Jonah: For a full answer, you will, ahem*, have to read Chapter 8 of my forthcoming book (chapter title: “The Leap into the Fourth Dimension”).

In very brief: There is no such thing as “the fourth dimension.” There are ENTIRELY ABSTRACT mathematical objects called “manifolds.”** “Dimension” is one property of a manifold. Manifolds are very handy for modeling physical systems, including our familiar 3-D space, and also the 4-D spacetime of Relativity Theory. The dimension of a manifold can be any positive whole number. Manifolds are, though, I repeat, abstract mathematical objects, like numbers or points.

And the manifolds used to model those two physical “realities” are very different kinds of manifold. The first one is Euclidean, which means that Euclid’s geometry all works in it — the angles of a triangle add up to 180 degrees etc., etc. The manifolds used to model 4-D spacetime, however (not even to mention the 10- or 11-dimensional ones used to model String Theory) are radically non-Euclidean. To find your way around them, for instance, you have to use imaginary numbers.

While Kathryn’s letting me blog math, here is a solution to the brainteaser in my January diary.

* Kathryn, are we still permitted to ahem? I think I have pretty much stopped er-ing, but I’m not sure about other interjections, and I’ve lost my NR style book.

** This word, by the way, was introduced into math my hero, the ineffably great Bernhard Riemann, though he said it in German: “Mannigfaltigkeit.”