Fermat cubic

File:3D model of Fermat cubic.stl In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by

$x^3+y^3+z^3=1.$

Methods of algebraic geometry provide the following parameterization of Fermat's cubic:

$x(s,t)=\frac{3t-\frac{1}{3}(s^2+st+t^2{)}^2}{t(s^2+st+t^2)-3}$

$y(s,t)=\frac{3s+3t+\frac{1}{3}(s^2+st+t^2{)}^2}{t(s^2+st+t^2)-3}$

$z(s,t)=\frac{-3-(s^2+st+t^2)(s+t)}{t(s^2+st+t^2)-3}.$

In projective space the Fermat cubic is given by

$w^3+x^3+y^3+z^3=0.$

The 27 lines lying on the Fermat cubic are easy to describe explicitly: they are the 9 lines of the form (w : aw : y : by) where a and b are fixed numbers with cube −1, and their 18 conjugates under permutations of coordinates.

Real points of Fermat cubic surface.

References

• Ness, Linda (1978), "Curvature on the Fermat cubic", Duke Mathematical Journal 45 (4): 797–807, doi:10.1215/s0012-7094-78-04537-4, ISSN 0012-7094, https://projecteuclid.org/euclid.dmj/1077313099 
• Elkies, Noam. "Complete cubic parameterization of the Fermat cubic surface". http://www.math.harvard.edu/~elkies/4cubes.html.