Gauss Animations

Gauss Surfaces

(* 
      And this is a Gauss Surface *) 
t2  =  Table[
ParametricPlot3D[{{r Cos[ϕ],  
                r Sin[ϕ], 
    (r^5) Cos[
                      5  ϕ]  +  6 (r^4) Cos[4  ϕ]   +  
                  50 (r^3) Cos[3  ϕ]  +   80 (r^2) Cos[
                      2 ϕ] },   
{r Cos[ϕ],  r Sin[ϕ],  0}},  {r,  0,  range},  {ϕ,  0,  2 Pi}, 
            Ticks -> False,  Boxed -> False,  
            ImageSize -> {600,  300}, PlotPoints -> 32,  
            BoxRatios -> {1,  1,  1}], 
{range,  
            0.5,  15,  0.5}];

Gauss Curves

(* Animation of Gauss curves as R gets larger,  
      to see that at a large enough circle,  
      you do indeed get 2  
        m cuts on the rim of the circle. *) 
t1  =  
        Table[
ParametricPlot3D[
{
{r Cos[ϕ],  r Sin[ϕ], 
                (r^5) Cos[5  ϕ]  +  6 (r^4) Cos[4  ϕ]   +  
                  50 (r^3) Cos[3  ϕ]  +   80 (r^2) Cos[
                      2 ϕ] }, 
{r Cos[ϕ],  
                r Sin[ϕ],  0}   (* 
                Gauss surface and the plane to put it on *) 
}, 
{r,  0,  
              range}, 
 {ϕ,  0,  
              2 Pi}, 
 
            BoxRatios -> {1,  1,  0.05}, 
 
            PlotPoints -> 48, 
 
            PlotRange -> {{- range,  range},  {- range, 
                  range},  {-0.05,  
                  0.05}},  
ImageSize -> {600,  
                300}, 
 
            Ticks -> False, 
 
            Boxed -> False], 
{range,  0.25,  10,  
            0.25}];