Fermat cubic

{{Short description|Geometrical surface}}

File:3D model of Fermat cubic.stl

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

:$x^3\; +\; y^3\; +\; z^3\; =\; 1.\; \backslash$

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

:$x(s,t)\; =\; \{3\; t\; -\; \{1\backslash over\; 3\}\; (s^2\; +\; s\; t\; +\; t^2)^2\; \backslash over\; t\; (s^2\; +\; s\; t\; +\; t^2)\; -\; 3\}$

:$y(s,t)\; =\; \{3\; s\; +\; 3\; t\; +\; \{1\backslash over\; 3\}\; (s^2\; +\; s\; t\; +\; t^2)^2\; \backslash over\; t\; (s^2\; +\; s\; t\; +\; t^2)\; -\; 3\}$

:$z(s,t)\; =\; \{-3\; -\; (s^2\; +\; s\; t\; +\; t^2)\; (s\; +\; t)\; \backslash over\; t\; (s^2\; +\; s\; t\; +\; t^2)\; -\; 3\}.$

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.

Image:FermatCubicSurface.PNG

::::Real points of Fermat cubic surface.