I once made a discovery about modified Fibonacci sequences.
Let's say the usual Fibonacci series is the 2-nacci series where F(n) (n-th number of the series is recursivley defined by(n>=2): F(n)=F(n-2)+F(n-1) and F(0)=0 F(1)=1. We define any F(m) with m<0 as 0. In words: every 2-nacci number is the sum of the two prior 2-nacci numbers.
We can now, more generally, define k-nacci numbers as Fk(n)=Fk(n-k)+Fk(n-k+1)+...+Fk(n-2)+Fk(n-1), with Fk(m)=0 for m<=0 and Fk(1)=1
Now if we increase k, we can make an observation: the sequence transforms into something else: for any k>=2, the first k powers of 2(starting from 2^0) appear. For a sufficiently high k, Fk(n)=2^(n-1). In short: You can generate the powers of 2 by (ab)using the fibonacci sequence

File: nautilus-spiral-vs-golden(...).gif (64 KB, 348x509)

64 KB GIF

>>16788181
>>Used in trades
In a contrived manner. Most tradies don't pay it any mind and the ones that do are only doing so because it's "muh golden ratio."

>>Is found in nature
Not really. There's a lot of variability in nature so you'll only find coincidental matches. The association people make between the golden ratio and spirals is just priming.
There is absolutely no reason you should think "golden ratio" every time you see a logarithmic spiral in nature.

See: pic related. Overlaying a golden spiral on a nautilus shell reveals how divergent the two spirals are. They have nothing to do with each other.

File: 1752282771018418.jpg (1.46 MB, 4181x1618)

1.46 MB JPG

>>16788236
that is really interesting, the ancient egyptians were obsessed with both the golden ratio and powers of 2

i'll need to explore this link you've created in further detail

I found out that tiles in my bathroom wall are almost golden rectangles. Measured the side ratio with a tape measure. It was only 1% off from the golden ratio.

File: Screenshot_20250918_220111.jpg (70 KB, 1080x814)

70 KB JPG

If you surround a circle with three equal sized circles and a straight line, the ratio between the radii of one of the big circles and the small circle randomly happens to be the golden ratio.

>>16789238
>If you surround a circle with three equal sized circles and a straight line, the ratio between the radii of one of the big circles and the small circle randomly happens to be the golden ratio.
What happens if you surround the 3 big circles with even bigger circles? :^)