Lamé's special quartic explained

Lamé's special quartic, named after Gabriel Lamé, is the graph of the equation

x ^{4} + y ^{4} = r ^{4}

where

r > 0 .^[1] It looks like a rounded square with "sides" of length 2 r and centered on the origin. This curve is a squircle centered on the origin, and it is a special case of a superellipse.^[2]

Because of Pierre de Fermat's only surviving proof, that of the n = 4 case of Fermat's Last Theorem, if r is rational there is no non-trivial rational point (x, y) on this curve (that is, no point for which both x and y are non-zero rational numbers).

Notes and References

1. .
2. .