from Hacker News

Going beyond the Golden Ratio

by Sukotto on 3/12/19, 6:50 AM with 32 comments

  • by svat on 3/12/19, 12:48 PM

    This is a lovely and gentle (you hardly realize it) to a lot of very deep mathematics... great post, thanks to the author! There's a lot I learned and would love to look up and continue to explore.

    ---

    As an aside, one thing I like to point out though is that the definition of “good approximation” seems to some extent determined by what has the cleanest theory, than what one may naively desire, as in this paragraph from the article:

    > Emily consider ways of giving each answer a score. Initially, she thought that for each fraction, the score could be the (absolute) difference between her number and the proposed fraction, and then multiplied by the denominator. (The lower the better). However, after talking to some of her tech friends, she decided to make it even stricter [...] denominator squared.

    A similar thing comes up in many expositions of “best rational approximation” in books and on the internet, where instead of |x-p/q| we use |q(x-p/q)| = |qx-p|, and here in this post for even cleaner theory we're using |q(qx-p)|. A post I wrote a while ago to clarify this issue, with a small C program: https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...

  • by slazaro on 3/12/19, 10:26 AM

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

  • by extremelearning on 3/12/19, 9:40 AM

    Author here. Happy to try to answer any questions any one might have on this post or topic. )

  • by Yajirobe on 3/12/19, 10:14 AM

    850/10 equals 85, not 8.5. 425/5 is equal to 85, not 8.5.

  • by twic on 3/12/19, 2:24 PM

    > Emily had stumbled on a very counter-intuitive pattern first discovered by Markoff (in this very specific field of maths his name is traditionally spelled ‘Markoff’ but in all other areas, it is usually spelled ‘Markov’).

    Sounds like he had a badly approximable name.