We know that $\pi$ is defined as the ratio of circumference and diameter of a circle. Also the circle is defined as collection of points at a fixed distance from a given point (centre). Here let us assume the centre to be origin for sake of simplicity.
We know that, in euclidean space the distance is defined as $\sqrt{x^2+y^2}$. Now if we redefine the distance as $\sqrt[n]{x^n+y^n}$ the shape of the circle (as defined above) would change and so would the value of $\pi$. Can someone derive the formula for any general $n$.
I proceeded with taking a circle of 1 unit radius. Equation of this circle would be $x^n+y^n=1$. We just have to find the perimeter of this circle and divide by $2$ to get the required $\pi$.
But the catch here is that the distance is redefined. So, if we try to use integration to calculate perimeter by taking small elements, the length of this small elements would be $\sqrt[n]{x^n+y^n}$ and not $\sqrt{x^2+y^2}$.
Due to this I get a complex integral which I am unable to solve analytically. Is there any other way round?
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Aayush Dwivedi
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This concept is related to the concept of the $p$-norm in finite dimensions. – Mark Schultz-Wu Mar 08 '17 at 10:11
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This is also very related to this question. – Mark Schultz-Wu Mar 08 '17 at 10:13
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Thanks. But does that mean that analytically solving $\pi$ is not possible yet? – Aayush Dwivedi Mar 08 '17 at 10:14
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I'm unsure, as I know nothing about this field. This answer discusses a paper which involves numerically computing it for various $p$, that paper may have references to the infeasibility of the analytic solution. – Mark Schultz-Wu Mar 08 '17 at 10:16
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"Is there any other way round?" Is that a pun? – Gerry Myerson Mar 08 '17 at 11:56
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Haha @GerryMyerson . Let's say it is. – Aayush Dwivedi Mar 11 '17 at 11:59
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The definition of $\pi$ is not
"the ratio of circumference and diameter of a circle"
One of its possible definitions is
the ratio of circumference and diameter of a circle in the Euclidean geometry.
In hyperbolic geometry (as an example) the circumference of a circle of radius $r$ is
$$2\pi \sinh(r).$$
(Here the parameter of the hyperbolic plane is $1$.)
So the "hyperbolic $\pi$" is not
$$\frac{2\pi \sinh(r)}{2r}=\frac{\pi\sinh(r)}r.$$
Such a "$\pi$" would not even be a constant. It is an Euclidean specialty that the circles are similar and, as a result, we can define $\pi$ as constant. The real surprise is that this rudimentarily Euclidean constant appears so may times without Euclidean geometry.
zoli
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I'm not sure this is a suitable answer - the question might misuse $\pi$, but it does ask for calculation of a particular constant, which is merely motivated by the definition of $\pi$ from Euclidean geometry. – Milo Brandt Mar 09 '17 at 16:07
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@MiloBrandt: I understand. So, consider my answer to be a comment, too long to be a comment comment. The OP may have used the Euclidean definition only to motivate a calculation, yes, then my answer did not help him/her. – zoli Mar 09 '17 at 16:11
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Thanks @MiloBrandt . I always thought that $\pi$ is defined the way I said above. Although It did not help me much regarding the question asked but it is good to know that $\pi$ is a variable in some geometries. – Aayush Dwivedi Mar 11 '17 at 12:05