691.7274 cents) sound dissonant?

This was explained in the video. φ is maximally irrational precisely because its continued fraction coefficients are all the lowest possible value: 1. Any natural number denominator greater than one will tend towards dominating the result, such that you could take just the terms up to that point, and use them to construct a rational number that is a "ridiculously good" approximation of the irrational number, so that it would take many additional terms to calculate a significantly better approximation.Dan wrote: I saw this Numberphile video and thought it was interesting.

https://www.youtube.com/watch?v=sj8Sg8qnjOg

Of course, I already knew that φ equals the continued fraction [1; 1, 1, 1, 1, ...], but didn't realize what this had to do with "optimizing" the seed arrangement of a sunflower.

If φ is "the most irrational number", it raises the question of how to quantify what makes one real number "more irrational" than another. Any ideas for how to define "relative irrationality"?

So I suppose we could define a relative scale of "rationality" by weighting the coefficients by their size and perhaps their position in the series (giving greater weight to earlier coefficients).