A periodic minimal surface is a minimal surface in three-dimensional Euclidean space that is invariant
under a nonzero translation. If 
denotes the surface and 
is a period vector, then
 |
In the translational classification, a connected nonplanar periodic minimal surface with a discrete group of translations is called a singly periodic minimal surface, a doubly periodic minimal surface, or a triply periodic minimal surface according to whether the maximum number of linearly independent period vectors is 1, 2, or 3. A plane has continuous translational symmetries and is treated separately from this discrete classification. A surface can also be studied modulo a smaller subgroup of its periods, and broader usage of singly periodic allows a screw motion in place of a translation (Meeks and Pérez 2012, pp. 19-21).
The helicoid has translational periodicity in its axial direction. Scherk's minimal surfaces include both singly and doubly periodic examples. The Schwarz P surface, Schwarz D surface, and gyroid are triply periodic examples (Schoen 1970, pp. 1-6, 13-17; Meeks and Pérez 2012, pp. 20-21, 25-26).
