Here are rhombs drawn on a plane and on a parametric torus using Mesh
:
Clear[x, y, i, j, i0, j0]
myTorus[x_, y_] = {Cos[x] (16 + Cos[y]), Sin[x] (16 + Cos[y]), Sin[y]};
ParametricPlot3D[{x, y, 0}, {x, 0, 2 Pi}, {y, 0, 2 Pi},
MeshFunctions -> {2 #4 + #5 &, 2 #4 - #5 &},
Mesh -> {Range[0, 6 Pi, Pi/4], Range[-2 Pi, 5 Pi, Pi/4]}]
ParametricPlot3D[myTorus[x, y], {x, 0, 2 Pi}, {y, 0, 2 Pi},
MeshFunctions -> {2 #4 + #5 &, 2 #4 - #5 &},
Mesh -> {Range[0, 5 Pi, Pi/4], Range[-2 Pi, 5 Pi, Pi/4]},
PlotPoints -> 100]
and next as a collection of 3D polygons having vertices in the mesh intersections:
vert[i0_, j0_] =
First@SolveValues[{2 x + y == Pi/4 i0,
2 x - y == -2 Pi + Pi/4 j0}, {x, y}] // Simplify;
rhombTile[i_, j_] =
Line[{vert[i, j], vert[i + 1, j], vert[i + 1, j + 1],
vert[i, j + 1]}];
Graphics3D[
Union[Flatten@Table[rhombTile[i, j], {i, 0, 40}, {j, 0, 8}] /.
Line[pts_] :> Polygon[MapApply[myTorus, pts]]] /.
plg_Polygon :> {RandomColor[], plg}]