Function kind of cross between a helicoid and a complex plane wave? I would like to translate a mental picture into a mathematical expression if possible. The picture might be roughly thought of as a cross between a complex plane wave and a helicoid. A construction I think goes as follows, take some complex scalar plane wave in 4 dimensional spacetime Ψ = exp(-i[p⋅x]) where p is the energy-momentum 4-vector for a massless and spinless particle and x is the spacetime 4-vector. Consider an infinite line, L, parallel to the 3-momentum vector together with the time axis. Consider the infinite half-plane, S, defined by those two lines where the "edge" of the half-plane is the line L. Edit, sorry for my mistake but the half plane I'm thinking of is defined by the line L above and a ray that starts at a point on L and is perpendicular to L in space and is not the time axis. 2nd Edit, seems I cut some of the origional, sorry. Let this half-plane S define a cutting of our function Ψ, called Ψ_cut. Now deform Ψ_cut as follows, shift one surface defined by this cutting forwards in time by 1/2 period and shift the other surface backwards in time by 1/2 period. Now glue the surfaces back together and allow the Ψ to "relax" (minimize curvature in some unique way?). Was my description clear enough so that Ψ might now be given as a mathematical expression and be defined almost everywhere? Suggestions on how to come up with the expression would be appreciated. Now I would like to do the same thing again by shifting the two surfaces defined by this cutting forwards and backwards in time by 1/2 period but then wrap the surfaces around the line L till they meet and glue the surfaces back together and again (if done properly I think we create two "sheets"?). Now allow the Ψ to "relax". Have I given a construction that could be expressed mathematically? Thanks for any help.