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SURFACE WITH ROTATIONAL SYMMETRY
In the most general meaning, a surface with rotational symmetry is a surface invariant under the action of a symmetry of a nonzero angle.
A surface is said to have rotational symmetry of order n if it is invariant under the action of a rotation by an nth of a turn (and not invariant by a rotation of a smaller angle).
A surface with rotational symmetry of order n might not have a plane of symmetry (see for example the Klein bottles of order n), but the surfaces with rotational symmetry of order n that have a plane of symmetry are the surfaces that have exactly n planes of symmetry that pass through one axis. Then, they have (at least) the symmetries of a regular pyramid with n lateral faces.
It is the generalization to space of the notion of curve with rotational symmetry (and all the sections by planes perpendicular to the axis of rotation are such curves).
General cylindrical equation of a surface with rotational symmetry of order a multiple of n:  Corresponding Cartesian equation: 
with f of smallest period with respect to equal to ,  with

Special case: (has n planes of symmetry)  with 
The case n = 2 is studied at
flippable surface (surface invariant under the action of a halfturn).
Any cylinder or right cone with directrix a curve with rotational symmetry is a surface with rotational symmetry.
Examples of families of algebraic surfaces with rotational symmetry:
Name  Polar equation  Cartesian equation  n = 2  n = 3  n = 4 
Plücker's conoid 
algebraic surface of degree n+1 for n even, of degree 2n +2 for n odd 
remark: the symmetry is of order 6 

Octopus saddle 
algebraic surface of degree n 



Sinusoidal cone 
algebraic surface of degree n for n odd, of degree 2n for n even 
remark: symmetry of order 4 
remark: symmetry of order 8 

?? 
algebraic surface of degree n+1 for n even, of degree 2n +2 for n odd 
parabolic cyclide 

Costa's algebraic surface 
algebraic surface of degree n+1 for n even, of degree 2n +2 for n odd 
Regarding the parametrized surfaces, we get a large family of surfaces with rotational symmetry of order n by taking where f is a complex function  periodic with respect to its first variable, and g, as well, but real.
For example, for , we get the opposite surface, in which appears a trapezohedron. Morin's surface of order n is also such a surface. 
Several families of minimal surfaces have rotational symmetry of order n:
the Enneper surfaces, the Richmond surfaces, the nnoids, and the CostaHoffmanMeeks surfaces.
For other surfaces with symmetries, see Goursat surface.
Surface with rotational symmetry of order 5, by Patrice Jeener, with his kind authorization
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© Robert FERRÉOL
2017