Both of those claims have been broadly debunked, and the so-called golden ratio is grossly oversold.
It also shows up anywhere with pentagonal (or icosahedral) symmetry, since the golden ratio is the ratio of the diagonal to the side of a pentagon. So that includes e.g. various viruses with a shape based on icosahedral symmetry.
The stuff about shells, galaxies, etc. (and the supposed advantages of this specific aspect ratio for design / visual art) is generally bullshit.
But that doesn’t mean that there’s some magical feature of hats that produces transcendent art.
Hambridge's theories were quite controversial, but the book is an enjoyable read anyway. https://archive.org/details/ElementsOfDynamicSymmetryHambidg...
I mostly just use frameworks that think about this stuff for me while working hard to keep consistency in spacing (padding, margins, etc) which functional ones like basscss and tachyons do a good job of pushing you towards (much more than Bootcamp and others).
The golden rectangle and spiral represent this best to me. The whole is found in the parts, infinitely, the further you iterate, like looking down a well and seeing all the way to the bottom. It's gratifying to conceive of more by doing less. I wish I were better with words, but I do feel φ's beauty somehow boils down to efficiency of representation.
With a large enough population, you are going to end up with the same representation (structure of neurons, glial cells etc) in some subset of brains.
As a painter, I am very aware that taste is not as universal as is constantly advertised. Some people like the shades of blue I use and some don't or are indifferent. Why I chose that blue and why they liked that blue is cause we probably share, in some small corner of a trillion brain cells, a small clump of structure that's exactly alike.
Also, to be 100% accurate, I should say that technically evolution is a form of a genetic algorithm which is slightly different and more constrained than a pure random search, but it's not that big of a deal with respect to this conversation.
Also it is turned up to an extreme degree here. I'm using Firefox and the scrolljacking doesn't happen in FF, but when I open it up in another browser, a scrollwheel flick that would normally scroll down 1/4rd of the page is enough to continue down all the way to the very end. Even with smaller scrolls, it goes around 1.5-2.5x the expected distance.
Scrolling is muscle memory and the consistent behaviour of it makes us have expectations for what scrolling should do, flagrantly destroying those will cause unease.
While the main brainwave bands are harmonics (theta is roughly double delta, alpha is roughly double theta, beta is roughly double alpha and so on), the band widths have a ratio of the golden ratio. That's because the golden ratio is the most irrational number. While harmonic oscillations will overlap regularly, irrational frequency ratios will never overlap. This allows for multiplexing in the brain, EMG , where high theta and low theta don't interfere with one another.
Pletzer, B., Kerschbaum, H., & Klimesch, W. (2010). When frequencies never synchronize: the golden mean and the resting EEG. Brain research, 1335, 91-102.
This is why it creates some distinctive patterns when applied to circular/spiral shapes.
Discussion at: https://www.reddit.com/r/math/comments/2zop5o/i_realised_the...
This relation is seen in the coordinates of the vertices of the solid: all permutations of (0, ±1, ±golden ratio).
They can't into math, can they? Or is it a special notation of some sort?
a / b = (a + b)/a = 1 + b / a = 1.618..
(a) / (b) = ((a) + (b)) / (a)
> Let’s imagine that you need to start your design by creating a line. Next, you copy it and divide it into two parts, getting two shapes a) the first line, and b) the second one.
What does this even mean? What are the constraints between these two lines that leads to that ratio to be the golden ratio?!?