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How to graph this:

\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 $k = 8$:

A three-dimensional rose with eight lobes

The underlying idea is to take $\rho = \cos(m\theta)\cos^{k}(n\phi)$ in spherical coordinates $$ (x, y, z) = (\rho\cos\theta \cos\phi, \rho\sin\theta \cos\phi, \rho\sin\phi). $$

Source

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  • 1
    $\begingroup$ I'm voting to close this question as off-topic because questions about how to use Wolfram|Alpha are explicitly off topic $\endgroup$ – Michael E2 Jun 13 '16 at 1:13
  • 3
    $\begingroup$ Stink, I will just remove the WOlfram Alpha part. $\endgroup$ – Dale Jun 13 '16 at 1:16
  • 1
    $\begingroup$ Possible duplicate of problem with coloring spherical harmonics $\endgroup$ – Jens Jun 13 '16 at 4:04
  • 1
    $\begingroup$ vote to re-open? $\endgroup$ – Dale Jun 14 '16 at 19:56
  • $\begingroup$ But have you looked up SphericalPlot3D[] in the docs? $\endgroup$ – J. M. will be back soon Jun 14 '16 at 23:43
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{a, m, n, k} = {1, 4, 1, 8};

ParametricPlot3D[a Cos[m t] Cos[n s]^k Cos[s] { Cos[t], Sin[t], Sin[s]/Cos[s]}, 
    {s, -Pi, Pi}, {t, -Pi, Pi}, PlotRange -> All, Mesh -> 40]

Mathematica graphics

Alternatively,

x[s_, t_] := a Cos[m t] Cos[n s]^k Cos[t] Cos[s];
y[s_, t_] := a Cos[m t] Cos[n s]^k Sin[t] Cos[s];
z[s_, t_] := a Cos[m t] Cos[n s]^k Sin[s];
ParametricPlot3D[{x[s, t], y[s, t], z[s, t]}, {s, -Pi, Pi}, {t, -Pi, 
  Pi}, PlotRange -> All, Mesh -> 40]
(* ==> same picture *)
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