
Going beyond the Golden Ratio - Sukotto
http://extremelearning.com.au/going-beyond-golden-ratio/
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svat
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...](https://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-
approximations-are-the-convergents-of-the-continued-fraction/)

~~~
ComplexSystems
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
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. ;)

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slazaro
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)

~~~
okmokmz
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.

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

~~~
numbergeek666
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?

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qubex
It’s called ‘ _coincidence_ ’: you have so many sites ( _N_ ) to choose from,
and there are _N_ ²︎ 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
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
Oh boy: [https://www.goldennumber.net/golden-ratio-of-
earth/](https://www.goldennumber.net/golden-ratio-of-earth/)

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

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

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

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

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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.

