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**mikau****Member**- Registered: 2005-08-22
- Posts: 1,504

Me and my sister were at a mall in New Jersey when we wandered into a boxstore. I quickly made my way to the science section to see if they had any good math or science books. I picked up a book called "The Golden Ratio." not knowing what it was. But below the title it said "the story of Phi" now I'm not sure who this mario livio is, but anyone who would write an entire book about a single number, is definitly the kind of author who's work I'd like to read!

I read perhaps half of the first chapter. Quite interesting. It explained that if you had a line ABC such that BC/AB = ABC/BC then phi is equal to both these ratio's. I've not read of phi in any of my mathbooks yet, though I've heard it spoken of a great deal, I never really knew what its definition was.

Anyways, the book doesn't simply discuss the mathematical properties of phi, but goes on to demonstrate the various places phi appears in nature. In an apple, in a sea shell, and many other places. Fascinating stuff! I wish I'd had the money on me to buy it, but I'll probably order it one of these days.

http://www.amazon.com/gp/product/0767908155/102-6063661-4252923?v=glance&n=283155 if you click on "search inside" you can read a bit of the begining.

*Last edited by mikau (2006-06-18 04:47:34)*

A logarithm is just a misspelled algorithm.

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**mikau****Member**- Registered: 2005-08-22
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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 mathematiciansto theologians have pondered the myseriousrelationship between numbers and the natureof reality. In this fascinating book, Mario Livio tells the tale of a numberat the heart of this mystery: phi, or 1.6180339887... This curiousmathematical relationship, widely known as "Golden Ratio," was definedby Euclid more then two thousand years ago because of its crucial rolein the construction of the pentagram, to whichmagical properties have been attibuted.Since then it has shown a propensity to appearin a most astonishing variety of places - from mollusk shells, sunflower florets, and the crystals of some materials, to the shapes of galaxiescontaining 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 (2006-06-18 04:42:57)*

A logarithm is just a misspelled algorithm.

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**luca-deltodesco****Member**- Registered: 2006-05-05
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a formula for phi is also sqrt(5)/2 + 0.5, and im not sure if it says it in the book or not, it probably does

the fibonacci series u[n] = u[n-2]+u[n-1], u[0] = 0, u[1] = 1

if you divide u[n] by u[n-1], as n increases, this value becomes increasingly closer to phi

phi = 1.6180339887498948482045868343656

fib = 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377

fib[2]/fib[1] = undefined

fib[3]/fib[2] = 1

fib[6]/fib[5] = 1.666666666

fib[7]/fib[6] = 1.6

fib[11]/fib[10] = 1.6176470588235294117647058823529

fib[12]/fib[11] = 1.6181818181818181818181818181818

fib[15]/fib[14] = 1.6180257510729613733905579399142

fib[40]/fib[39] = 63245986/39088169 = 1.61803398874989 51409056791583151

so even after only fib[15], you have phi to 4 decimal places, and pretty close to actual value

and after fib[40] we have phi to 14 decimal places

*Last edited by luca-deltodesco (2006-06-18 05:12:52)*

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The End Of All Things To Come.

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**mikau****Member**- Registered: 2005-08-22
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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!

A logarithm is just a misspelled algorithm.

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**luca-deltodesco****Member**- Registered: 2006-05-05
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i still find the most remarkable thing about phi

phi^2 = phi + 1

1/phi = phi - 1

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**mikau****Member**- Registered: 2005-08-22
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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.

A logarithm is just a misspelled algorithm.

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**Zhylliolom****Real Member**- Registered: 2005-09-05
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Don't mess with that number! It is the Devil's constant!

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**mikau****Member**- Registered: 2005-08-22
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hehehe! Creepy!

A logarithm is just a misspelled algorithm.

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**Zhylliolom****Real Member**- Registered: 2005-09-05
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This post has inspired me to work out some expressions for φ. Being a fan of infinite series representations of special constants, I chose to focus on that area:

In the process of deriving the third equation above, I came across a noteworthy sum:

I'll spend some more time finding expressions for φ and other constants later.

Edit: For some reason I had put the negative in formula two outside of the first sum. Interestingly enough, this made it an expression for -1/φ.

*Last edited by Zhylliolom (2006-06-19 00:08:09)*

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**Zhylliolom****Real Member**- Registered: 2005-09-05
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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 (2006-06-18 19:03:22)*

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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Impressive sums Zhylliolom.

I have heard someone say (I think here on this forum) that they believe φ to be more important than π. But π turns up in the most interesting places.

In a totally subjective manner I rank π at the top, then * e*, then φ, then

But then, 0 and 1 are quire useful

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**mikau****Member**- Registered: 2005-08-22
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is i ever usefull?

A logarithm is just a misspelled algorithm.

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**luca-deltodesco****Member**- Registered: 2006-05-05
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Zhylliolom wrote:

Don't mess with that number! It is the Devil's constant!

no it isnt

-sin(666) - cos(216) = 0.735626729782984

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**Ricky****Moderator**- Registered: 2005-12-04
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Did you do that in degrees or radians? I get 1.618... and my calculator even gives me:

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**Ricky****Moderator**- Registered: 2005-12-04
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mikau wrote:

is i ever usefull?

Sure it is. Actually, I was just reading about this today. I was thinking of starting a new topic, but I'll just post it here.

The most common number systems we have are here because of polynomials. That is, solving equations.

We start of with the natural numbers. These numbers, as the name implies, are natural. That is, they come from counting. You want to be able to say how many of a certain thing you have.

But they aren't good enough. We can to be able to have a group. The one thing that is missing from natural numbers are inverses. That is, if we have a number a, we want to also have a number b so that a + b = 0. And thus, the integers are born. The integers form a group with repsect to addition. They are "nice". We know many things about them because of this.

But wait. Lets say we have an equation ax = b, and are given a and b. For example, 3x = 5. We want to be able to solve this equation. But with integers, we can't do it always. So we need a new number. In comes the rational numbers. With rationals, we can solve any equation in this form, so long as a and b are rational, a not equal to 0.

So are we done? Nope, not quite. What about equations like x² = 2. We can't solve that with rational numbers. And so we make the real numbers. Finally, x² = -1. We need to be able to solve this as well. That means we need imaginary numbers.

And now we can say we are done. That is, any polynomial must have at least one complex solution (remember, complex numbers include the reals).

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**mikau****Member**- Registered: 2005-08-22
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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?

A logarithm is just a misspelled algorithm.

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**Ricky****Moderator**- Registered: 2005-12-04
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I guess my first question to you would be, who cares about the real world?

But yes. First, let me say I haven't had much experience in complex analysis. Ok, that's not so true, I've had just about none. But I know certain equations in differential calculus reduce to formulas with complex numbers, specifically dealing with a bouncing spring. Those can then be reduced even further thanks to the properties of i, to equations of real values.

Now I seem to remember a function for predicting the placement of primes using complex values, but I can't seem to remember the name. Anyone? Anyone?

"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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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.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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**Ricky****Moderator**- Registered: 2005-12-04
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Sure. I didn't really take my time on that post, more or less just ran through it. There is a lot more detail, but most of it involves groups and fields, which are hard to explain in a page or two. I'll see if I can come up with an easy way to explain those.

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**mikau****Member**- Registered: 2005-08-22
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"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.

A logarithm is just a misspelled algorithm.

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**Zhylliolom****Real Member**- Registered: 2005-09-05
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mikau wrote:

is i ever usefull?

*i* has many applications outside of pure mathematics. In special relativity, time is treated as imaginary in Minkowski space in order to make it symmetric to space: the fourth dimension, time, is given as *x*[sub]4[/sub] = *ict*. Quantum mechanics is littered with imaginary numbers. The example of *i* from quantum mechanics which springs to my mind instantly is Schrödinger's equation:

I know imaginary numbers play a role in electrical engineering as well, but sadly I do not have as much experience as I wish to in that area.

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**Ricky****Moderator**- Registered: 2005-12-04
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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.

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**mikau****Member**- Registered: 2005-08-22
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agreed!

A logarithm is just a misspelled algorithm.

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**luca-deltodesco****Member**- Registered: 2006-05-05
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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 (2006-06-20 03:33:10)*

The Beginning Of All Things To End.

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**MathsIsFun****Administrator**- Registered: 2005-01-21
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And we could all be simulations in that very computer, and not even know it!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..." - Leon M. Lederman

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