How to graph this on wolfram, google, or other free software:
\begin{alignat*}{3} x(s, t) &= a\cos(mt) \cos^{k}(ns) &&\cos(t) &&\cos(s), \\ y(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(t) &&\cos(s), \\ z(s, t) &= a\cos(mt) \cos^{k}(ns) &&\sin(s) && \end{alignat*}
The following is: $m = 4$, $n = 1$, and $k = 8$:
The underlying idea is to take $\rho = \cos(m\theta)\cos^{k}(n\phi)$ in spherical coordinates $$ (x, y, z) = (\rho\cos\theta \cos\phi, \rho\sin\theta \cos\phi, \rho\sin\phi). $$
Edit: I tried modifying a parametricplot3D that was pre-existing:
This works in wolfram: Bold = changed.
parametricplot3d[{cos(3*u)*cos(4*v)^2*cos(u)*cos(v), (1+ cos(u/2)*sin(v)- sin(u/2)*sin(2*v))*sin(u), sin(u/2)*sin(v)+ cos(u/2)*sin(2*v)}, {u,0,2*Pi},{v,0,2*Pi}]
This Barely Does: parametricplot3d[{cos(3*u)*cos(4*v)^2*cos(u)*cos(v), cos(3*u)*cos(4*v)^2*cos(u)*cos(v), sin(u/2)*sin(v)+ cos(u/2)*sin(2*v)}, {u,0,2*Pi},{v,0,2*Pi}]
This Doesn't: parametricplot3d[{cos(3*u)*cos(4*v)^2*cos(u)*cos(v), (cos(3*u)*cos(4*v)^2*cos(u)*cos(v), (cos(3*u)*cos(4*v)^2*sin(v), {u,0,2*Pi},{v,0,2*Pi}]
Is the Problem Computation time?