And at long last, the crucial connection. 4. Something in the Woodshed ____________________________ And there's another devil lurking: the extents S_i are (reciprocal) roots of an elegant polynomial equation ||B(1/S)|| = 0, with B(T) = <B>_0 + <B>_2 T + <B>_4 T^2 + ... representing a kind of Taylor series in the variable scalar T. The associated axis L_i is given by the bivector (grade-2) part of the singular versor scalar + L_i = B(T) (d/dT)B(T) evaluated at T = 1/S_i --- at least, provided L_i is not parabolic (S_i = 0), and B is not "isoclinic" (S_i multiple root). [This mysterious expression parrots the formula for the tangent to the space curve defined by a vector function of a single parameter.] The equation in S has only even powers; when a root (S_i)^2 > 0, the associated rotation S + L is hyperbolic, and setting S = coth(t/2) converts it into a continuous motion of time t; when (S_i)^2 < 0, S + L is elliptic, and S = cot(t/2) for angle t. [Yes, I know there's an \I = sqrt(-1) gone missing above, and honestly, I can explain, only for the time being I'm not going to, so there!] Suppose the equation in S^2 had a pair of conjugate complex roots? Since this never seemed to happen in most of these geometries --- though I should very much like to know, I actually have no idea why not --- as far as I am aware, nobody else bothered about this possibility much. But I bothered about it. For a start --- if such an exotic beast existed --- what physical meaning might be assigned to its single complex extent S, an apparently inextricable entanglement of angle and length? And behold: it actually does exist, in n-space Lie-sphere geometry, whose isometries are represented by the algebra Cl(n+1,2). The associated pair of rotations is also conjugate complex, their eigenvectors E + \I F, G + \I H combining in conjugate pairs across rotations, rather than (as it were) along a single elliptic rotation. The canonical grade-4 example has eigenvectors \x +/- \I\w, \y +/- \I\v, and associated conjugate complex rotation axes (\x\y + \v\w) -/+ \I(\x\v + \y\w), with product essentially \x\y\v\w. This can now be transmogrified into a continuous motion with two real parameters, by attaching complex extents S = 2(a + b\I), S^- = 2(a - b\I) to the axes, yielding A = (S + \x\y + \v\w - \I\x\v - \I\y\w)(S^- + \x\y + \v\w + \I\x\v + \I\y\w) = |S|^2 + 2 Re(S)(\x\v - \y\w) + 2 Im(S)(\x\y + \v\w) + 4\x\y\v\w = (a^2+b^2) + b\x\y + a\x\v - a\y\w + b\v\w + \x\y\v\w (dropping scalar factor 4), as previously presented. Isometries of this type turn out to be surprisingly common: an early attempt at implementation of factorisation into orthogonal rotations encountered a specimen as its very first random test datum, causing the hapless programmer (me) to suspect the early onset of senile dementia. Conjugates of A, along with compositions of hyperbolic-hyperbolic, hyperbolic-elliptic, and elliptic-elliptic of two types, are the only types of bi-rotation in Lie-sphere geometry having positive density [Bertrand's paradox preventing any more specific assertion concerning the relative values involved]. WFL