Looking At Things Four-Dimensionally
I’d like to close out the month by pointing to 4D Visualization, a web site set up by … well, I’m not sure the person, but the contact e-mail address is 4d ( at ) eusebeia.dyndns.org for whatever that’s worth. (Worse, I can not remember what site led me to it; if you’re out there, referent, please say so so I can thank you properly. In the meantime, thank you.) The author takes eleven chapters to discuss ways to visualize four-dimensional structures, and does quite a nice job at it. The ways we visualize three-dimensional structures are used heavily for analogies, and the illustrations — static and animated — build what feels like an intuitive bridge to me, at least.
Eusebeia (if I may use that as a name) goes through cross-sections, which are generally simple to render but which tax the imagination to put together1, and projections, and the subtleties in rendering two-dimensional images of three-dimensional projections of four-dimensional structures so that they’re sensible. It’s all quite good and I’m just sorry that my belief in the promise “More chapters coming soon!” clashes with the notice, “Last updated 13 Oct 2008”.
The main page is still being updated regularly, including a Polytope Of The Month feature. A polytope is what people call a polygon or polyhedron if they don’t want their discussion to carry the connotation of being about a two- or three-dimensional figure. It’s kind of the way someone in celestial mechanics talking about the orbit of an object around another might say periapsis and apoapsis, instead of perigee and apogee or perihelion and aphelion, although as far as I can tell people in celestial mechanics are only that precise if they suspect someone pedantic is watching them. I’m not well-versed enough to say how much polytope is used compared to polyhedron.
Anyway, for those looking for the chance to poke around higher dimensions, consider giving this a try; it’s a good read.
[1: I knew that a three-dimensional cube has, on the right slice, a hexagonal cross-section. It’s something I discovered while fiddling around with the problem of charged particles on a conductive-particule sphere, believe it or not. ]