Laying Down The Fourth-Dimensional Real-Vector Mack

Minkowski described a special form of geometric space that lent itself easily to Einstein's Special Theory of Relativity, being a space defined by four mutually orthoganal vectors (e₀, e₁, e₂, e₃) such that −(e₀)² = (e₁)² = (e₂)² = (e₃)² = 1.  It is a pseudo-Euclidian space that also accounts for time as a fourth dimension.

This is all, as you can well imagine, a pain in the ass to anyone who isn't a Lithuanian mathematician, or at the very least mildly engaged in some nonrelative physics now and again; it does, however, conjure a couple of immediate puns and a bit of semantic chicanery here and there, so it is infinitely useful for my regular purposes.

Take for example the concept of the learning curve.  I've always seen that plotted in two dimensions (measure of performance vs. number of attempts), giving no thought to the geometric concept of torsion on a curve that exists in Euclidian space.  It is unfathomable in that model, apparently, that in addition to moving up that curvature from a slow beginning to an accelerated learning state before gently curving back to a plateau, that the line might deviate from true on some Z-axis (like whether or not the learner actually gives a shit about the subject).  This is exactly the kind of tragic, incomplete analogy that occurs when psych majors appropriate slang from physicists.

To continue, let's add a 4th dimension of time to our model; I would suggest that the number of attempts is often interpreted as a timeline, but strictly speaking that isn't true.  Think of it like picking up strangers in a bar; sometimes "Nice shoes..." works on the first chick, and sometimes you end up telling eight people a twenty-minute story about how your dog died and you need to be held.  Number of attempts is in no way synonymous with time invested.

I propose the influence of the Z-axis (everything orthoganal to the learning process) and the number of attemps within a given space both impact the learning curve, and if the activity of learning is assumed to be utterly unrelated to all other activities and the iterations of learning attempts are spaced irregularly or inadequately, the 2-dimensional model doesn't plot anything meaningful.

It's also worth noting that the unrelated fractal known as Minkowski's Curve is also called Minkowski's Sausage, and that somewhere there is undoubtedly a cognitive psych / discrete mathematics double major that will respond well to that veiled reference if you buy them a beer.