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Going beyond the Golden Ratio

extremelearning.com.au

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Re: Going beyond the Golden Ratio

#21

Post by extremelearning »

svat wrote: post #18

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 f…

I pondered this issue for what seemed like an inordinate amount of time namely: on how to describe this subtle but key difference.

Unfortunately, I couldn't find a nice way, so i glossed over this point, which you correctly say makes many expressions and theorems cleaner and more elegant.

Furthermore, this difference helps explains why q^2 is a natural choice, which some other readers on this thread have enquired about.

Re: Going beyond the Golden Ratio

#22

Post by ComplexSystems »

svat wrote: post #18

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 f…

How do the results change if you use multiplicative error rather than additive error? That is, rather than |x-p/q|, you use max(x/(p/q), (p/q)/x). This is sometimes useful when trying to approximate rationals.

Re: Going beyond the Golden Ratio

#23

Post by extremelearning »

ComplexSystems wrote: post #22
svat wrote: post #18

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 f…

How do the results change if you use multiplicative error rather than additive error? That is, rather than |x-p/q|, you use max(x/(p/q), (p/q)/x). This is sometimes useful when trying to approximate rationals.

Good question, but I don't know and haven't really done anything substantially related that might even give us a hint.

Hopefully someone else chime in on this thread. ;)

Re: Going beyond the Golden Ratio

#24

Post by ml_basics »

extremelearning wrote: post #2

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

Really great post! One question. What exactly is meant by the following?

> the critical score... separates the world of infinite rationals with merely a finite number of rationals

I'm not sure that I understand what's being said here. There are countably (i.e. infinitely) many rationals, so is this saying that there is some particular finite set of rationals that are particularly relevant to the critical score?

Re: Going beyond the Golden Ratio

#25

Post by twic »

> 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.

Re: Going beyond the Golden Ratio

#26

Post by okmokmz »

slazaro wrote: post #5

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

I found this channel a while back and spent almost an entire day watching their videos. It's fascinating stuff, and pretty easily digestible even if math isn't necessarily your thing.

Re: Going beyond the Golden Ratio

#27

Post by extremelearning »

ml_basics wrote: post #24
extremelearning wrote: post #2

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

Really great post! One question. What exactly is meant by the following? > the critical score... separates the world of infinite rationals with merely a finite number of rationals I'm not sure that I understand what's being said here. There are countably (i.e. infinitely) many rationals, so is this saying that there is some particular finite set of rationals that are particularly relevant to the critical score?

Yes.

Consider π. For any S>0, you can construct an infinite number of rational approximations that have a score of less than S.

But for any quadratic irrational (surd), as the depth of the corresponding continued fraction increases, the score will converge (in an alternating manner) to a critical score, S.

This means that for any score S , there is only a finite set of rational approximations that have a score of less than S.

For example, in figure 3, for S=0.4 Hope that helps!

Re: Going beyond the Golden Ratio

#28

Post by JadeNB »

extremelearning wrote: post #15
onedognight wrote: post #10

Earlier quoted context omitted.

Why not penalize the error by the cube or some other higher power of the denominator?

Generally my answer is that this is for the same reason that fitting lines of best fit to data is nearly always done via a least-squares fitting. Squaring has a few major benefits. The first is that is never negative. Therefore, one might ask why don't we just take absolute value (1-norm)? It turns out that the absolute function makes many calculus expressions very messy. Thus, ironically, when analysing these concep…

Any answer to "why squaring?" that doesn't reference the usual Euclidean distance, even if indirectly via, say, the Pythagorean theorem, seems to be missing an obvious avenue of (if I may coin an awful word) intuitive-isation.

Re: Going beyond the Golden Ratio

#30

Post by extremelearning »

JadeNB wrote: post #29
extremelearning wrote: post #4

Earlier quoted context omitted.

LOL! You're totally right. It should be 85/10 and 425/50. Now fixed.

I still see 850/10 and 425/5. EDIT: OK, it seems to have been a cache issue as extremelearning supposed.

i suspect a caching issue. I cleared my wordpress cache, so hopefully it will appear correct to others soon. ;)
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