This is actually two questions about the Wolfram MathWorld article on torsion here.
1: What is a geometric interpretation of torsion? For curvature I understand it as the reciprocal of the radius of the circle around which a point is moving at a given instant. But the description in the link says torsion "is the rate of change of the curve's osculating plane". I know what the osculating plane is, but what does it mean by its "rate of change"?
2: This is a small point, but how do they get from the equation $$\tau = \frac{\left|\dot{x} \,\ddot{x} \, \dddot{x}\right|}{\left|\dot{x} \times \ddot{x}\right|^2}$$ to this? $$\tau = \rho^2 \,\left|\dot{x} \,\ddot{x} \, \dddot{x}\right|$$ (where $|x\,y\,z|$ is the triple product and $\rho$ is the radius of curvature)
Is the following correct? $$\rho = \frac{\left| \dot{x} \right|^3}{\left|\dot{x} \times \ddot{x}\right|}$$
It seems like there is a factor of $\left| \dot{x} \right|^6$ unaccounted for.
$\newcommand{\Vec}[1]{\mathbf{#1}}\newcommand{\Tgt}{\Vec{T}}\newcommand{\Nml}{\Vec{N}}\newcommand{\Bnm}{\Vec{B}}$Here's one way to make "rate of change of the osculating plane" precise. A path $\gamma$ parametrized by arc length may be expanded as a Taylor polynomial in the arc length $s$. Using subscripts to denote evaluation at $s = 0$, assuming $\kappa_{0} > 0$, and letting $\{\Tgt_{0}, \Nml_{0}, \Bnm_{0}\}$ denote the Frenet frame at $\gamma_{0}$, $$ \gamma(s) = \gamma_{0} + s\, \Tgt_{0} + \kappa_{0} \tfrac{1}{2}s^{2}\, \Nml_{0} + \kappa_{0} \tau_{0} \tfrac{1}{6}s^{3}\, \Bnm_{0} + \text{higher order terms.} $$
In answer to your second question, probably the Wolfram page assumes the path is parametrized by arc length.