Fermat cubic

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3D model of Fermat cubic (real points)

In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by

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

In projective space the Fermat cubic is given by

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.


  • 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, MR 0518106
  • Elkies, Noam. "Complete cubic parameterization of the Fermat cubic surface".