The facts about $\varphi$ [closed]

Question

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?

Answer 1

(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.

Answer 2

Some special thingies:

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

Answer 3

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.

Answer 4

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.