Visualizing Higher Dimensions

Clayton Shonkwiler

shonkwiler.org

What do we mean, “higher dimensions”?

“Visualizing higher dimensions is an exercise in creative lying.”

—Herman Gluck

Projection to 1D

Rotating

Singularity

Projection to 2D

Rotation

The Sphere

Slicing

What are the coordinates of a cube?

How about a hypercube?

Projection to 3D

Rotations

Slicing

Unfolding

Can you build it?

Practice: 5-Cell

  • Build a tetrahedron.

  • How does it project to the plane?

  • What if it’s rotating?

  • Visualize the 4-dimensional version of the tetrahedron (called the 5-cell).

Some more visualizations of higher dimensions

Stereo[p_] := p[[;; -2]]/(1 + p[[-1]]);
vertices =
  Normalize /@
   DeleteDuplicates[
    Flatten[
     Permutations /@ ({-1, -1, 0, 0}^Join[#, {1, 1}] & /@ 
        Tuples[{0, 1}, 2]), 1]];
edges = Select[Subsets[vertices, {2}],
   #[[1]] != -#[[2]] && HammingDistance[#[[1]], #[[2]]] == 2 &];
smootheststep[t_] := -20 t^7 + 70 t^6 - 84 t^5 + 35 t^4;

With[{viewpoint = 100 Normalize[{1, 1 - Sqrt[2], 0}], 
  cols = RGBColor /@ {"#0CCA98", "#5E366A", "#201940"}},
 Manipulate[
  Graphics3D[{Thickness[.004], 
    Tube[#, .05] & /@
     (Stereo[
           RotationTransform[π/2 smootheststep[t], {{1, 1, 0, 
               0}, {0, 0, 1, 1}}][#]]
          & /@ # & /@ edges)},
   PlotRange -> 3.1, ImageSize -> {800, 600}, ViewPoint -> viewpoint, 
   ViewVertical -> {0, 0, 1}, Boxed -> False, ViewAngle -> π/300,
   Lighting -> {{"Point", cols[[1]], {0, -3, 0}}, {"Point", 
      cols[[2]], {3, 0, 0}}, {"Ambient", cols[[-1]], viewpoint}}, 
   Background -> cols[[-1]]],
  {t, 0, 1}]]

Web: shonkwiler.org

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Mastodon: @shonk@mathstodon.xyz

These slides: shonkwiler.org/hypercube

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Visualizing Higher Dimensions

By Clayton Shonkwiler

Visualizing Higher Dimensions

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