I recently added the National Geographic book to my library. It has broader coverage than some books on the topic which are strictly mathematical, which is good. At the risk of being self-promoting, I’ll add that if you like that book you’ll likely like mine, which on Amazon it has hundreds of times as many reviews, at about 82% 5 stars. here:

https://bit.ly/goldenratiobook

Re the “usual trench warfare” I moderate every comment that is approved for publication on the site to maintain the overall quality, so I try to eliminate those that don’t add anything to the discussion or that take it off in a completely irrelevant direction. My choices won’t be the same as everyone’s, but this is not a site where people can just come and say anything and hijack the post comments for their own agenda.

Good video. It makes the same point that I make in various places: Spirals in nature are generally equiangular (aka exponential or logarithmic) spirals, which is not the same as a golden spiral. This results in misinformation that leads to a lot of confusion about spirals and the golden ratio in general. See:

https://www.goldennumber.net/spirals/
https://www.goldennumber.net/nautilus-spiral-golden-ratio/

Re Japanese architecture and root 2, you may also be interested in this article on Orthogons:

https://www.goldennumber.net/orthogons/

Interesting link that has a good description of the metallic ratios. The latter part of the article is a little too advanced for me at the moment.

I found this very interesting video on metallic ratios and related ones. Much of classic Japanese architecture is based on the “Japanese” ratio (the sq root of 2). Also the narrator shows how a peregrine falcon uses silver (I think) ratio spirals to catch its prey. The advantage of using a spiral as it closes in is the falcon doesn’t have to take its eyes off the prey.

Metallic ratios (other than the golden one) on their own are not that interesting but when converted into spirals come into their own.

Hi Gary, completely agree re: discussing the golden ratio and all its implications but I do get nervous because religious discussions can get a bit heated on both sides and in a way clog up any meaningful discussion I’m fine with those discussions as long as they add something and is not the usual trench warfare we see between atheists and non atheists.

Thanks for the link. I’ll have a look at it.

I’m with you s regards mathematics – I’m only interested in mathematics that is applied or abstract mathematics that can be applied such as the square root of -1. Von Neumann wrote an interesting essay called The Mathematician where he critiqued abstract mathematics which was interesting considering a lot oh his applied maths is based on abstract mathematics.

I came to the Golden Mean and your website through reading a book called The Golden Ratio which is part of a 64 volume series titled Our Mathematical World (published by National Geographic) which deals with applied mathematics but also discusses concepts such as Squaring the Circle.

Hi Frank. Good question. I’ve not covered the metallic ratios in my research or writing, but they would certainly be worth an article on the site. In the mean time, here’s another good article:

https://rosettacode.org/wiki/Metallic_ratios

From what I see, they’re interesting as a mathematical exercise but I’ve not seen much in the way of meaningful application.

As site owner and moderator, that’s why I don’t mind discussion of the meaning of the golden ratio with respect to design in life and implications about the existence of God. Without that, it seems a bit like reading a book and discussing only its technical aspects of grammar, structure and style rather than the meaning and message of the story, and what knowledge and insights can be learned from the author.

My question to Gary and anyone else is:

What about the other metallic ratios? How relevant are they?

Here’s an interesting article about them.

https://www.popularmechanics.com/science/a30714119/silver-ratio-math/

How kind of you. Thanks for the unexpectedly quick reply.

My English is not good enough – I still can’t find (or understand) the golden ANGLE, just the golden ratio. I guess I’m asking for a lot – where exactly can I read the description of golden angle of 137,51°?  🙂

Thank you for feeling me. I wish you all the best.
Olga

Absolutely. That’s how it’s been done for thousands of years. I show two constructions, animated to show the steps, on my page at https://www.goldennumber.net/geometry/. There’s also another couple using circles for added interest at https://www.goldennumber.net/phi-formula-geometry/.

I admire your work.

Please, is there a way to construct a golden angle using a compass and a ruler? (without using protractor)
I didn’t find it anywhere and it would help me when working with children.

Many thanks from my hart.

Olga