Mathologer
3 HN recommenders · recommended in 5 months, 2019–2026
13:29 · 658K views · Jul 30, 2016
Mathologer uses continued fractions to show why the golden ratio is the most irrational number, comparing it with root 2, e and pi and pointing to where this shows up in nature.
Recommended as a clear, visual explanation of continued fractions and why phi is hardest to approximate by fractions.
5 comments from Hacker News
Pretty good Mathologer video about continued fractions: this video One neat trivium I like to quiz the mathematically-minded on is: what number is the most irrational , i.e., toward which number do rational approximations converge the slowest ? The answer can be determined from continued fractions and is discussed in the video near the end.
"same with golden ratio (repeating 1s)" Mathologer has a good video in which he uses this to justify calling the golden ratio the "most irrational" number because of this expansion: this video The video also goes over the details of the pi expansion.
I was going to post this Mathologer video ( this video ) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi. But its still a really interesting video!
Edit: years of searches and minutes after I post this I found this video thanks to using "continued fraction" in my search instead of "infinite series" X( Original: Tangentially, for a few years I've been looking for a Youtube video, I think by Mathologer [1], that explained (geometrically?) how the Golden Ratio was the limit of the continued fraction 1+1/(1+1/(1+1/(...))). Anyone know what I'm talking about? I know Mathologer had a conflict with his editor at one point that may have sown chaos on his channel. [1] [LINK]
The golden ratio is the most irrational number: this video This is why it creates some distinctive patterns when applied to circular/spiral shapes.