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programming - Morphing a "sheet of paper" into a torus


How can I visualize the standard topological "rubber-sheet" construction of a torus, that is, morphing a square into a torus?



How can I start or are there any examples in the Mathematica documentation respectively?



Answer



Edit I had some time so I've added full surface torus. Old code in edit history.


  DynamicModule[{x = 2., l = 100., x2 = 2., l2 = 100., grid, fast, slow},
    Grid[{{
    Graphics3D[{
      Dynamic[Map[{Blue, Polygon[#[[{1, 2, 4, 3}]]]} &, 
                  Join @@@ (Join @@ Partition[#, {2, 2}, 1])
                 ]&[
                   ControlActive[fast[l, l2], slow[l, l2]]]

                   ]

                }, PlotRange -> {{-7, 7}, {-7, 7}, {-1, 2}}, ImageSize -> 600, 
                   Axes -> True, BaseStyle -> 18]
    ,
    Column[{
      Slider[Dynamic[x, (l = 10.^#; x = #) &], {.0001, 2.}],
      Slider[Dynamic[x2, (l2 = 10.^#; x2 = #) &], {.0001, 2.}] }]
    }}]
 ,

 Initialization :> (

 grid[l_, l2_, n_, m_] := Outer[Compose,         
   Array[RotationTransform[# Pi/l2, {0, 0, 1.}, {0, -l2, 0}] &, n, {-1, 1}],
   Array[RotationTransform[# Pi/l, {1., 0, 0}, {0, 2, l}][{0, 2, 0}] &, m, {-1, 1}], 
   1];

   fast[l_, l2_] = grid[l, l2, 10, 10];
   slow[l_, l2_] = grid[l, l2, 50, 25];
   )]


enter image description here


For < V.9 please switch Array to Table. This syntax for Array was introduced, silently, in V.9. linespace equivalent in MMA


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