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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?

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    J’ai trouvé une merveilleuse réponse à cette question, mais la marge est trop étroite pour la contenir. :)2011-05-22

3 Answers 3

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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.

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Sometimes it's called a superellipse - see, e.g., http://en.wikipedia.org/wiki/Superellipse

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    I like these: $ A x^4 + B x^2 y^2 + C y^4 = 1, $ with A,C, 4 A C - B^2 > 0.2011-05-22
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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 (but not for $x \in [0,r]$, since $y(r)=0$). This accounts for the square shape.