The 8×2ⁿ Dimensions of the Particles: From Structural Mechanics to Particle Physics through Octonion Mathematics

Table of Contents
PREFACE
INTRODlTCTION
OVERVIEW
	1. Octonion Mathematics for Physics
	2. The Deformable Continuum
	3. The Particle
	4. One-dimensional Aggregates of Two-di mentional Particles
	5. Two-dimensional Aggregates of One-dimensional Particles
	6. Three-dimensional Aggregates of Particles
1. OCTONIONS
	1.1 OCTONION ALGEBRA
		1.1.1 Linear Algebras without Divisors of Zero (LAwDoZ)
		1.1.2 Octonion Product
		1.1.3 Octonion Tensors
		1.1.4 Polar and Axial Vectors
		1.1.5 Orthogonal Transformations
		1.1.6 Position Vectors
		1.1.7 Position Vectors
		1.1.8 Free Polar Vectors
		1.1.9 The Polar Position
		1.1.10 Orientation Vectors
	1.2 OCTONION ANALYSIS
		1.2.1 Octonion Functions
		1.2.2 Partial derivatives
		1.2.3 Derivatives with respect to an octonion
		1.2.4 Extend Cauchy - Riemann Equations
		1.2.5 Octonion Potential Functions
		1.2.6 Octonion Line Integrals
		1.2.7 Extended Cauchy's Integral Theorem
		1.2.8 Extended Cauchy's Integral Formula
		1.2.9 Formulas of Gauss, Stokes and Green
2. THE DEFORMABLE CONTINUUM
	2.1. The Concept of the Deformable Continuum
	2.2. Translations and Strains
	2.3 Orbital Velocities and Strain Rates
	2.4 Forces and Stresses
	2.5 Orbital Momentums and Impulses
	2.6 Constitutive Laws of Solids
	2.7 Constitutive Laws of Liquids
	2.8 Constitutive Laws of Gauges
	2.9 A Vacuum Hypothesis
3. THE PARTICLE
	3.1 The Concept of Particle
	3.2 Motives, Composants, Shape and Lever-arm Functions
	3.3 Invariants
	3.4 Conformal and Contra-formal Motions
	3.5 The Octal Decomposition
	3.6 Polar Vector Composants and Motives
	3.7 Axial Vector Composants and Motives
	3.8 Induction Moments
	3.9 Real Scalar Composants and Motives
	3.10 Imaginary Scalar Composants and Motives
	3.11 Spin and Bi-moment
	3.12 The Complete System of Composants and Motives
4. ONE-DIMENSIONAL AGGREGATES OF TWO-DIMENSIONAL PARTICLES
	4.1 Beams and String
	4.2 Reference Systems
	4.3 The Bi-quaternal Decomposition
	4.4 Lateral Force Composants and Motives
		4.4.1 Polar Vector Composants and Motives
		4.4.2 Axial Vector Composants and Motives
		4.4.3 Real Scalar Composants and Motives
	4.5 Normal Force Composants and Motives
		4.5.1 Polar Vector Composants and Motives
		4.5.2 Axial Vector Composants and Motives
		4.5.3 Imaginary Scalar Composanls and Motives
		4.5.4 Contra-formal Normal Composauts and Motives
	4.6 The Complete System of State Parameters
	4.7 Loads and Stresses
	4.8 Interacting Composants
		4.8.1 Normal Interacting Composants
		4.8.2 Lateral Interacting Composants
	4.9 Hidden Composants
		4.9.1 Normal Hidden Composants
		4.9.2 Lateral Hidden Composants
		4.9.3 Universal Strains and Stresses
5. TWO DIMENSIONAL AGGREGATES OF DIMENSIONAL PARTICLES
	5.1 Surface Systems
	5.2 Reference System
	5.3 The Tri-Dual Decomposition
	5.4 Loads and Stresses
	5.5 Interacting Composants
	5.6 Hidden Cornposants
	5.7 Inter-composant Relations
		5.7.1 Inter-tangential Composants Relations
		5.7.2 Axial to Polar Vector Composants Relations
		5.7.3 Polar Vector Composants to Loads Relations
		5.7.4 Real Scalar lnteracting to Hidden Composants Relation
	5.8 Motives
	5.9 Universal Strains and Stresses
6. THREE-DIMENSIONAL AGGREGATES OF PARTICLES
	6.1 Zero-Dimensional Particles
	6.2 Solid Particles of Finite Size
	6.3 Solid Particles of Infinite Size
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