A singly periodic minimal surface, in the translational sense, is a periodic minimal surface with period vectors in one direction
but no two linearly independent translational
periods. Its periods form a discrete group generated
by a single nonzero vector 
,
 |
The quotient identifies points differing by an integer multiple of 
and gives a minimal surface
in 
(Meeks and Pérez 2012, pp. 19-21).
The helicoid, illustrated above, is an example. In the parameterization 
for real 
, 
,
and 
with 
, replacing 
by 
translates the surface by
. The second of Scherk's
minimal surfaces is another singly periodic example (Meeks and Pérez 2012,
pp. 20-21, 25).
The Riemann minimal examples and Karcher saddle towers supply further singly periodic examples.
More generally, singly periodic minimal surfaces may be defined using a discrete cyclic group generated by a screw motion, combining a rotation about an axis with a translation along that axis. This broader convention need not give a pure translational period (Meeks and Pérez 2012, pp. 19, 25).
