A tessellation for Fermat surfaces in \(\mathbb C\mathbb P^3\) (Q1015155)

Fermat surfaces are 4-manifolds whose equations can be written in the form \(z_0^n+z_1^n+z_2^n+z_3^n=0\) in the standard homogeneous coordinates \([z_0,z_1,z_2,z_3]\). In this paper, for each positive integer \(n\), we present a tessellation of \(\mathbb C\mathbb P^3\) that can be lifted, through the branched covering, to a symmetric tessellation of degree \(n\) in \(\mathbb C\mathbb P^3\). The process is systematic and symbolically algebraic. Each four-cell in the tessellation is bounded by four pentahedrons, and each pentahedron has four triangular faces and one quadrilateral face. Graphically, one can produce the entire surface from one single four-cell using translations generated by permutations and phase multiplications of the homogeneous coordinates of \(\mathbb C\mathbb P^3\). Note that the tessellation of the Fermat surface of degree 4, a \(K3\) surface, has exactly 24 vertices. The algorithm is useful in order to create visual images of this type of surfaces.