6 ms·

Going beyond the Golden Ratio

143 points by Sukotto 8y ago

extremelearning 8y ago
Author here. Happy to try to answer any questions any one might have on this post or topic. )
- VorticesRcool 8y ago
  Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them. Do you know what the metallic means are bounded by? Are they as bad as the silver ratio/(1+sqrt(2))? These most irrational numbers, (9+sqrt(221))/10, (13+sqrt(1517))/26... how interesting that they are not just the simple generalization of the continued fraction for the golden ratio.
  - extremelearning 8y ago
    What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio. As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m. For example, m=1 is the golden ratio, m=2 is the silver ratio,... But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m. In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3]. And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers". It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions. [1] https://en.wikipedia.org/wiki/Metallic_mean https://en.wikipedia.org/wiki/Metallic_mean [2] http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/ http://extremelearning.com.au/unreasonable-effectiveness-of-... [3] https://en.wikipedia.org/wiki/Plastic_number https://en.wikipedia.org/wiki/Plastic_number
- numbergeek666 8y ago
  open google earth use ruler for below miles from Angkor Wat to Giza pyramid 4754 miles. This multiplied by the glden ratio of 1.618 give 7692 miles which is the distance from Giza to Nazca . Now 7692 miles multiplied by the golden ratio again gives 12446, which is the distance from Nazca to Angkor Wat why?
  - qubex 8y ago
    It’s called ‘coincidence’: you have so many sites (N) to choose from, and there are N²︎ connections between them. To some degree of accuracy you’re going to find ratios between some of these that are ‘close’ to apparently ‘important’ numbers (and there’s plenty of those, and of course integer multiples thereof, which seem to catch just as much attention). It’s just a numbers game (excuse the pun). It’s just pure numerology. And an overabundance of ratios and constants and multiples thereof to choose from. It would be pretty unlikely that no such coincidental values would turn up.
    - mxfh 8y ago
      In this case, it's all about how triangles behave on a unit sphere, if one edge gets close to the length of π, or half a circumference. For Earth and Miles, r is 3963 and r * π = 12450, which is awfully close to 12446. We are effectively looking at a https://en.wikipedia.org/wiki/Spherical_lune https://en.wikipedia.org/wiki/Spherical_lune here. The dihedral angle can be chosen freely. One half great circle is going directly between the antipodal points, while the other half great circle is intersected at the ratio into two edges. So all you need to find are two antipodal points. Then any point lying on the two "small" circles defined by the ratio in either direction of the half great circle fullfills this condition. Helpful if you have bit of wiggle room with a place like Nazca. If we take for simplicity the North and South Pole then any point at the latitude 21.25 North or South would fullfill this condition. Mecca at 21.4N would come within 15 km of that band already.
      - mxfh 8y ago
        Oh boy: https://www.goldennumber.net/golden-ratio-of-earth/ https://www.goldennumber.net/golden-ratio-of-earth/
- onedognight 8y ago
  Why not penalize the error by the cube or some other higher power of the denominator?
  - extremelearning 8y ago
    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 concepts theoreticlly/algebraically it is usually easier to square the errors (use the 2-norm), rather than the 1-norm. The x^2 function is a very elegant function that smoothly curves. The |x| function has a pointy corner at x=0, which causes many analytical headaches. (Although, I must admit that in recent years with large-scale computing, errors based on the absolute value are making a notable comeback, especially in machine learning!) Secondly, history seems to have shown that squaring is frequently the simplest transformation that leads to non-trivial results. Thus, the principle of Occam's razor, would suggest that 2 is a very good place to begin and end. Finally, if we consider higher powers, it makes sense to ensure our errors are not negative, so that generally rules out cubes. Finding square roots, and roots of quadratic equations is relatively simple, but finding roots of degree 4 polynomials is very tough, and finding roots of higher even degree polynomials is usually intractable. Hope that helps!
    - JadeNB 8y ago
      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.
  - extremelearning 8y ago
    @svat's comment and link may also be helpful in this regard.
- typothrowaway 8y ago
  Very nice. I think there are some typos though, below "Here were some of them, along with their continued fractions." Two of the shorthand notations for the continued fractions and the last decimal expansion are wrong. Or I'm crazy.
  - extremelearning 8y ago
    Thanks. No, you aren't crazy, but maybe my typos made you crazy! ;) I have now fixed a couple of typos in the grammar and continued fractions expressions for that section.
- ml_basics 8y ago
  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?
  - extremelearning 8y ago
    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 < 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 < 1/√5 ≃ 0.447, there is only one fraction that gives a score of less than S=0.4. Hope that helps!
Yajirobe 8y ago
850/10 equals 85, not 8.5. 425/5 is equal to 85, not 8.5.
- extremelearning 8y ago
  LOL! You're totally right. It should be 85/10 and 425/50. Now fixed.
  - JadeNB 8y ago
    I still see 850/10 and 425/5. EDIT: OK, it seems to have been a cache issue as extremelearning supposed.
    - extremelearning 8y ago
      i suspect a caching issue. I cleared my wordpress cache, so hopefully it will appear correct to others soon. ;)
slazaro 8y ago
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 https://www.youtube.com/watch?v=sj8Sg8qnjOg
- extremelearning 8y ago
  Absolutely! Everybody loves the numberphile videos. They frequently distil deep maths topics into very intuitive and visual explanations. ;)
- okmokmz 8y ago
  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.
svat 8y ago
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-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...
- extremelearning 8y ago
  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.
- ComplexSystems 8y ago
  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.
  - extremelearning 8y ago
    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. ;)
twic 8y ago
> 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.

143 points by Sukotto 8y ago

extremelearning 8y ago

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

VorticesRcool 8y ago
Fascinating post. I had always assumed that the 3rd most irrational number would be the third metallic mean given by n = (n+ sqrt(n^2+4))/2, and subsequently the fourth metallic mean etc. The metallic means also pack the disks nicely. I have recently had my interest in them sparked after I came across solution to point vortex equilibria involving them. Do you know what the metallic means are bounded by? Are they as bad as the silver ratio/(1+sqrt(2))? These most irrational numbers, (9+sqrt(221))/10, (13+sqrt(1517))/26... how interesting that they are not just the simple generalization of the continued fraction for the golden ratio.
- extremelearning 8y ago
  What I find fascinating is that there seem to be so many valid ways to generalize the Golden Ratio. As you say, the "metallic means" [1] are quite well-known, and relate to the recurrence relation via: T(n) = m *T(n-1)+ T(n-2), for some constant integer m. For example, m=1 is the golden ratio, m=2 is the silver ratio,... But one of my other posts [2], generalizes the Golden ratio via the "Harmonious Numbers", as defined by the lagged recurrence, T(n+m) = T(n)+T(n-1), for some constant m. In this case, m=1 relates to the Golden Ratio, and m=2 relates to the Plastic Number [3]. And then finally, this post explores generalizing it via a completely different perspective, that of "Lagrange Numbers". It seems that we need to 'think outside the box' a litte when generalizing the Golden ratio, as there is not single obvious way to generalise continued fractions. [1] https://en.wikipedia.org/wiki/Metallic_mean https://en.wikipedia.org/wiki/Metallic_mean [2] http://extremelearning.com.au/unreasonable-effectiveness-of-quasirandom-sequences/ http://extremelearning.com.au/unreasonable-effectiveness-of-... [3] https://en.wikipedia.org/wiki/Plastic_number https://en.wikipedia.org/wiki/Plastic_number
numbergeek666 8y ago
open google earth use ruler for below miles from Angkor Wat to Giza pyramid 4754 miles. This multiplied by the glden ratio of 1.618 give 7692 miles which is the distance from Giza to Nazca . Now 7692 miles multiplied by the golden ratio again gives 12446, which is the distance from Nazca to Angkor Wat why?
- qubex 8y ago
  It’s called ‘coincidence’: you have so many sites (N) to choose from, and there are N²︎ connections between them. To some degree of accuracy you’re going to find ratios between some of these that are ‘close’ to apparently ‘important’ numbers (and there’s plenty of those, and of course integer multiples thereof, which seem to catch just as much attention). It’s just a numbers game (excuse the pun). It’s just pure numerology. And an overabundance of ratios and constants and multiples thereof to choose from. It would be pretty unlikely that no such coincidental values would turn up.
  - mxfh 8y ago
    In this case, it's all about how triangles behave on a unit sphere, if one edge gets close to the length of π, or half a circumference. For Earth and Miles, r is 3963 and r * π = 12450, which is awfully close to 12446. We are effectively looking at a https://en.wikipedia.org/wiki/Spherical_lune https://en.wikipedia.org/wiki/Spherical_lune here. The dihedral angle can be chosen freely. One half great circle is going directly between the antipodal points, while the other half great circle is intersected at the ratio into two edges. So all you need to find are two antipodal points. Then any point lying on the two "small" circles defined by the ratio in either direction of the half great circle fullfills this condition. Helpful if you have bit of wiggle room with a place like Nazca. If we take for simplicity the North and South Pole then any point at the latitude 21.25 North or South would fullfill this condition. Mecca at 21.4N would come within 15 km of that band already.
    - mxfh 8y ago
      Oh boy: https://www.goldennumber.net/golden-ratio-of-earth/ https://www.goldennumber.net/golden-ratio-of-earth/
onedognight 8y ago
Why not penalize the error by the cube or some other higher power of the denominator?
- extremelearning 8y ago
  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 concepts theoreticlly/algebraically it is usually easier to square the errors (use the 2-norm), rather than the 1-norm. The x^2 function is a very elegant function that smoothly curves. The |x| function has a pointy corner at x=0, which causes many analytical headaches. (Although, I must admit that in recent years with large-scale computing, errors based on the absolute value are making a notable comeback, especially in machine learning!) Secondly, history seems to have shown that squaring is frequently the simplest transformation that leads to non-trivial results. Thus, the principle of Occam's razor, would suggest that 2 is a very good place to begin and end. Finally, if we consider higher powers, it makes sense to ensure our errors are not negative, so that generally rules out cubes. Finding square roots, and roots of quadratic equations is relatively simple, but finding roots of degree 4 polynomials is very tough, and finding roots of higher even degree polynomials is usually intractable. Hope that helps!
  - JadeNB 8y ago
    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.
- extremelearning 8y ago
  @svat's comment and link may also be helpful in this regard.
typothrowaway 8y ago
Very nice. I think there are some typos though, below "Here were some of them, along with their continued fractions." Two of the shorthand notations for the continued fractions and the last decimal expansion are wrong. Or I'm crazy.
- extremelearning 8y ago
  Thanks. No, you aren't crazy, but maybe my typos made you crazy! ;) I have now fixed a couple of typos in the grammar and continued fractions expressions for that section.
ml_basics 8y ago
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?
- extremelearning 8y ago
  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 < 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 < 1/√5 ≃ 0.447, there is only one fraction that gives a score of less than S=0.4. Hope that helps!

Yajirobe 8y ago

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

extremelearning 8y ago
LOL! You're totally right. It should be 85/10 and 425/50. Now fixed.
- JadeNB 8y ago
  I still see 850/10 and 425/5. EDIT: OK, it seems to have been a cache issue as extremelearning supposed.
  - extremelearning 8y ago
    i suspect a caching issue. I cleared my wordpress cache, so hopefully it will appear correct to others soon. ;)

slazaro 8y ago

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

extremelearning 8y ago
Absolutely! Everybody loves the numberphile videos. They frequently distil deep maths topics into very intuitive and visual explanations. ;)
okmokmz 8y ago
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.

svat 8y ago

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-rational-approximations-are-the-convergents-of-the-continued-fraction/ https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rat...

extremelearning 8y ago
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.
ComplexSystems 8y ago
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.
- extremelearning 8y ago
  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. ;)

twic 8y ago

> 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.