Visualizing Higher Dimensions
Clayton Shonkwiler
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
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Build a tetrahedron.
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How does it project to the plane?
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What if it’s rotating?
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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}]]