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Going beyond the Golden Ratio (extremelearning.com.au)
143 points by Sukotto on March 12, 2019 | hide | past | favorite | 32 comments

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...

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.

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. ;)

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.

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

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

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.

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

[2] http://extremelearning.com.au/unreasonable-effectiveness-of-...

[3] https://en.wikipedia.org/wiki/Plastic_number

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?


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!

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.

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.

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 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!

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.

@svat's comment and link may also be helpful in this regard.

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


Because our brain is a superb pattern matcher. It can see pattern almost everywhere, even when there are none. That has helped our ancestors find animals that can be hunted, and prepare for flooding events timed at certain intervals (12 full moons until the next harvest/sowing season). But that also leads to situations where coincidental patterns are found, where the constituent parts have no casual relationship to one another.

I bet you can find even more patterns, where there are none: http://www.tylervigen.com/spurious-correlations

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.

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 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.

Pleased to make your acquaintance, geometer! I’m of the algebraic faction myself. I bid you a pleasant day.

The distance from Angkor Wat to Giza is: 4754 miles. And the distance from Gize to Nazca is: 4754 x φ miles.

By adding these, you get that the distance from Angkor Wat to Nazca is 4754 (1+φ) miles.

But φ is defined such that 1+φ = φ², [Verify for yourself that 1.61803398875² = 2.61803398875]

So thus, the distance from Ankor Wat to Nazca can also be described as 4754 φ² miles.

Also were talking spheroid surface distances aka geodesics here. So unless three places lie on a great circle their distances are not cummutative, since they form a geodesic triangle.

Google Earth Engine's geometry constructors build geodesic geometries by default


A geodesic is a locally length-minimizing curve. Equivalently, it is a path that a particle which is not accelerating would follow. In the plane, the geodesics are straight lines. On the sphere, the geodesics are great circles (like the equator). The geodesics in a space depend on the Riemannian metric, which affects the notions of distance and acceleration

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

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. ;)

> 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.

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