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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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  • $\begingroup$ In this question,is $\varphi$ the golden ratio? $\endgroup$
    – Roby5
    Mar 25, 2016 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, 2016 at 17:47
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    $\begingroup$ @Bob I'm shaking my head at you. $\endgroup$
    – question_asker
    Mar 25, 2016 at 17:50
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    $\begingroup$ @question_asker I bite my thumb at you. $\endgroup$
    – Bob
    Mar 25, 2016 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, 2016 at 18:41

5 Answers 5

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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, 2016 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, 2016 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, 2016 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? $\endgroup$
    – Roby5
    Mar 25, 2016 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, 2016 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, 2016 at 17:26

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