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I came across a post on the geometric interpretation of $x^3+y^3+z^3 = t^3$ asking if it was a higher analogue of the Pythagorean theorem.

It made me wonder: if we plot $x^2+y^2+z^2=1$, of course we get a sphere. But if we plot $x^4+y^4+z^4=1$, we get this solid instead:

$\hskip1.8in$enter image description here

I know by a result of Elkies that this has infinitely many rational points $x,y,z$. What else is known about this solid?

  1. For one, what is its name?
  2. Also, if we snugly enclose it in a box of unit length, what is its volume?

P.S. For the case $x^5+y^5+z^5 = 1$

$\hskip1.8in$enter image description here

Tito Piezas III
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    The question in your title is different from the two questions you ask in the body... (Also: probably you shoud explain what it means for a box to have unit length (unit side?) and why doesn't the volume of the (cubic?) box is not determined by the side!) – Mariano Suárez-Álvarez Jan 04 '18 at 16:09
  • Question 2 confuses me a little. This solid has a well-defined volume, regardless of whether or not "we snugly enclose it in a box of unit length". Did you mean to ask something different, or is the unit length box a red herring? – Lee Mosher Jan 04 '18 at 16:11
  • You may call these bodies "N-dimensional ball in p-norm". (Here: N=3, p=4). Their general volumes are given here: https://math.stackexchange.com/questions/301506/ – Andreas Jan 04 '18 at 16:12
  • @Andreas: Ah, should have known this would be well-studied. – Tito Piezas III Jan 04 '18 at 16:17
  • @MarianoSuárez-Álvarez: Turns out it is a duplicate. Should I delete it? – Tito Piezas III Jan 04 '18 at 16:18
  • This shape is also a superellipsoid and a superquadric. – Travis Willse Jan 04 '18 at 16:22
  • @Andreas: Can you look at this plot for $x^5+y^5+z^5 = 1$? It looks more like a sheet than a ball. Is there something wrong with WolframAlpha's rendering? – Tito Piezas III Jan 04 '18 at 16:28
  • You want $|x|^5+|y|^5+|z|^5 = 1 $ for proper balls. Without the absolute values, you get "sheets" due to negative signs in other than the first orthant. – Andreas Jan 04 '18 at 16:41
  • @LeeMosher: It didn't seem to have a "radius", and it had sloping corners so how to measure its length? So the nearest thing that I could think of was to constrain it in a box of unit length. – Tito Piezas III Jan 04 '18 at 16:42
  • @Andreas: That makes sense. – Tito Piezas III Jan 04 '18 at 16:43

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