In the definition (in one spatial dimension) of $\Delta \tau$ there is the relation:
$(\Delta \tau)^2 = (\Delta t)^2 - (\Delta x)^2$ which is invariant. If $(\Delta x)^2 > (\Delta t)^2$ then there is the characterization "spacelike."
In this case $\Delta \tau$ will be an imaginary number.
My question is: what is the intuitive physical or geometric meaning of imaginary in this context?
(Just a guess; is it related to hyperbolic functions and the Lorentz transformations?)