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 A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

defined by :

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

Methods of

algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...

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