As a contrast, some motivational background ... 3. A Modicum of Isometry ________________________ Why, might one ask, should this horrible object be interesting? Clifford algebras provide homogeneous coordinates for the isometries and subspaces of a number of intuitively appealing geometric spaces, composition corresponding to product: spherical, Euclidean, Moebius (conformal, inversive), Laguerre (equilong), and Lie-sphere geometry in particular. [So of course do matrices --- but that's another story.] For most purposes we can restrict consideration to isometries which are "kinematic" (aka "Spin-up"), connected to the identity and so --- by simultaneously reducing their extent and increasing iteration --- capable of extension in the limit into continuous motion. A kinematic versor B is characterised algebraically by constraints ||B|| > 0 --- which might as well be rescaled to ||B|| = 1; and B is even (comprises only even-grade terms). The isometry structure of the more familiar geometries above is reasonably well-documented, at least in low dimension. Briefly, a kinematic isometry B of (maximum) grade 2l can uniquely be factorised into orthogonal grade-2 "rotations" B = B_1 B_2 ... B_l, where B_i = S_i + L_i; here S_i is a scalar representing the extent of the rotation, and L_i a versor purely of grade 2 representing its (coline) axis. The orthogonality constraint implies that the Clifford product of any k-subset of the axes L_i is "exterior", purely of grade 2k. Each axis can now in turn be factorised uniquely into a pair of eigenvectors L = F G with an explicit geometric interpretation --- as S varies through the limiting range, under the associated motion any (object represented by a) vector proceeds continuously from the source F to the sink G. [In Euclidean 3-space there is insufficient room for a pair of finite axes: the most general kinematic isometry is a "heliform" bi-rotation, composing a rotation about a finite axis line with a (parabolic) translation along it and about an orthogonal axis lying at infinity.] What a nice theory --- but there are devils in the detail, for both decompositions rely on solving quadratic equations. One devil concerns the sign of ||L||. When ||L|| = -1, L is "hyperbolic" and F,G are real and visible: for example, dilation (in the Moebius group) has eigenvectors at the origin and infinity. When ||L|| = +1, L is "elliptic" and F,G are conjugate complex and invisible: for example, in Euclidean rotation there is plainly no actual source or sink. When ||L|| = 0, L is "parabolic" and F,G are coincident --- while we'd prefer not to have to go there too soon, these are difficult to avoid --- one example is Euclidean translation, with double eigenvector at infinity. WFL