Arithmoí - A Study of Mathematical Cohesiveness in Physics and Metaphysics
Volume I, Part 1: Mathematical Foundations and the Singularity
Table of Contents
Introduction… 9
Acknowledgments… 10
1.1.1 - Matrix Group Theory… 11
1.1.2 - Matrix Group Properties… 13
1.1.3 - General Linear Group… 13
1.1.4 - Special Linear Group… 14
1.1.5 - Orthogonal Groups… 15
1.1.6 - The Linear Transformation… 15
1.1.7 - Group Exponentiation… 15
1.1.8 - The Singular Value Decomposition… 17
1.1.9 - The Matrix Transpose… 18
1.1.10 - Calculating the SVD… 19
1.1.11 - The Commutator… 20
1.1.12 - Complex Powers and Logarithms… 20
1.1.13 - Matrix Powers and Logarithms… 21
1.1.14 - Symmetric and Dihedral Groups… 22
1.1.15 - The Symmetric Groups… 23
1.1.16 - The Dihedral Groups… 23
1.1.17 - The Symmetric/Dihedral Isomorphism… 25
1.1.18 - The Symmetric/Dihedral Group in 3R... 25
1.1.19 - The DihedralGroup and Symmetry Transformations... 26
1.1.20 - The Symmetry Group... 30
1.1.21 - Polygonal Rotations in U(3)… 34
1.1.22 - Hilbert Spaces… 37
1.1.23 - Geometric Visualization of the Inner Product… 42
1.1.24 - The Cartan Subalgebra of U(3)… 44
1.1.25 - The Anti-Diagonal Symmetry of the Cartan Subalgebra… 47
1.1.26 - The Symmetric/Dihedral Cone Group… 47
1.1.27 - Projective Forms… 50
1.2.1 – Tangent Spaces and Differential Forms… 56
1.2.2 - The Cyclic Trigonometric Functions and their Derivatives… 59
1.2.3 - The Dihedral Differential Forms in 3C… 62
1.2.4 - The Dihedral Differential Group in 3R... 66
1.2.5 - The Dihedral Cone Differential Group in 3C… 68
1.2.6 - The Lie Algebra and Differential Forms... 71
1.2.7 - Tangent Spaces... 76
1.2.8 - Dihedral Group Tangent Spaces... 77
1.2.9 - Numeri Mirabiles... 81
1.2.10 - Chapter 2 Summary... 95
1.3.1 - Curvature and Light... 96
1.3.2 - The Space of Angles... 98
1.3.3 - Vector Representations in Complex-Angled Space... 101
1.3.4 - Curvature... 103
1.3.5 - Circular Curvature and the Wave Equation... 109
1.3.6 - Hyperbolic Curvature... 116
1.3.7 - Complex Curvature... 122
1.3.8 - Geometric Algebra... 125
1.3.9 - The Asymptotic Light Functions... 129
1.3.10 - The Curvature of Light... 148
1.3.10 - Chapter 3 Summary… 153
1.4.1 - Matrix Group Conic Sections... 155
1.4.2 - SU(2) as the Inscribed Sphere of the Light Cone... 157
1.4.3 - The Tangent Space of the Cone... 163
1.4.4 - Null Curves and the Cone Group… 166
1.4.5 - Number Forms, Odd and Even Partition Geometry, and Minimal Elements... 176
1.4.6 - The Conic Group... 180
1.4.7 - The Axis Group... 182
1.4.8 - The Subgroup of the Symmetric Conic Group... 186
1.4.9 - The SU(2) Subgroup of the Symmetric Conic Group... 188
1.4.10 - The Pauli Matrices Subgroup of the Symmetric Conic Group… 193
1.4.11 - Geometric Visualization of Matrix Group Conics... 195
1.4.12 - Chapter 4 Summary… 207
1.5.1 - Tangent Space Angles and Path-Connectedness in the Symmetric Cone Group... 209
1.5.2 - The Symmetric Cone Group Lie Algebra... 210
1.5.3 - The Matrix Logarithm… 211
1.5.4 - The Baker-Campbell-Hausdorff Formula... 213
1.5.5 - The Matrix-Matrix Exponential… 217
1.5.6 - The Matrix-Matrix Exponential and Smoothing the Tangent Space... 219
1.5.7 - Smoothing the Minimal Element Basis… 227
1.5.8 - The Infinite Symmetry Group and the Subtangent Space... 231
1.5.9 - Subtangent Vectors… 236
1.5.10 - Chapter 5 Summary… 239
1.6.1 - The Singularity… 240
1.6.2 - Infinite Differentiation and the Matrix Logarithm… 242
1.6.3 - Matrix-Based Infinite Integration and Differentiation… 247
1.6.4 - Harmonic Logarithms and the Infinite Subtangent Space… 255
1.6.5 - Outer Products and Infinities… 260
1.6.6 - The Alternating Harmonic Series and its Inverse Analog… 272
1.6.7 - Transformations and Representation Theory… 277
1.6.8 - Matrix Determinants and the Infinite Symmetry Group… 283
1.6.9 - The Infinite Sphere of Tangent Spaces… 284
1.6.10 - Chapter 6 Summary… 293
1.7.1 - Prime Numbers and the Geometry of the Minimal Element… 294
1.7.2 – The Geometric Primes and the Higher-Ordered Forms… 315
1.7.3 - Prime Number Theory and the Geometry of Groups… 334
1.7.4 - Primality… 349
1.7.5 - Chapter 7 Discussion and Summary... 354
1.8.1 - The Flowing Point and the Theory of Light… 359
1.8.2 - The Flowing Point of Axial Light… 359
1.8.3 - The Quantum Flowing Point… 364
1.8.4 - Relativistic Mechanics and the Unified Field… 371
1.8.5 - The Unit Invariant Mass and the Flowing Point Ideals… 389
1.8.6 - The Theory of Light… 393
1.8.7 - Chapter 8 Summary… 423
