Shadows from higher dimensions (Flatland, 3D, 4D and beyond)

Friday, 9 December 2005

In the Keynote (Apple’s wonderful alternative to PowerPoint) presentations I’ve been giving over the last couple of years, I have been fortunate to have been given permission to use some imagery from some serious polytope pioneers. What is a polytope, you ask? The mathematical answer (hopefully I’m not too far off with my paraphrasing, for those with far more expertise in this arena than I) is that it is a general term for certain types of shapes of arbitrary dimension. The easiest ones to discuss are the “regular” ones; those with all the angles and side lengths the same.

Let’s start with regular polytopes in two dimensions; these are called polygons, and are probably the most familiar to most folks. In 2D, you have polygons like triangles, squares, pentagons, hexagons, heptagons, octagons, nonagons, decagons, etc. to infinity… In 3D, there are only 5 regular polyhedra, called the Platonic Solids. My sacred geometry tutorial at http://www.intent.com/sg (and various links on the site) has lots of info about these classic shapes that have amazing connections to every aspect of our usual (illusory?) material world that we humans hang out it most of the time. 🙂 Once we get past three dimensions, however, our minds have a bit of trouble grokking the hyper-jump into 4D. There are actually 6 regular polytopes in 4 dimensions and these are called polychora. Need more help with this concept?

I recently put up lots of links on my my links page. Scroll down to “Links to sites about polyhedra & higher dimensional polytopes.” At the top of that section are links to Russell Towle’s 4D shadows (and animations) as well as lots of other related links. I had the pleasure of spending the better part of a day at Russell’s place near where I lived in California last year, and he showed me a plethora of exotic and wonderful imagery created with Mathematica, POV-Ray and other tools that dazzled my imagination. Accordingly, I put his links at the top of the list, although the other links in that section are also a cornucopia of great imagery, animations and info as well.

If the 4D math is too intimidating, I still recommend visiting these sites just for the imagery! Perhaps the flatland-pencil metaphor is in order for the those of us who minds turn to putty trying to wrap around even the most basic polychora. Imagine us 3D folks being somehow magically trapped in 2D (flat-land), and stuck on a piece of paper. If some 3D soul with a slightly mischievious bent wanted to have a little fun, all they would have to do is pass a pencil through our “world” (the paper) and we would have these bizarre hallucinations (quick phone the Dimensional Enquirer; we have their lead paranormal story!) At first we might see a black circle (graphite), then a light brown circle (shaved cedar), then a yellow hexagon (the paint on the pencil), then another circle (the metal holding the eraser), then a pink, slightly squishy circle (the eraser). To complete our hallucination, the pink circle would disappear almost instantly, and we’d immediately question reality, our sanity or perhaps our optometrist… 🙂

So if string theory suggests there are at least 11 dimensions (and this stuff is all completely consistent and robust in terms of the physics and math from what I understand … which isn’t a lot), then perhaps this metaphor might go a LONG way toward explaining a whole lot in our “reality” that is currently relegated to “meta-” stuff.

Those mystics among us might not be quite so crazy after all… I’m sure I’ll have more to share on this subject later, but that’s enough for now, and do check out the links, especially Russell’s awesome animations (each frame of which is like a “3D slice” of a 4D object.) If you have more links (or ideas along these lines to suggest), please contact me via the “Contact..” link on my site.


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