geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the Fermat cubic, named after

Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...

, 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) =

y(s,t) =

z(s,t) = .

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

* * Algebraic surfaces {{algebraic-geometry-stub