Math by Diane Demers

A Paper on a Nonassociative, Clifford-Like Algebra

The Orientation Congruent Algebra

Abstract.  The correlated grade form of twisted blades (twisted simple multivectors) faithfully renders in symbols their native geometric structure. The discovery of this paper's nonassociative Clifford-like algebra was driven by trying to calculate exterior products of straight and twisted multivectors directly in a basis of this form. The key was found to be the orientation congruent (OC) algebra. This paper is being published electronically in about ten sections, each offered as soon as written. In this first section we axiomatize the orientation congruent algebra by generators and relations. The next section derives the sign factor function sigma and proves that the Clifford product times it is the multiplication of an explicitly Clifford-like algebra isomorphic to the orientation congruent algebra. Later sections are planned to show how to calculate the OC product in Mathematica and Clical; to define the orientation congruent contraction operators, deduce their properties, derive other expressions for them, and use them to compute the OC product within the exterior algebra using a modified Cartan decomposition formula; to develop the algebra's product sequence graph with labeled edges; to derive a predictor of a null associator as a function of the grades of the three elements in it; to prove the associomediative property of the algebra's counit; to develop matrix representations (under a nonassociative matrix product) of the orientation congruent product; and to discuss the motivating application per se and as inspiration for the first set of axioms.

This paper is available for downloading or reading online. It will be split by section into separate PDF files. Each file will be offered as soon as it is written. As the paper approaches completion the finished sections may be combined into one file. Revisions may occur so be sure to get the latest versions of each section.

Abstract.  Similar to Version 1.3.

A Brief Explanation of the Motivating Application

The OC Algebra Generalizes MacFarlane's Hyperbolic Quaternions

The OC₂ algebra is isomorphic to the hyperbolic quaternions of Alexander MacFarlane. MacFarlane published a paper "Hyperbolic Quaternions," (1989-1900, Proc. Royal Society of Edinburgh 23:169-181) on his discovery of "a system of quaternions complementary to that of Hamilton, which is capable of expressing trigonometry on the surface of the equilateral hyperboloids." According to the brief biographical essay on MacFarlane Emil Borel later observed that the 3D hyperboloid model of hyperbolic space describes the kinematic velocity space of special relativity. Also see the Wikipedia article:  Hyperbolic Quaternions.

The series of orientation congruent algebras OC_n starting with n = 1 and increasing are isomorphic to the the double or split-complex numbers for n = 1, the hyperbolic quaternions for n = 2, and their generalizations to higher dimensions for n > 2.

Links to a Few Clifford Algebra Related Sites

• Advances in Applied Clifford Algebras:  An online journal.
• University of Amsterdam Geometric Algebra Group:  Leo Dorst and colleagues.
• Ian Bell:  Maths for (Games) Programmers:  Section 4 - Multivector Methods.
• James E. Beichler's journal "YGGDRASIL":  Links essay "Twist til' we tear the house down!" on Clifford's math and paraphysics.
• John Browne:  "Grassmann Algebra" (book draft); Chapter 12 is "Exploring Clifford Algebra." Also working on Mathematica implementation of Grassmann algebra.
• Cambridge University Geometric Algebra Research Group
• Clyde Davenport:  Author of the privately published book "A Commutative Hypercomplex Calculus with Applications to Special Relativity."
• Bertfried Fauser:  Site of Extension. He applies a Hopf gebraic formulation of the Clifford algebra of a not necessarily symmetric bilinear form to quantum physics using commutative and tangle diagrams. Contains his papers, talks, and links to "Clifford people" (many broken).
• Ghent University Clifford Research Group
• David Hestenes:  Geometric Calculus Research and Development. Promoter of the application of Clifford algebra to physics.
• International Clifford Algebra Society:  abstracts, bulletins, and conferences
• Pertti Lounesto:  Now deceased. This is a mirror of his last site provided by Perttu Puska.
• Perttu Puska:  Algebras of electromagnetics. This site contains a Clifford algebra section.
• Patrick Reany:  Clifford Algebra Papers. Contains his papers and some links.
• Garret Sobczyk:  Coauthor of classic "Clifford Algebra to Geometric Calculus."  Contains a Cliffordian approach to linear algebra and other papers.

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