Authors: Robert B. Easter
This note very briefly describes or sketches the general ideas of some applications of the G(p,q) Geometric Algebra (GA) of a complex vector space C^(p,q) of signature (p,q), which is also known as the Clifford algebra Cl(p,q). Complex number scalars are only used for the anisotropic dilation (directed scaling) operation and to represent infinite distances, but otherwise only real number scalars are used. The anisotropic dilation operation is implemented in Minkowski spacetime as hyperbolic rotation (boost) by an imaginary rapidity (+/-)f = atanh(sqrt(1-d^2)) for dilation factor d>1, using +f in the Minkowski spacetime of signature (1,n) and -f in the signature (n,1). The G(k(p+q+2),k(q+p+2)) Mother Algebra of CGA (k-MACGA) is a generalization of G(p+1,q+1) Conformal Geometric Algebra (CGA) having k orthogonal G(p+1,q+1):p>q Euclidean CGA (ECGA) subalgebras and k orthogonal G(q+1,p+1) anti-Euclidean CGA (ACGA) subalgebras with opposite signature. Any k-MACGA has an even 2k total count of orthogonal subalgebras and cannot have an odd 2k+1 total count of orthogonal subalgebras. The more generalized G(l(p+1)+m(q+1),l (q+1)+m(p+1)):p>q k-CGA algebra, for even or odd k=l+m, has any l orthogonal G(p+1,q+1) ECGA subalgebras and any m orthogonal G(q+1,p+1) ACGA subalgebras with opposite signature. Any 2k-CGA with even 2k orthogonal subalgebras can be represented as a k-MACGA with different signature, requiring some sign changes. All of the orthogonal CGA subalgebras are corresponding by representing the same vectors, geometric entities, and transformation versors in each CGA subalgebra, which may differ only by some sign changes. A k-MACGA or a 2k-CGA has even-grade 2k-vector geometric inner product null space (GIPNS) entities representing general even-degree 2k polynomial implicit hypersurface functions F for even-degree 2k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. Only a k-CGA with odd k has odd-grade k-vector GIPNS entities representing general odd-degree k polynomial implicit hypersurface functions F for odd-degree k hypersurfaces, usually in a p-dimensional space or (p+1)-spacetime. In any k-CGA, there are k-blade GIPNS entities representing the usual G(p+1,q+1) CGA GIPNS 1-blade entities, but which are representing an implicit hypersurface function F^k with multiplicity k and the k-CGA null point entity is a k-point entity. In the conformal Minkowski spacetime algebras G(p+1,2) and G(2,p+1), the null 1-blade point embedding is a GOPNS null 1-blade point entity but is a GIPNS null 1-blade hypercone entity.
Comments: 8 Pages.
[v1] 2017-02-18 21:44:17
Unique-IP document downloads: 115 times
Vixra.org is a pre-print repository rather than a journal. Articles hosted may not yet have been verified by peer-review and should be treated as preliminary. In particular, anything that appears to include financial or legal advice or proposed medical treatments should be treated with due caution. Vixra.org will not be responsible for any consequences of actions that result from any form of use of any documents on this website.
Add your own feedback and questions here:
You are equally welcome to be positive or negative about any paper but please be polite. If you are being critical you must mention at least one specific error, otherwise your comment will be deleted as unhelpful.