moebiusstrip[u_, v_] := {Cos[u] + v Cos[u/2] Cos[u], 
        Sin[u] + v Cos[u/2] Sin[u], v Sin[u/2]}

ParametricPlot3D[
      moebiusstrip[u, v], {u, 0, 2  Pi}, {v, -0.5, 0.5},  
      PlotPoints -> {40, 5}, Axes -> None, Boxed -> False, 
      ViewPoint -> {1.449,  -2.437,  1.847}]

kleinbotle[a_][u_, 
        v_] := {(a + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]) 
          Cos[u], (a + Cos[u/2] Sin[v] - Sin[u/2] Sin[2 v]) Sin[u], 
        Sin[u/2] Sin[v] + Cos[u/2] Sin[2  v]}

ParametricPlot3D[kleinbotle[2][u, v] // Evaluate, {u, 0, 
        3  Pi/2}, {v, -Pi/4, 3  Pi/2}, Axes -> None, 
      Boxed -> False]

\(\(kleinbottlebis[e_, f_]\)[u_, v_] := 
      If[u < 0, 
\ \ \ \ \ \ \ \ {f*
            Sin[u]/2, \((\(-e\)\  - \ f/2\  + \ f*Cos[u]/2)\)*
            Cos[v], 
\ \ \ \ \ \ \ \ \ \((e\  + \ f/2\  - \ f*Cos[u]/2)\)*
            Sin[v]}, 
\ \ \ \ \ \ \ \ {10*Sin[0.105*u]\  + \ 
            2*\((e\  + \ 
                  0.033*f*u)\)*
\ \ \ \ \ \ \ \ \ \((Cos[0.105*u]\  + \ 
                  4*Cos[0.21*u])\)*
              Cos[v]*
\ \ \ \ \ \ \ \ \ \((Sin[0.105*u]\  + \ 
                    2*Sin[0.21*u])\)/
\ \ \ \ \ \ \ \ \ Sqrt[
                  100*Cos[0.105*u]^2\  + \ 
                    4*\((Cos[0.105*u]\  + \ 
\ \ \ \ \ \ \ \ \ \ 4*
                              Cos[0.21*u])\)^2*\((Sin[0.105*u]\  + \ 
                            2*Sin[0.21*u])\)^2], 
\ \ \ \ \ \ \ \ \ \((Sin[
                    0.105*u]\  + \ 
                  2*Sin[0.21*
                        u])\)^2\  - \ 
\ \ \ \ \ \ \ \ \ \((10*\((e\  + \ 
                      0.033*f*u)\)*Cos[0.105*u]*
                  Cos[v])\)/
\ \ \ \ \ \ \ \ \ Sqrt[
                100*Cos[0.105*u]^2\  + \ 
                  4*\((Cos[0.105*u]\  + \ 
\ \ \ \ \ \ \ \ \ \ 4*
                            Cos[0.21*u])\)^2*\((Sin[0.105*u]\  + \ 
                          2*Sin[0.21*u])\)^2], 
\ \ \ \ \ \ \ \ \ \((e\  + \ 
                0.033*f*u)\)*Sin[v]}]\)

\(\(ParametricPlot3D[
        Evaluate[
\(kleinbottlebis[1, 2]\)[u, v]], 
{u, \(-Pi\), 
          29.9517}, 
{v, \(-Pi\), Pi}, 
ViewPoint -> {\(-2\), 0, \(-3\)}, 

        PlotPoints -> {49, 19}, 
Axes -> None, Boxed -> False];\)\)

http: // 
      beltrami . sc . unica . it/curve - 
        superfici/index . html

    SPAZIO PROIETTIVO

roman[u_, v_] := {Sin[2  u] Cos[v]^2, Sin[u] Sin[2  v], 
        Cos[u] Sin[2  v]}

\(\(ParametricPlot3D[

        roman[u, v], 
{u, 0, Pi}, 
{v, \(-Pi\)/2, Pi/2}, 
Axes -> None, 
        Boxed -> False];\)\)

ParametricPlot3D[
      Append[moebiusstrip[u, v], 
          FaceForm[RGBColor[0.27451,  0.92549,  1], White]] // 
        Evaluate, {u, 0, 2  Pi}, {v, -0.5, 0.5},  
      PlotPoints -> {40, 5}, Axes -> None, Boxed -> False, 
      Lighting -> False, 
      ViewPoint -> {1.449,  -2.437,  1.847}]

\(\(ParametricPlot3D[

        Append[roman[u, v], 
            FaceForm[RGBColor[0.847059, \ 0.0235294, \ 0.290196], 
              RGBColor[0.807843, \ 0.972549, \ 0.290196]]] // Evaluate, 
{u, 
          0, Pi}, 
{v, \(-Pi\)/2, Pi/2}, 
Axes -> None, Boxed -> False, 
        Lighting -> False];\)\)

IMPLICITPLOT

<< 
      Graphics`ContourPlot3D`

ContourPlot3D[
      x^2  y^2 + y^2  z^2 + z^2  x^2 - 1, {x, -2, 2}, {y, -2, 
        2}, {z, -2, 2}, PlotPoints -> {5, 5}, Boxed -> False, 
      ViewPoint -> {-0.045,  0.001,  
          3.383}
]

\(kummerimplicit[x_, y_, z_] := \ 
      x^4\  + \ y^4\  + \ 
        z^4\  - 
\ \ \ \ \ \ \((y^2\ z^2\  + \ z^2\ x^2\  + \ 
            x^2\ y^2)\)\  - \ \((x^2\  + \ y^2\  + \ z^2)\)\  + \ 
        1\)

\(\(kummerimp = 
        ContourPlot3D[

          kummerimplicit[x, y, z] // Evaluate, 
{x, \(-3\), 3}, {y, \(-3\), 
            3}, {z, \(-3\), 3}, 
PlotPoints -> {5, 6}, 

          DisplayFunction -> dip[EdgeForm[]]];\)\)