In what field does π²=2?

Answer 1

The field is cooking.
$pi^2$ milliliters = 2 teaspoons
https://www.google.ch/search?hl=en&q=pi+squared+milliliters+in+teaspoons

Answer 2

The field spherical geometry.
For a small circle of radius subtending angle $theta = 2.010311...$ radians at the centre of the sphere, the ratio between the circumference and the diameter measured on the surface of the sphere is $sqrt2$. In this precise case we could say $pi^2=2$.

Mathematically, I think the answer is

where this denotes

of

This is a complete field (the question asks "in what field does..." and "I am looking for a complete answer") in which $pi^2=2$.

---

My first idea was simply

where

This isn't as silly as it looks, since the notation $pi$ is often used in algebraic number theory for elements of $p$-adic completions of number fields rather than for $3.14159265358979...$. But I amended it as suggested by @Meelo since I think it makes slightly more sense with the actual number $pi=3.14159265358979...$ identified with $sqrt{2}$ via quotienting.

The Zeta function ζ(2) = π^2 $ζ(2) = π^2/6$ but then you want to know the field, the Riemann zeta function is used in quantum theory so that's your field.

EDIT

stupid me, I went to look for a reference to Wikipedia for the answer and found out I was almost right:

I forgot i had to divide it by 6.

BONUS (unrelated to question):

the Quantum theory often makes odd statements which turn out to be true for example: $1 + 2 + 3 + 4 + 5 + .... = -1/12$ (so counting up all integers to infinity equals minus 1/12th) Link for explanation
DISCLAIMER
As enforced by the police I have to add this is only true when used in several techniques called analytic continuation and Ramanujan sums. So stay in school kids, else the police will find you and they will correct you!

The answer is Cooking, where the pie has a piece cut out...

Not a full circle, so $pi^2$ = 2

In physics, we sometimes use $pi$ to mean the permutation operator that swaps two particles. If you swap two particles and then swap them again, you get back to the same state as before (...usually). Therefore, $pi^2 = 1$.

I like commenter Lopsy's suggestion of this

which seems to be a circle in Taxicab geometry AKA $L^1$ space. In this case however, $pi = 2^2$, not $pi^2 = 2$. It's possible that the OP made a mistake.

(Lopsy, you can see that $pi=4$ not $2$ or $2sqrt{2}$ because the length of each diagonal side is equal to $r+r$)

I tried to find a value of $p$ that made $pi = sqrt{2}$ in $L^p$ space but found $pi$ was minimized at $p=2$ with the usual value of $approx 3.1416$. It is equal to $4$ at $p=1$ and $p=infty$ and diverges to $infty$ as $p to 0$. I don't think a circle is well-defined for $p leq 0$.

The simplest answer would seem to be "boxing", if we consider a ring to be synonymous with a circle.

If the diameter of the square is its diagonal and the circumference is its perimeter then the square of their ratio is 2.

Likely wrong, but I wanted to add a different angle to the question, i.e. move a bit further outside the box.

Could be the touch-field of a pocket-calculator or other technichal device where touching the "Pi" key twice gives you 2 ? (Haven't found an according calculator though, yet.)

Here's an answer that goes in and out of figuration, becoming more figurative than literal and then more literal than figurative, stimulated by the hint.

The answer we arrrive at is that

The working is as follows.

We want $pi^2$, i.e.

OK so

${}$

Then

We get

We have

Now

Note:

Or any other similar non-euclidean geometry that is sufficiently warped.

This video explains it with a nice visual demonstration using strechy fabric.

I haven't looked at all the existing answers yet.
Has somebody already discovered this?

Don't know if correct, because " π is the ratio of the circumference of a circle to its diameter".