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$a^2+b^2=c^2$ is related to right triangles - how the sums of the squares of the legs equals the square of the hypotenuse.

What about the following sum of cubes? $$x^3+y^3+z^3=k^3$$
Is there a geometric interpretation for this, like there is with the Pythagorean theorem? Is there some shape that embodies this equation, like a right triangle does for $a^2+b^2=c^2$? Would it involve sums of volumes of certain cubes?

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The answer it's a sphere, but not with the usual Euclidian metric, but instead is the $L^3$ metric in a 3D space. In the Euclidian or $L^2$ metric, in $n$ dimensions, the norm is given by $$||X||_2=(x_1^2+x_2^2+...)^{1/2}$$ In general, the distance in the $L^p$ metric is given by the above equation by replacing $2$ with $p$ $$||X||_p=(x_1^p+x_2^p+...)^{1/p}$$ You can find more details on wikipedia or there is a nice movie on youtube (the shape of $L^3$ circle is shown at 6 minutes and 17 seconds in this clip).

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