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...
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.
Hopefully someone else chime in on this thread. ;)
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.
As you say, the "metallic means"  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 , 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 .
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.
> 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!
I have now fixed a couple of typos in the grammar and continued fractions expressions for that section.
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.
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
I bet you can find even more patterns, where there are none: http://www.tylervigen.com/spurious-correlations
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.
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.
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.
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
Sounds like he had a badly approximable name.