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What areas of math are related to both $\pi$ and $\varphi$?

There are at least two answers.

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closed as too broad by Gabriel Romon, Bob, Deusovi♦, Nautilus, DylanSp Mar 25 '16 at 20:02

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ In this question,is $\varphi$ the golden ratio? $\endgroup$ – Dragonemperor42 Mar 25 '16 at 17:26
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    $\begingroup$ "There are at least two answers." - How is this question not too broad? $\endgroup$ – Bob Mar 25 '16 at 17:47
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    $\begingroup$ @Bob I'm shaking my head at you. $\endgroup$ – question_asker Mar 25 '16 at 17:50
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    $\begingroup$ @question_asker I bite my thumb at you. $\endgroup$ – Bob Mar 25 '16 at 18:20
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    $\begingroup$ I'm voting to close this question as off-topic because it's not related to puzzling. $\endgroup$ – Deusovi♦ Mar 25 '16 at 18:41
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Number theory,

because

in number theory, $\pi(n)$ denotes the prime-counting function (the number of primes $\le n$), and $\varphi(n)$ denotes the Euler totient function (the number of integers between $1$ and $n-1$ that are relatively prime to $n$).

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  • $\begingroup$ This was the intended explanation for Number Theory. $\endgroup$ – wythagoras Mar 25 '16 at 20:09
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I hope I'm missing the point, but I'll just answer this like it was a normal question:

Geometry and Number Theory

Because

Pi and the Golden Ratio show up in both Geometry and Number Theory. Though I feel like if this is a valid answer, there are dozens of other valid answers

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  • $\begingroup$ This were the two things I had in mind, but I don't think that Pi and the Golden Ratio are very common in Number Theory. However, there are other reasons why it is related to Number Theory, for which I refer to the answer by Big Black Box. $\endgroup$ – wythagoras Mar 25 '16 at 20:08
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    $\begingroup$ Ok, but I'll make a couple comments: 1)This isn't really a puzzle. 2)The greek letters pi and phi are used in basically every branch of mathematics and physics with various meanings. I was really hoping the real answer would be a joke or a pun of some sort. $\endgroup$ – Solocutor Mar 25 '16 at 20:38
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The area of a circle with radius $\phi$ is $\pi \phi^2$, which is very obviously related to both numbers.

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In algebra (specfically polynomials) , the $n$-th cyclotomic polynomial is defined as $$\Phi_n(x) = \prod_{\stackrel{1\leq k \leq n}{\gcd(k,n)=1}} \left(x-e^{2i\pi\frac{k}{n}}\right)$$

Also,

In probability theory, the probability density function of the normal distribution. $$\phi(x) = (2\pi)^{-1/2}\cdot e^{-x^2/2}$$

Please see that $\varphi$ is the same as $ϕ$ for both the above mentioned functions.

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    $\begingroup$ Why did I get a down vote? $\endgroup$ – Dragonemperor42 Mar 25 '16 at 17:49
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Are they both

Transcendental

numbers?

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  • $\begingroup$ Assuming you take $\varphi$ to be the golden ratio, it is not transcedental number, since $\varphi^2-\varphi-1=0$. They are both irrational though. Also, your answer is not really an area of math. $\endgroup$ – wythagoras Mar 25 '16 at 17:25
  • $\begingroup$ Hm. Then I'm out, but I'll leave it up. Unless of course the answer is a man. $\endgroup$ – Raystafarian Mar 25 '16 at 17:26

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