Visualizing Higher Dimensions

Hypersphere hijinks

Part 1 — Visualize

People are visual creatures, and when reasoning about about high dimensions, one of the first considerations will likely be an attempt to imagine a visualization using metaphors from our shared three dimensional surroundings. Unfortunately, there are some elements of higher dimension that just don’t directly translate to 3D. Consider the difference between a 0D point, 1D line, 2D square, and 3D cube, and how each tier results in a increased number of vertices (1/2/4/8), edges (0/1/4/12), and faces (0/0/1/6). We can see that some of these elements are climbing in a non-linear manner, and so any attempt to visualize a progression may quickly become overwhelmed in a maze of features. As used here, a vertex is like a point intersection between edges, an edge is like a line intersection between planes, and a face is like a plane intersection between 3 dimensional elements. We can reasonably infer that at higher dimensions a similar progression of lower dimensioned features may emerge, for example a tesseract (a 4D hypercube) may have some set of three dimensional cube elements as an equivalents to a face’s plane (8 of them in fact), and a 5-cube may have some set of tesseract features. However since visualizing much beyond a tesseract is itself a challenge, that observation may not be helpful for our purposes.

Figure 1: hypercubes in 0–4 dimensions
Figure 2: hypercube graphs in 0–9 dimensions — Table[HypercubeGraph[k], {k,0,9}]
Figure 3: hypercube graphs in 0–4 dimensions, adjacency matrix — Table[AdjacencyMatrix[HypercubeGraph[k]]//MatrixForm, {k,0,4}]
Figure 4: hypercube graphs in 0–9 dimensions, adjacency matrix plot — Table[MatrixPlot[AdjacencyMatrix[HypercubeGraph[k]]] // MatrixForm, {k,0,9}]
Figure 5: Slices of Tesseract rotations projected to 2D image [Noll, 1967]
Figure 6: Hopf fibration map of a 3D sphere [Walczyk, 2022]
Figure 7: Hopf fibration map of a 4-sphere [Walczyk, 2022]
Figure 8: Hopf map of 3D sphere points to 4d sphere circles [Belmonte, 2022]

Part 2 — Imagine

Hypercubes and spheres aren’t the only shapes possible at high dimension, at a minimum their study has probably originated from their being the two simplest examples for cases of continuous verses disjointed geometries, sort of like how in machine learning tanh and ReLU activation functions are considered two different regimes of smooth or disjointed (the latter of which opened the door to deep learning as an aside). Even though tanh and ReLU are different regimes, we previously noted that a 3 layer ReLU network is capable of modeling a function generated by a smaller depth smooth activation. These type of equivalency considerations are also studied in high geometries for geometric figures like spheres and cubes.

Figure 9: Number of unique shapes with common faces to each side by dimension count
Figure 10: Unique shapes with common faces retained through dimensions
Figure 11: 2-sphere packing as grid and hexagonal
Figure 12: Sphere packed cube thought experiment
Debussy’s Minstrel Prelude — Nicholas Teague

References

Belmonte, Nico. Hopf Fibrations. (2022) http://philogb.github.io/page/hopf/#

The Art of Doing Science and Engineering — Richard Hamming

The Art of Doing Science and Engineering

Flatland — Edwin Abbott

Flatland

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Nicholas Teague

Writing for fun and because it helps me organize my thoughts. I also write software to prepare data for machine learning at automunge.com. Consistently unique.