## Table of Contents

**Preface**

**About the Author**

**Part I. The Atomic Structure**

**Chapter 1. Spherical Quantities in Two-Dimensional Spacetime**

1.1. Introduction

1.2. Euclidean Space and Newtonian Time

1.3. Two-Dimensional Spacetime

1.3.1. General Harmonic Oscillation

1.3.2. Duality in a Helium Shell

1.3.3. Duality in an LC Circuit

1.3.4. The Duality Equation vs. SchrĂ¶dingerâ€™s Equation

1.4. Dynamic Calculus

1.4.1. Infinitesimal Calculus Definition

1.4.2. Dynamic Calculus Definition

1.4.3. Spacetime Continuity

1.4.4. Properties of Spherical Quantity

1.5. Linear Interpretation of Spherical Quantity

1.5.1. Spherical Quantity as a Rotatory Vector

1.5.2. Dimension Concepts of Space and Time

1.5.3. Spherical Quantities in Euclidean Space

**Conclusion**

**References**

**Chapter 2. The Theory of Quaternity**

2.1. Introduction to Four Space Dimensions

2.1.1. Four Geometric Elements in a Sphere

2.1.2. Orthogonality and Scalability of Four Space Dimensions

2.2. Quaternity Spacetime

2.2.1. Symmetry and Complementarity of Time and Space

2.2.2. Rotatory Operations in the Quaternity Equation

2.2.3. Electronic Orbitals in a Neon Shell

2.2.4. Dimension Encapsulation Mechanism

2.2.5. The Differential Chain Rule

2.3. Four-Dimensional Harmonic Oscillations

2.3.1. Harmonic Oscillation in One Dimension

2.3.2. Harmonic Oscillation in Two Dimensions

2.3.3. Harmonic Oscillation in Three Dimensions

2.3.4. Harmonic Oscillation in Four Dimensions

2.3.5. Four Kinds of Orbital Geometries

2.4. Compatibility with Classical and Quantum Mechanics

**Conclusion**

**References**

**Chapter 3. Dynamic Calculus in Vector Calculus**

3.1. Introduction

3.2. Vector Calculus in One-Dimensional Space

3.3. Vector Calculus in Quaternity Space

3.3.1. The Polor in the First Dimension

3.3.2. The Metor in the Second Dimension

3.3.3. The Vitor in the Third Dimension

3.3.4. The Scalor in the Fourth Dimension

3.4. Orbitals Synchronization

3.4.1. Greenâ€™s Theorem

3.4.2. Stokesâ€™ Theorem

3.4.3. Gaussâ€™s Theorem

3.5. Electronic Interaction through Virtual Photons

3.6. About Quaternity Space

**Conclusion**

**References**

**Chapter 4. Quaternity, Relativity, and Quantum Mechanics**

4.1. Introduction

4.2. Scope of Views

4.3. Quaternity on EPR Paradox

4.4. Special Relativity in Quaternity

4.5. Quaternity Solutions to SchrĂ¶dingerâ€™s Equations

4.5.1. Duality as a Solution to One-dimensional SchrĂ¶dingerâ€™s Equation

4.5.2. Quaternity as a Solution to Three-dimensional SchrĂ¶dingerâ€™s Equation

**Conclusion**

**References**

**Chapter 5. The Structure of Atomic Shells**

5.1. Introduction

5.2. Electronic Orbitals in an Argon Shell

5.2.1. Spherical Quantities for 3s3p Electrons

5.2.2. Complementary Orbitals due to a Clockwise Direction Shift

5.2.3. Dimension Factors as Operators

5.2.4. Electronic Ropes

5.3. Electronic Orbitals in a Krypton Shell

5.3.1. Coordination of Circular Complex Functions

5.3.2. Circular Complex Functions as Exponential Functions

5.3.3. Circular Complex Functions as Radian Angles

5.3.4. Spherical Quantities for 3d Electrons

5.3.5. Complementary Orbitals due to a Standpoint Shift

5.4. Electronic Orbitals in a Radon Shell and Beyond

5.4.1. Spherical Quantities for 4f Electrons

5.4.2. The Quaternity Periodic Table of Elements

5.4.3. Electronic Interactions and Waves in Atoms

5.4.4. The Atomic Shell Octet

**Conclusion**

**References**

**Intermission**

**Part II. The Natural Pattern**

**Chapter 6. Anisotropic 2p Orbitals and a Chiral Carbon**

6.1. Introduction to Anisotropic 2p Orbitals

6.1.1. Geometries of 2p Orbitals

6.1.2. Covalent Bonds in Methane, Ethene, and Ethyne

6.1.3. Nucleophilic Substitutions of Alkyl Halides

6.1.4. Covalent Bonds in Aromatic Rings

6.1.5. Electrophilic Substitutions of Mono-Substituted Benzenes

6.2. Molecular Chirality

6.2.1. The Density Gradient of 2s2p Orbitals

6.2.2. The Origin of Molecular Chirality

6.2.3. The Physical Principle of Optical Rotation

6.2.4. Chiral Disposition Induced by Electrophilic Groups

6.2.5. Selective Electron Transfer through Chiral Centers

**Conclusion**

**References**

**Chapter 7. DNA Circuits and Waves**

7.1. Introduction

7.2. DNA Electric Conductivity

7.2.1. DNA Electronic Components and Circuitry

7.2.2. Circuit Model Discussion and Prediction

7.3. DNA Wave Functions

7.3.1. Harmonic Oscillation between Base Pairs

7.3.2. Mathematical Formula of DNA.

**Conclusion**

**References**

**Chapter 8. The Principle of Cells and Life**

8.1. Introduction

8.2. The Helium Model for Rope Structure

8.3. The Carbon Model for Gene Expression

8.3.1. DNA Replication (1s to 1s)

8.3.2. DNA Transcription (2s to 2px)

8.3.3. RNA Translation (2px to 2py)

8.3.4. Enzymatic Catalyses (2py to 2pz)

8.3.5. Metabolism and ATP Cycle (2pz to 2s)

8.4. The Zinc Model for Endosymbiotic Organelles

8.5. The DNA Circuitry for the Human Lineage

8.6. Wave Functions as Biological Variables

8.7. The Rotatory Operation for the Mind and Body Relationship

8.8. The Poisson Distribution and Evolution

8.9. The Law of Nature and General Gyroscope

8.10. At the Dawn of Calculus Biology

**Conclusion**

**References**

**Chapter 9. Circular Motion and Central Force**

9.1. Introduction

9.2. Keplerâ€™s Laws in Atomic Spacetime

9.3. The Energy Density and Coulombâ€™s Law.

**Conclusion**

**References**

**Chapter 10. The Order of Physical Quantities**

10.1. Introduction

10.2. Dimensions of Physical Quantities

10.3. Interpretation of Electromagnetic Quantities

10.4. The Unification of Physical Quantities

10.5. Dimensions of Thermodynamic Quantities

**Conclusion**

**References**

**Chapter 11. The Rational Core of Chinese Medicine**

11.1. Introduction to Yin and Yang

11.2. The Common Origin of â€śYi Jingâ€ť and TCM

11.3. Literal Meanings of Four Spherical Quadrants

11.4. Geometrical Interpretation of Five Elements

11.5. Quaternity Interpretation of Eight Trigrams

**Conclusion**

**References**

**Appendix I. Review Questions**

**Appendix II. Glossary of Key Concepts**

**Appendix III. The Law of Nature in Mathematics**

**Appendix IV. Special Meanings of Symbols**

**Index**

#### Author’s ORCID iD

**Kunming Xu** – 0000-0002-5418-6133