phi=[-1+sqrt(5)]/2. is ome of the quadratic solutions of 1/f=1+f. It is also the limit of the ratios of two sucessive numbers in the Fibonacci sequence* (as it goes to infinity) and the ratio of two sucessive nested pentagrams. In art, phi is the ratio of the sides of the golden rectangle.
Interesting natural number, even more so since it is simultaneiusly algebraic and transcendal.
Rap
F(n+1)=F(n)+F(n-1)
First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34...............
and
phi=lim F(n+1)/F(n) as n->infinity
I was always impressed with the many graphical solutions of phi by furniture makers,
Why does a Federal Highboy look balanced, yet an Indonesian knock-off looks top heavy?
Thats right, phi.
Maybe I am I am having a hard time seeing it.
Raprap, no wonder Dali had help in the Geometry part of the paintings, sounds very complicated.
Farmerman, your quote sounds like something Madame Blavatsky would say.
Thank you all,
AE
That the key--not that it is all that complicated, phi is a nexus of many solutions.
Mona Lisa is framed in a golden rectangle from eyebrow to frame. Leonardo knew phi, so did Titian. There is a famous Lithograph by Durer that expounds upon the science of art.
Rap
Thanks raprap for the info. I did not know that it was used way back then. I just love Durer!
But wait a minute, I just remembered. I'm always buying books, sometimes they are not what I thought they would be. I keep the books because they are interesting, and I always learn from them. I bought this book called "The Earliest Irish and English Bookarts" Visual and Poetic forms before A.D. 1000, thinking it was a book of poetry forms. They used phi (the monks did) to make the religious books and to write the poems or prayers. The book also gives you the numerical formulas to create the images and texts. Very interesting.
AE
