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Table of Contents
1 Introduction and Orientation; 2 Units and Orders of Magnitude; 2.1 External Interactions and Newton's Unit; 2.2 Speed of Light as Space-Time Hinge; 2.3 Gravity and Maximal Speed of Action; 2.4 Quantum Structure and Planck's Unit; 2.5 Minimal Quantum of Action and Maximal Speed of Action; 2.6 Intrinsic Units; 2.7 The (hbar,c, G)-System
Universal Units?; 2.8 Electrodynamics and Sommerfeld's Fine-Structure Constant; 2.9 Binding Energies and Couplings; 2.10 Electroweak Interactions; 2.11 Units and Symmetry Normalizations; 3 How Complex Is Nature?; 3.1 Numbers for Objects and Time
3.2 Numbers as Operations
From the Naturals to the Rationals3.3 Discrete and Continuous
From the Rationals to the Reals; 3.4 Constancy in Change
From the Real to the Complex Numbers; 3.5 Dynamics as Time Action; 3.6 Modalities; 3.7 Non-commutative Numbers
Quaternions; 3.8 Non-commutative Complex Operations; 3.9 Causal Minkowski Spacetime; 3.10 Hypercharge Group and Isospin Group; 3.11 Lorentz Group and Poincaré Group; 3.12 Internal Scalar Product and External Lorentz Metric; 3.13 Operational Spacetime; 4 Plato's Beautiful Symmetry
4.1 Regular Polygons and Platonic Triangles4.2 Platonic Solids; 4.3 Plato's Immaterial Basic Physics; 4.4 Dual Platonic Solids; 4.5 Soccer Molecules; 4.6 Coordinates for Platonic Solids; 4.7 Kepler's Mysterium Cosmographicum; 4.8 Polytopes and Mosaics; 4.9 Orthogonal Symmetry Groups; 4.10 Erlangen Program of Geometry; 4.11 Erlangen Program of Flat Space-Time; 4.12 Simple Lie Operations; 4.13 Lie Symmetries and Metrical Structures; 4.14 The Simplest Simple Symmetry; 4.15 Quantum Numbers and Weights of Simple Lie Algebras; 4.16 Weights, Roots, and Weyl Mirrors
4.17 Decomposition of Product Representations4.18 Squares and the Physics of Atoms; 4.19 Plato's Polytopes and Cartan's Weight Diagrams; 4.20 Quark Flavors; 4.21 Plato's Triangles and Dynkin's Diagrams; 4.22 Fundamental Representations; 4.23 Quantum Theory and Symmetry; 4.24 Basic Internal Symmetry Program; 4.25 Basic External Symmetry Program; 5 Circles and Winding Numbers; 5.1 Circles on the Earth and in the Sky; 5.2 Winding Numbers and Unit Roots; 5.3 Electromagnetic Winding Numbers; 5.4 Energies and Time-Winding Numbers; 5.5 Spin Winding Numbers; 5.6 Windings of the Binary Alternative
5.7 Charge Numbers of Leptons and Quarks5.8 Winding Number Matrices and Representation Weights; 5.9 Fractionality of Unitary Group Representations; 5.10 Adjoint Winding Numbers and Adjutopes; 5.11 Additive Unification of Unitary Symmetries; 5.12 Fractionality Correlations for Unitary Groups; 5.13 Broken Winding Numbers; 5.14 Broken Hypercharges and Nonabelian Internal Symmetry; 5.15 Young Frames for SU(n)-Representations; 5.16 Young Tableaux for Young Frames; 5.17 Box-Matrix Algebras; 5.18 Algebras for Finite Groups; 6 The Hall of Mirrors; 6.1 Reflection Group and Orthogonality
Universal Units?; 2.8 Electrodynamics and Sommerfeld's Fine-Structure Constant; 2.9 Binding Energies and Couplings; 2.10 Electroweak Interactions; 2.11 Units and Symmetry Normalizations; 3 How Complex Is Nature?; 3.1 Numbers for Objects and Time
3.2 Numbers as Operations
From the Naturals to the Rationals3.3 Discrete and Continuous
From the Rationals to the Reals; 3.4 Constancy in Change
From the Real to the Complex Numbers; 3.5 Dynamics as Time Action; 3.6 Modalities; 3.7 Non-commutative Numbers
Quaternions; 3.8 Non-commutative Complex Operations; 3.9 Causal Minkowski Spacetime; 3.10 Hypercharge Group and Isospin Group; 3.11 Lorentz Group and Poincaré Group; 3.12 Internal Scalar Product and External Lorentz Metric; 3.13 Operational Spacetime; 4 Plato's Beautiful Symmetry
4.1 Regular Polygons and Platonic Triangles4.2 Platonic Solids; 4.3 Plato's Immaterial Basic Physics; 4.4 Dual Platonic Solids; 4.5 Soccer Molecules; 4.6 Coordinates for Platonic Solids; 4.7 Kepler's Mysterium Cosmographicum; 4.8 Polytopes and Mosaics; 4.9 Orthogonal Symmetry Groups; 4.10 Erlangen Program of Geometry; 4.11 Erlangen Program of Flat Space-Time; 4.12 Simple Lie Operations; 4.13 Lie Symmetries and Metrical Structures; 4.14 The Simplest Simple Symmetry; 4.15 Quantum Numbers and Weights of Simple Lie Algebras; 4.16 Weights, Roots, and Weyl Mirrors
4.17 Decomposition of Product Representations4.18 Squares and the Physics of Atoms; 4.19 Plato's Polytopes and Cartan's Weight Diagrams; 4.20 Quark Flavors; 4.21 Plato's Triangles and Dynkin's Diagrams; 4.22 Fundamental Representations; 4.23 Quantum Theory and Symmetry; 4.24 Basic Internal Symmetry Program; 4.25 Basic External Symmetry Program; 5 Circles and Winding Numbers; 5.1 Circles on the Earth and in the Sky; 5.2 Winding Numbers and Unit Roots; 5.3 Electromagnetic Winding Numbers; 5.4 Energies and Time-Winding Numbers; 5.5 Spin Winding Numbers; 5.6 Windings of the Binary Alternative
5.7 Charge Numbers of Leptons and Quarks5.8 Winding Number Matrices and Representation Weights; 5.9 Fractionality of Unitary Group Representations; 5.10 Adjoint Winding Numbers and Adjutopes; 5.11 Additive Unification of Unitary Symmetries; 5.12 Fractionality Correlations for Unitary Groups; 5.13 Broken Winding Numbers; 5.14 Broken Hypercharges and Nonabelian Internal Symmetry; 5.15 Young Frames for SU(n)-Representations; 5.16 Young Tableaux for Young Frames; 5.17 Box-Matrix Algebras; 5.18 Algebras for Finite Groups; 6 The Hall of Mirrors; 6.1 Reflection Group and Orthogonality