# The facts about $\varphi$ [closed]

A lot of people believe there is something special about the number $\varphi= \frac {1+ \sqrt5}{2}$.

However, I can only think of cultural explanations for looking at each property of $\varphi$ as if it was unique.

What interesting facts do you know about it? Do you know any fact that, because of its mathematical nature, is not well-known to most people?

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## closed as not a real question by rschwieb, Brett Frankel, Ross Millikan, Alexander Gruber♦, Davide GiraudoJan 15 '13 at 21:38

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"Besides the facts we can read everywhere"? I suspect everything worth saying about $\varphi$ has been said a great many times already. –  Chris Eagle Jan 15 '13 at 20:58
What I ask is if there is some connection between those facts, or is just us, as human beings being fascinated about some ''coincidences'' and then trying to find such coincidences everywhere –  Zango Lotino Jan 15 '13 at 21:00
@ZangoLotino: There are some that have reasonable accounts of what the call Fibonacci Flim Flam and others that seem really out there. I think this gives a realistic view on the matter. I was going to give a talk on this, but changed my mind because it seemed impossible to separate fact from fiction. Look at the Golden Ratio images. –  Amzoti Jan 15 '13 at 21:06
Seeing this and your previous question, you really need to work on formulating your questions more precisely! If you're looking for mathematically interesting objective facts about the golden ratio, surely you'll find a lot of them in the mathematics section of Wikipedia's article about it. I don't know if Wikipedia counts among "the facts we can read everywhere" that you don't want. –  Rahul Jan 15 '13 at 21:08
Tons of things are special about any number, and phi is no exception. It appears in relation to certain recurrence relations which sometimes appear in nature. It is thoroughly revered by numerologists. However, we must not forget that special things pop up whenever we look for them; for any number. And we must not forget the most special number of all, Ϝ <-- this is a link –  Greg Ros Jan 15 '13 at 21:34

(This is not meant very seriously.)

For any algebraic number $\xi\in{\mathbb R}$ there is a unique polynomial $$p_\xi(x)=a_n x^n+a_{n-1}x^{n-1}+\ldots+a_1 x+a_0\ ,\qquad a_k\in{\mathbb Z}\quad (0\leq k\leq n)\ ,$$ with minimal $n\geq 1$ and minimal $a_n>0$ such that $p_\xi(\xi)=0$. Call $$N(\xi):=\sum_{k=0}^n \bigl(1+\lfloor\log_2\bigl( |a_k|\bigr)\rfloor\bigr)$$ (or some similar quantity) the numerical depth of $\xi$.

The golden ratio number $\phi$ satisfies the equation $x^2-x-1=0$ and no equation of lesser degree. Therefore its numerical depth is $3$. There are not many numbers of this or lesser depth.

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What do you mean by "not many numbers of this or lesser depth"? I can think of infinitely many. –  Raskolnikov Jan 15 '13 at 21:27
@Raskolnikov: Sorry for the slip. I've edited that. –  Christian Blatter Jan 16 '13 at 9:09
Thank you! I find this answer nice. Also good sense of humour. It might no be a serious answer but it proves how easy you can find special things about whatever and so discover they are not so special. That was what I was looking for. –  Zango Lotino Jan 16 '13 at 13:19

Some special thingies:

\begin{align*}(1)&\;1-\varphi&=&\;\;\;\;\;-\frac{1}{\varphi}\\(2)&\;F_n&=&\;\;\;\;\;\frac{1}{\sqrt 5}\left(\varphi^n-(1-\varphi)^n\right)\,\,\,,\,\,F_n=\,\text{the n-th Fibonacci number}\\(3)&\;\varphi&=&\;\;\;\;\;\;[1;1,1,1,...]=\,\text{continued fraction}\end{align*}

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Mathematically, $\varphi$ is special because it is the unique positive number such that $\varphi^2 = \varphi + 1$, which means that when you cut the largest possible square out of a rectangle the part that's left over is a congruent rectangle. It is also, in a certain sense, the most "difficult" irrational number to estimate with rational numbers. It has been suggested that this is the reason for some of the places it shows up in nature (e.g. in the angle of leaves around a stalk, this gives minimal overlap). $\varphi$ does legitimately show show in nature, though not in all the places it is sometimes said to show up.

I'm not aware of connections to $\varphi$ in finance. Most of the connections to music I've seen seem to be coincidence or bogus. I don't find the connections to aesthetic very convincing, either--either bogus, coincidence, or, like you suggested, a sort of cultural feedback loop.

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Being a particular root of a polynomial doesn't really seem all that distinctive. There is also a unique positive number such that $\alpha^3=1-\alpha$, but nobody talks about it... I guess you can say that the particular equation is somehow interesting, where squaring is the same as adding 1, so maybe this is what you mean. –  rschwieb Jan 15 '13 at 21:17

Well, it is golden. Like, made of gold.

But seriously, this question borders on numerology. There is nothing mystical about $\varphi$, and just about anything that can be said mathematically about it would fall in the category of "facts one can read anywhere."

Culturally it is a very old number, having been studied for thousands of years. Algebraically, $\varphi$, which is one of the solutions to $x^2-x-1=0$, is not any more important than the roots of $x^2-x-3=0$ or $x^2-x-4=0$, etc. In fact, $x^2-x-1=0$ even has another solution, $(1-\sqrt{5})/2$, which could just as well have been called the Golden Ratio instead of $\varphi=(1+\sqrt{5})/2$.

That said, if you really want to learn more about the mathematics behind the golden ratio, the Wikipedia is very comprehensive and in my opinion is actually quite well written, especially concerning its relation to the Fibonacci sequence.

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Of course, I don't think there is something mystical about $\varphi$... but if you see ''coincidences'' you don't think of them as coincidences first and then, if you are scientific in some way, you seek what's behind those ''coincidences'' –  Zango Lotino Jan 15 '13 at 21:26
Which might result only in cultural explanation –  Zango Lotino Jan 15 '13 at 21:26