Graphs of Complex Valued Functions

Definitions of Projections and Rotations

proj[w_, o_]  :=  {w . o[[2]], w . o[[3]], 
          w . o[[4]]};

rot1[
          u_] := {{Cos[u], -Sin[u], 0, 0}, {Sin[u], Cos[u], 0, 0}, {0, 0, 
            1, 0}, {0, 0, 0, 1}};
rot2[
          u_] := {{Cos[u], 0, -Sin[u], 0}, {0, 1, 0, 0}, {Sin[u], 0, 
            Cos[u], 0}, {0, 0, 0, 1}};
rot3[
          u_] := {{Cos[u], 0, 0, -Sin[u]}, {0, 1, 0, 0}, {0, 0, 1, 
            0}, {Sin[u], 0, 0, Cos[u]}};
rot4[
          u_] := {{1, 0, 0, 0}, {0, Cos[u], -Sin[u], 0}, {0, Sin[u], 
            Cos[u], 0}, {0, 0, 0, 1}};
rot5[
          u_] := {{1, 0, 0, 0}, {0, Cos[u], 0, -Sin[u]}, {0, 0, 1, 
            0}, {0, Sin[u], 0, Cos[u]}};
rot6[
          u_] := {{1, 0, 0, 0}, {0, 1, 0, 0}, {0, 0, 
            Cos[u], -Sin[u]}, {0, 0, Sin[u], Cos[u]}};
rot[t_] :=  
      rot1[t[[1]]] . rot2[t[[2]]] . rot3[t[[3]]] . 
        rot4[t[[4]]] . rot5[t[[5]]] . rot6[t[[6]]]

sixrots  =  
        If[u < 1, {2  π u, 0, 0, 0, 0, 0}, 
If[
            u < 2, {0, 2  π (u - 1), 0, 0, 0, 
              0}, 
If[
              u < 3, {0, 0, 2  π (u - 2), 0, 0, 
                0}, 
If[
                u < 4, {0, 0, 0, 2  π (u - 3), 0, 
                  0}, 
If[
                  u < 5, {0, 0, 0, 0, 2  π (u - 4), 
                    0}, 
If[
                    u <= 6, {0, 0, 0, 0, 0, 
                      2  π (u - 5)}]]]]]];

Graph of w = (z^2)

w  =  {x, y, (x^2) - (y^2), 2  x y};

Rotate simultaneously through six directions in (R^4).

o  =  rot[{u, u, u, u, u, u}];

Do[ParametricPlot3D[
        Evaluate[proj[w, o]], {x, -2, 2}, {y, -2, 2}, 
        PlotRange -> {{-8, 8}, {-8, 8}, {-8, 8}}], {u, 0, 
        2  π - π/32, π/32}]

Rotate separately through six directions in (R^4).

o  :=  rot[sixrots];

Do[ParametricPlot3D[
        Evaluate[proj[w, o]], {x, -2, 2}, {y, -2, 2}, 
        PlotRange -> {{-8, 8}, {-8, 8}, {-8, 8}}], {u, 0, 6, 
        1/8}]

Graph of w  =  (e^z)

w  =  {x, y, (E^x) Cos[y],  (E^x) Sin[y]};

Rotate simultaneously through six directions in (R^4).

o  =  rot[{u, u, u, u, u, u}];

Do[ParametricPlot3D[
        Evaluate[ proj[w, o]], {x, -2, 2}, {y, -2 π, 2  π}, 
        PlotPoints -> 30, 
        PlotRange -> {{-12, 12}, {-12, 12}, {-12, 12}}], {u, 
        0, 2  π - π/32, π/32}]

Rotate separately through six directions in (R^4).

o  :=  rot[sixrots];

Do[ParametricPlot3D[
        Evaluate[ proj[w, o]], {x, -2, 2}, {y, -2 π, 2  π}, 
        PlotPoints -> 30, 
        PlotRange -> {{-12, 12}, {-12, 12}, {-12, 12}}], {u, 
        0, 6, 1/12}]