The Golden Ratio (why it is so irrational) - Numberphile

a mind-rewiring video by numberphile concerning the golden ratio [0] made sense of all the spiral overlay art: yes, plants use it because it is an optimal solution for minimizing overlap when sprouting leaves during growth, maximizing spatial coverage but also, as a number that is least able to be approximated by rational, erm, ratios - it is a pattern that most closely approximates random patterns. So more than any other spiral, a fibonacci spiral has the highest likelihood of overlapping random points in 2D space... (and fibonacci series converges on the golden ratio because they are equivelent expressions of the continued fraction 1+(1/(1+(1/(1+(1/(1+(1/... [0] ▶ this video

Golden ratio is not arbitrary, it solves the problem: How to make sure nothing lines up. This link gets passed around a lot and it's very illuminating, demonstrating that the golden ratio is objectively "the most irrational number": ▶ this video

I learned about this not from Mathologer, but Numberphile [1]. The second half of the video is the continued fraction derivation. I remember this being the first time I appreciated the sense in which the phi was the most irrational number, which otherwise seemed like just a click-bait-y idea. But you've found an earlier (9 years ago vs 7) Mathologer video on the same topic. [1] ▶ this video

The article attempts to explain this property of phi: > Maybe you have a picture of a flower, and you want to implement “every time the user clicks the mouse, add a petal to the flower.” In that case you want to use the golden ratio: Make the angle from one petal to the next 360/phi and you can loop around the circle forever, adding petals, and the next petal will always fit neatly into the biggest gap and you’ll never loop back to your starting position. And points to a video. I think there's a much clearer explanation which shows how this is because (in some sense) phi is the "most irrational" ratio, stemming from its derivation from continued fractions: ▶ this video I freely admit I have not a clue how this does or doesn't improve hash algos, but it's a cool video! :)

Not directly related to the phenomenon in the article, Numberphile has a video[1] that goes into why pi has a surprising amount of regularity despite being transcendental. [1]: ▶ this video

I recently saw this [0] Numberphile video that touches some of the similar stuff at the end of this article, with the spirals being animated. [0] ▶ this video

Phi is the most irrational of the irrational numbers, meaning that successive rational approximations are just slightly better: 1/1, 1/2, 2/3, 3/5, 5/8, 8/13, etc. There are no rational approximations that "jump out" as being much better, so you won't find any unexpected patterns when using it. The Golden Ratio (why it is so irrational) - Numberphile ▶ this video