4-Dimensional (and beyond) Polytopes – Revisited

In a recent metaphysical Zoom meeting, a friend mentioned a story about 2-dimensional creatures which I recognized as the classic book: “Flatland: A Romance of Many Dimensions by Edwin A. Abbott” first published in 1884 – note the very archaic labels on the cover illustration. (We’ve made several other references to flatland before on the GeometryCode website.) We got into a fun and mind-stretching conversation about how we believe we’re 3-dimensional creatures almost completely oblivious about higher dimensions and probably just as mindless about 1-dimensional and 2-dimensional realms. And of course, what would a 0-dimensional – non-dual beyond spacetime? – world be? No self-respecting 3-D creature wants to go there!

My favorite analogy is imagining the plight of someone trapped in 2-D (like Abbott’s Flatland) and having a “paranormal” experience when someone in 3-D (what Abbott calls “Spaceland”) decides to play a prank and push a 3-D pencil through some unfortunate 2-D denizen’s world – Let’s call her Trudy. At first, Trudy sees (hallucinates?) a black dot growing steadily larger (the lead of the pencil) which turns brown as it grows (the wood of the pencil) then morphs into a yellow hexagon which Trudy can only detect because of the 6 edges. The apparition remains a hexagon for a while until it abruptly changes into a slightly larger metallic circle, then a slightly smaller flexible pink circle (the eraser), and then, just as mysteriously completely vanishes as the 3-D pencil leaves the 2-D plane of Trudy’s paper-thin world.  What recourse does she have now but to phone the Flatland equivalent of National Enquirer to report an alien encounter and/or phone her therapist?

So what if we go the other direction and explore spatial dimensions of 4 and beyond? A couple of decades ago I had the good fortune to live not far from fellow geometer Russell Towle who lived in Dutch Flat, California. Russell spent several hours with me showing his brilliant work on his Mac along with lots of other amazing things, such as zonohedra and similar work from other math-savvy colleagues who had explored these realms. In 2013, I made a short post “In memory of Russell Towle” when I learned of his passing.

Not long after this, Russell gave me this link to one of Mark Newbold‘s pages about Russell Towle’s 4D Star Polytope Animations and I shared it on my Resources page – scroll down to Links to sites about polyhedra and higher dimensional polytopes. When I went to do a screen share of Towle’s animations for my Zoom colleagues, I discovered that they were made with a version of QuickTime that is no longer supported, so I quickly converted them to mp4 files and uploaded them to the GeometryCode YouTube channel (“shorts”) category here. Here is an excerpt from Mark’s pages about Russell’s work:

“These may be the first animations ever made of the solid sections of four-dimensional star polytopes. To get a better idea of just what these “polytopes” are, one should read H.S.M. Coxeter‘s “Regular Polytopes” (Coxeter01). Briefly, plane polygons are two-dimensional polytopes, and polyhedra, three-dimensional polytopes. Where polygons are bounded by line segments, and polyhedra by polygons, a 4-polytope is bounded by polyhedra.
Just as we may have any number of planes in three dimensions, in 4-space we may have any number of 3-spaces. Two 3-spaces might be a millionth of an inch apart and yet have no common point (thus the popular idea of parallel universes). It follows that, given a fixed direction in the 4-space, we can take solid sections of objects in the 4-space, perpendicular to that direction.

If you find these concepts difficult, you are not alone. Even when a person is blessed with some extraordinary faculty for visualizing objects in higher space–as was Alicia Boole Stott, a century ago–it is a matter of years, and considerable patience, before much progress is made in the subject.

In these animations, a 3-space is passed from one vertex of each star polytope, to the opposite vertex, and sections taken at small intervals. The star polytopes were constructed, and the sections found, using Mathematica 4.0. The sections were rendered in POV-Ray (a freeware ray-tracer).”

I had a copy of Regular Polytopes by Coxeter for several years, and I think I understood a small amount of it (on probably a very superficial level), but most of it was beyond my comprehension, yet fascinating!

Towle was also able to plumb the mental realms of 4+ dimensions further than I will likely ever venture and made some amazing computer animations of morphing polyhedra that represent projections of a 4-D polytope (a.k.a. polychoron) onto 3-D polyhedra, further projected down onto 2-D by representing 3-D polyhedra as 2-D animation frames. These amazing short animations may be metaphorically somewhat akin to consecutive 3-dimensional “slices” through a 4-dimensional shape, with each frame of the video being the next adjacent slice. Enjoy!

• Russell Towle’s 4D Star Polytope Animation {52,3,5}vert

  

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• Russell Towle’s 4D Star Polytope Animation {3,3,52}vert

  

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• Russell Towle’s 4D Star Polytope Animation {5,3,52}vert

  

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• Russell Towle’s 4D Star Polytope Animation {5,52,3}vert

  

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• Russell Towle’s 4D Star Polytope Animation {5,52,5}vert

  

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• Russell Towle’s 4D Star Polytope Animation {52,3,3}vert