Going beyond the Golden Ratio • Hacker News Board

Re: Going beyond the Golden Ratio

#11

Post by qubex » Tue, Mar 12, 2019, 11:17 AM UTC

numbergeek666 wrote: post #9

extremelearning wrote: post #2

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

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?

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.

Re: Going beyond the Golden Ratio

#12

Post by typothrowaway » Tue, Mar 12, 2019, 11:22 AM UTC

extremelearning wrote: post #2

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

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.

Re: Going beyond the Golden Ratio

#13

Post by extremelearning » Tue, Mar 12, 2019, 11:24 AM UTC

numbergeek666 wrote: post #9

extremelearning wrote: post #2

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

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?

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.

Re: Going beyond the Golden Ratio

#14

Post by mxfh » Tue, Mar 12, 2019, 11:45 AM UTC

extremelearning wrote: post #13

numbergeek666 wrote: post #9

Earlier quoted context omitted.

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?

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.

Re: Going beyond the Golden Ratio

#15

Post by extremelearning » Tue, Mar 12, 2019, 11:50 AM UTC

onedognight wrote: post #10

extremelearning wrote: post #2

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

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!

Re: Going beyond the Golden Ratio

#16

Post by extremelearning » Tue, Mar 12, 2019, 11:57 AM UTC

typothrowaway wrote: post #12

extremelearning wrote: post #2

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

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.

Re: Going beyond the Golden Ratio

#17

Post by numbergeek666 » Tue, Mar 12, 2019, 11:58 AM UTC

mxfh wrote: post #14

extremelearning wrote: post #13

Earlier quoted context omitted.

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

https://developers.google.com/earth-engine/geometries_planar...

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

Re: Going beyond the Golden Ratio

#18

Post by svat » Tue, Mar 12, 2019, 12:48 PM UTC

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.

Re: Going beyond the Golden Ratio

#19

Post by maze-le » Tue, Mar 12, 2019, 12:48 PM UTC

numbergeek666 wrote: post #9

extremelearning wrote: post #2

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

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?

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

Re: Going beyond the Golden Ratio

#20

Post by extremelearning » Tue, Mar 12, 2019, 12:53 PM UTC

onedognight wrote: post #10

extremelearning wrote: post #2

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

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

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