Copied from the front and back flap:

The first comprehensive book for the layperson on a
mathematical relationship that has obsessed mathematicians, philosophers, scienists and artists since ancient greece-
an omnipresent number considered to be "divine"

   Throughout history, thinkers from mathematicians
to theologians have pondered the myserious
relationship between numbers and the nature
of reality. In this fascinating book, Mario Livio tells the tale of a number
at the heart of this mystery: phi, or 1.6180339887... This curious
mathematical relationship, widely known as "Golden Ratio," was defined
by Euclid more then two thousand years ago because of its crucial role
in the construction of the pentagram, to which
magical properties have been attibuted.
Since then it has shown a propensity to appear
in a most astonishing variety of places - from mollusk shells, sunflower
florets, and the crystals of some materials, to the shapes of galaxies
containing billions of stars. Psychological studies have investigated whether the Golden Ratio is the most aesthetically pleasing proportion extant and it has been asserted that the creators of the Pyramids and the Parthenon employed it. It is believed to feature in works of art from da Vinci's "Mona Lisa" to Salvador Dali's "The Sacramant of the Last Super", and poets and composers have used it in their works. It has even been suggested that it is connected to the behavior of the stock market!
   The golden Ratio is a captivating journey through art and architecture, botany and biology, physics and mathematics. It tells the human story of numerous phi-fixated individuals, including the followers of Pythagoras, who believed that this proportion revealed the hand of God; astronomer Johannes Kepler, who saw phi as one of the greatures treasures of geometry; such medieval thinkers as mathematician Leonardo Fibonacci of Pisa;
and such masters of the modern world as Debussy, Le Corbusier, Bartok, and physicist Roger Penrose.
   Wherever his quest for the meaning of phi takes him, Mario Livio reveals the world as a place where order,
beauty and eternal mystery will always coexist.

Last edited by mikau (2006-06-19 02:42:57)

yeah I calculated that with the quadratic formula ( sqrt (5) + 1 )/ 2 but the fibonacci formula I'd never seen before! Remarkable!

Also I read and tested that it can be found with     sqrt ( sqrt  (sqrt( sqrt (2) + 1) + 1) + 1 ) and so on. Rather incredible!

cool, those appear to be rearranged forms of the equation (phi + 1)/phi = phi, which is the equation you get from the line segment definition.

hehehe! Creepy!

I suppose the "noteworthy sum" for i can be used to simplify the third expression. I don't know why I didn't notice this before. Perhaps I liked the more complex formula and didn't want to change it. Anyway:

On second thought, this formula is more beautiful due to the compactness compared to its other form. Also, I prefer the relation of φ with constants not including itself, at least in this case.

On the topic of the book, I have not read it. I am becoming more interested in φ though, and I see this book everytime I go to buy more math textbooks, so perhaps I will give in a purchase it in the future.

Edit: A period was not where it needed to be. I can't handle mistakes like those!

Last edited by Zhylliolom (2006-06-19 17:03:22)

is i ever usefull?

Did you do that in degrees or radians?  I get 1.618... and my calculator even gives me:

interesting, but that is not what I meant by usefull. Could 'i' ever be used to solve a real world problem? (in a formula to produce a real number solution) for instance, adding i to both sides or multiplying both sides by i allowing you to rearranging it into another form that eventuallly eliminates i giving you a real number solution?

As far as I've seen its only for math itself and has no practical application outside of math texts, and appears rather useless. Not true?

Ricky wrote:

And now we can say we are done...

... but not yet ...

I would like to make a page about this (maybe called "The Evolution of Numbers"), and your words are really good.

But I would like to continue to irrationals and transcendentals, and perhaps even into sets.

With some nice illustrations, and links to pages that cover more depth, it could be a really nice read for many people.

"I guess my first question to you would be, who cares about the real world?"

nobody, but when we do computer simulations of the real world we need it!

Its somewhat odd how we spend so much time working to duplicate the world onto computers when its already in place, free for everyone with impecable graphics, souround sound, steroscopic vision, character experience (so to speak ) and everything else we crave! But I guess the difference is its a world we can control.

Sorry for that little tangent there.

Until I can play a game where every object is made up of atoms which act exactly as atoms in the real world, I will always want more.

even then, i would prefer to have all those atoms also made of the sub atomic particles, all acting as they would in real life

theres just one problem....

along with that, you would have to have photons of light being sent out from the atoms when the electrons drop to a lower level for illumination

and basicly, well we would have to have a computer with near limitless power

Last edited by luca-deltodesco (2006-06-21 01:33:10)