The graph of $x^{n}+y^{n}=r^{n}$ for sufficiently large $n$ The graph of the function $x^{n}+y^{n}=r^{n}$ for certain large values of $n$ looks suspiciously like a square.
See this page from wolframalpha. Have any results been proven regarding this observation? What do we call this figure anyway?
 A: Sometimes it's called a superellipse - see, e.g., http://en.wikipedia.org/wiki/Superellipse
A: By symmetry, you can consider the equation $y^n+x^n=r^n$ for $0 \leq x \leq r$. Rewrite as
$$
y(x) = \sqrt[n]{{r^n  - x^n }} = \sqrt[n]{{r^n  - r^n \bigg(\frac{x}{r}\bigg)^n }} = r\sqrt[n]{{1 - \bigg(\frac{x}{r}\bigg)^n }},
$$
for $0 \leq x \leq r$. This shows that $y$ is strictly decreasing from $r$ to $0$ as $x$ varies from $0$ to $r$, respectively, and that the sequence of functions $y(x) = y_n (x)$ converges pointwise, as $n \to \infty$, to the function $f$ defined by $f(x)=r$ if $0 \leq x < r$ and $f(r)=0$; moreover, the convergence to $f$ is uniform for $x \in [0,a]$, for any $0 < a <r$ (but not for $x \in [0,r]$, since $y(r)=0$). This accounts for the square shape.
A: You have just rediscovered the max-norm.
More precisely, you have noted that as $p$ becomes large, the unit circle in the $l_p$ norm looks similar and similar to the one of the $l_\infty$ norm.
A: For the super circle defined by $$x^{n}+y^{n}=r^{n}$$ use the parametrization
$$x(t)=\pm r \cos ^{\frac{2}{n}}(t)\qquad \text{and} \qquad y(t)=\pm r \sin ^{\frac{2}{n}}(t)$$ for $0 \leq t \leq \frac \pi 2$.
Its area is given in terms of the gamma function by
$$A_n=4r^2\, \frac{\Big[\Gamma \left(1+\frac{1}{n}\right)\Big]^2}{\Gamma \left(1+\frac{2}{n}\right)}$$
If you consider large values of $n$
$$A_n=4r^2\left(1-\frac{\pi ^2}{6 n^2}\right)+O\left(\frac{1}{n^3}\right)$$ Using, as you did, $n=24$, $A_{24}=3.98924 r^2$
