- The cross product operation of GH is NONASSOCIATIVE. Not only is ASSOCIATIVITY convenient, but it is NECESSARY for an OPERATION to GENERATE a GROUP.
- A SYSTEM CLOSED UNDER AN OPERATION ("all-in-the-family") is a GROUPOID.
- An ASSOCIATIVE GROUPOID is A SEMIGROUP.
- A SEMIGROUP with an IDENTITY ELEMENT is a MONOID.
- A MONOID in which EVERY OPERATION HAS AN INVERSE is a GROUP.
So GH would only countenance a LOWLY GROUPOID -- along with many other such "failures" -- were it not for the next complaint.
- As previously noted, this cross product vector is INVARIANT UNDER INVERSION, UNLIKE ITS COMPONENT VECTORS, forcing GH to go "schizoid", distinguish "axial" (such as this one) from "polar" vectors. So GH is not even a GROUPOID, failing in CLOSURE for the best definition of vector. Furthermore, by a definition I find useful ( elsewhere), GH is not even an ALGEBRA, but a QUALGEBRA (QUASI-ALGEBRA).
- VECTORS MODEL ROTATION, and it's useful for VECTORS TO COMBINE INTO VECTORS (of "the same kind"). But this also fails in GH, causing it again to go "schizoid", distinguishing "fixed" vectors (needed for axis of rotation) from "free" (standard) vectors.
- A vector is 1-D, a DIRECTED LINE SEGMENT DISTINGUISHING THE LINE OF WHICH IT IS A PART. The COMBINATION OF TWO VECTORS IS, ordinarily, 2D, since some PLANE CAN BE FOUND CONTAINING BOTH. But this case is 3D, its RESULT IN A PLANE DISTINCT FROM THAT OF ITS COMPONENTS!
- NON-NULL VECTORS in GH HAVE NO INVERSES (not so for MULTIVECTORS), forcing a clumsy formulation of a RECIPROCAL SYSTEM.
- The REAL and COMPLEX DOMAINS remain SEPARATE (not so for MULTIVECTORS!) ignoring
- Failure of ASSOCIATIVITY and INVERSITY under PRODUCT require VECTOR SPACES TO BE DEFINED OVER A RING (ADDITIVE GROUP, MONOID for SCALAR MULTIPLICATION), rather than over a FIELD (GROUPS BOTH FOR ADDITION AND MULTIPLICATION), as is possible for VECTORS in MULIVECTOR THEORY, although this has not be pursued!
- GH has "nowhere to go". For example, CANNOT BE EMBEDDED IN A SUPERALGEBRA (as can OUTERPRODUCT).
However, the choice of a
(As a second opinion, G. P. Wilmot, "The Structure of Clifford Algebras", J. Math. Phys., 29(11), 1988: "... to interpret the geometrical content of the Gibbs' identities, we have placed ourselves in the double bind of being restricted to three dimensions and having basis-dependent identities. The Clifford algebraic approach to escape this bind is simply to interpret the polyvectors as geometric objects. The bivector ... represents the planar sbspce containing the two vectors, unless they are collinear.")