Golden Ratio using an equilateral triangle inscribed in a circle (opens in new tab)

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

I also talk about the Continued Fraction algorithm for factorising integers, which is still one of the fastest methods for numbers in a certain range.

Continued Fractions also give what is, to me, one of the nicest proofs that sqrt(2) is irrational.

https://youtu.be/ubHVK71F01M

This one actually has the geometric (rectangle subdivisions) animations I had in mind.

Sure, moving a heading slightly higher can make it look much better than if it was perfectly equidistant from the side and the top, but the precise amount depends on a million visual factors. The golden ratio might happen to work fine, but there's nothing magical about it.

Even temples that we thought followed the golden ratio for their dimensions have been measured better, and it turns out they don't. The civilizations back then knew enough so they could have made them very close to the golden ratio, but they didn't. Not always at least.

I find the aesthetic arguments for it very overrated, though. A clear case of a guy says a thing, and some other people say it too, and before you know it it's "received wisdom" even though it really isn't particularly true. Many examples of how important the "golden ratio" are are often simply wrong; it's not actually a golden ratio when actually measured, or it's nowhere near as important as presented. You can also squeeze more things into being a "golden ratio" if you are willing to let it be off by, say, 15%. That creates an awfully wide band.

Personally I think it's more a matter of, there is a range of useful and aesthetic ratios, and the "golden ratio" happens to fall in that range, but whether it's the "optimum" just because it's the golden ratio is often more an imposition on the data than something that comes from it.

It definitely does show up in nature, though. There are solid mathematical and engineering reasons why it is the optimal angle for growing leafs and other patterns, for instance. But there are other cases where people "find" it in nature where it clearly isn't there... one of my favorites is the sheer number of diagrams of the Nautilus shell, which allegedly is following the "golden ratio", where the diagram itself disproves the claim by clearly being nowhere near an optimal fit to the shell.

https://www.youtube.com/watch?v=8BqnN72OlqA

or the older black-and-white film which I was shown in school when I was young.

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(https://wonger.dev/enjoyables on desktop / wide viewport)

And yes, for the people who get hung up on what the Old Masters did, it’s mostly armature grids and not the golden ratio!

But if your units follow a golden ratio progression, you just need to "concatenate" 2 consecutive units (2 measuring sticks) in order to find the third. And so on.

https://en.wikipedia.org/wiki/Fibonacci_heap

It's probably no longer "Commercial In Confidence" ... I should probably write it up sometime.

Asking because to me, any mathematical abstraction is a natural construct. Math isn't invented, it's discovered.

> In this work, I propose a rigorous approach of this kind on the basis of algorithmic information theory. It is based on a single postulate: that universal induction determines the chances of what any observer sees next. That is, instead of a world or physical laws, it is the local state of the observer alone that determines those probabilities. Surprisingly, despite its solipsistic foundation, I show that the resulting theory recovers many features of our established physical worldview: it predicts that it appears to observers as if there was an external world that evolves according to simple, computable, probabilistic laws. In contrast to the standard view, objective reality is not assumed on this approach but rather provably emerges as an asymptotic statistical phenomenon. The resulting theory dissolves puzzles like cosmology’s Boltzmann brain problem, makes concrete predictions for thought experiments like the computer simulation of agents, and suggests novel phenomena such as “probabilistic zombies” governed by observer-dependent probabilistic chances. It also suggests that some basic phenomena of quantum theory (Bell inequality violation and no-signalling) might be understood as consequences of this framework.

You're welcome

We did some statistical analysis on the golden ratio and its use in art. It does indeed seem that artists gravitate away from regular geometry such as squares, thirds etc and towards recursive geometry such as the golden ratio and the root 2 rectangle. Most of our research was on old master paintings, so it might be argued that this was learned behavior, however one of our experiments seems to show that this preference is also present in those without any knowledge of such prescribed geometries.