Automatic visualization of high-dimensional objects

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Last night in the chat room, the task of visualizing high-dimensional objects was brought up. It seemed fun, so I wandered off into the computational universe in that direction to find a minimal, shareable result. The steps are:

  1. Define an object (points and edges)
  2. Generate some sliders to control rotation (choosing a number of perpendicular axes equal to the number of dimensions of the object is a good starting set)
  3. Generate a set of orthogonal projections by choosing all pairs of dimensions

This shows the popular tesseract or 4-dimensional cube.
temp.gif

points = Tuples[{-1, 1}, 4];
edges = Select[Subsets[Range@Length@points, {2}],
   Count[Subtract @@ points[[#]], 0] == 3 &];

colors = ColorData["AtlanticColors"] /@ Rescale@points[[All, 1]];
dimension = Length@First@points;

controls =
  Column[(angle@# = 0; Slider[Dynamic@angle@#, {0, 2 Pi}]) & /@
    Range@dimension];

views = Dynamic@
   GraphicsGrid@
    Partition[
     Graphics[{Thick,
         GraphicsComplex[#,
          Line[edges, VertexColors -> (colors[[#]] & /@ edges)]]},
        ImageSize -> {100, 100}] & /@
      Outer[#2[[#]] &, Subsets[Range@dimension, {2}],
       MapIndexed[RotationTransform[angle @@ #2, #] &,
         Composition @@ Partition[IdentityMatrix@dimension, 2, 1, 1]]@
        points, 1], 3, 3, 1, Null];

Panel@Row@{controls, Spacer@10, Panel@views}
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