# Mesh on the Umbilic torus

There is a $3D$ graphic of the Umbilic torus.

x[u_, v_] := Sin[u] (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v])
y[u_, v_] := Cos[u] (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v])
z[u_, v_] := Sin[u/3 - 2 v] + 2 Sin[u/3 + v]

ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, -Pi, Pi}, {v, -Pi, Pi},
PlotPoints -> 150, MaxRecursion -> 3, Boxed -> False, Axes -> False, Mesh -> 36]


but there is a gap on the mesh How can I make a mesh like this? • It depends on what you would like to do: Would this help you: mr = DiscretizeRegion[ ParametricRegion[{x[u, v], y[u, v], z[u, v]}, {{u, -Pi, Pi}, {v, -Pi, Pi}}]]; HighlightMesh[mr, {2}] – user21 Mar 31 '17 at 11:58
• That mesh irregularity seems to be inherent in the parametric equations you used; the situation is similar to that of the Möbius strip. To see this for yourself, add the settings Mesh -> {Subdivide[-π, π, 8]}, MeshFunctions -> {#5 &}. – J. M.'s technical difficulties Mar 31 '17 at 12:04

The parameterization is all fine. It is just that you need a number of rectangles in the v-direction that is divisable by 3. The issue here is that Mesh counts only the interior mesh lines, so that the value of Mesh needs to be smaller by 1 than the number of desired reactangles.

x[u_, v_] := Sin[u] (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v])
y[u_, v_] := Cos[u] (7 + Cos[u/3 - 2 v] + 2 Cos[u/3 + v])
z[u_, v_] := Sin[u/3 - 2 v] + 2 Sin[u/3 + v]

ParametricPlot3D[{x[u, v], y[u, v], z[u, v]}, {u, 0, 2 Pi}, {v, 0, 2 Pi},
PlotPoints -> {149, 149},
MaxRecursion -> 3,
Boxed -> False, Axes -> False,
Mesh -> {35, 35},
BoundaryStyle -> Directive[Thick, Black],
MeshStyle -> Directive[Thick, Black],
ViewPoint -> {0, 0, 4}
] 