Non-Euclidean Geometry in Materials of Living and Non-Living Matter in the Space of the Highest Dimension

Contents
Preface
Chapter 1
Euclidean Geometry and Non-Euclidean Geometry
Abstract
Introduction
1.1. Euclidean Geometry
1.2. Non-Euclidean Hyperbolic Geometry
1.3. Non-Euclidean Eleptical Geometry
1.4. Non-Euclidean Geometry of the Higher Dimension Polytopes
Conclusion
Chapter 2
Geometry of the Polytopes of Higher Dimensions
Abstract
Introduction
2.1. Structure of the Polytope Higher Dimension
2.1.1. The Structure of a N-Cube
2.1.2. The Structure of a N-Simplex
2.1.3. The Structure of a N-Cross-Polytope
2.1.4. Reconstruction of the Geometry of Polytopes of Higher Dimension
2.2. The Structure of the Cube with Center
2.3. The Structure of the Octahedron with Center
2.4. The Structure of the Tetrahedron with Center
2.5. Polytopic Prismahedrons
2.6. Structure of Polytopic Prisms
2.7. The Incidence Values in the Polytopic Prismahedrons
2.8. Poly-Incident and Dual Polytopes
2.8.1. Polytope Dual to the Product of Two Triangles
Conclusion
Chapter 3
Non-Euclidean Properties of the Geometry of Polytopes of Higher Dimension
Abstract
Introduction
3.1. Axioms of Multidimensional Euclidean Geometry
3.1.1. Connection Axioms
3.2. About the Impossibility of the Axiom Systems of the N-Dimensional Geometry of Euclides for Higher Dimensional Popytopes
3.3. Inappropriatness of the Basic Principles of Classical Mechanics for the Analysis of Motion in a Space of Higher Dimensions
3.3.1. Movement of a Material Point in Four – Dimensional Spaces
3.3.1.1. Movement of a Material Point in a 4-Cube
3.3.1.2. Movement of a Material Point in a 4-Simplex
3.3.1.3. Movement of a Material Point in a 4-Cross-Polytope
3.3.1.4. Movement of a Material Point in n – Dimensional Spaces (n > 4)
3.4. On the Possibile Electronic Structure of Atoms in a Space of Higher Dimension
3.4.1. The Stationary Schrödinger Equation in a P – Dimensional Metric Space
3.4.2. The Decision of the Stationary Schrödinger Equation in a P – Dimension Metric Space
3.4.3. Quantum Numbers of Solutions of the Schrödinger Equation in a Space of Higher Dimension
Conclusion
Chapter 4
Polytopes of the Highest Dimension of Inert Substances
Abstract
Introduction
4.1. The Dimension of Adamantane Molecules and Methods of Molecules Connecting with Each Other
4.1.1. The Dimension of the Adamantine Molecule
4.1.1.1. Theorem 4.1 (Zhizhin, 2014 а, b)
4.1.1.2. Proof
4.1.2. Connection Types of Adamantane Molecules
4.2. The Structure of Binary Natural Compounds
4.2.1. The Dimension of the Wurtzite
4.2.2. The Dimension of the Fluorite
4.3. The Structure of Natural Compounds with a Large Number Types of Atoms
4.3.1. Theorem 4.2
4.3.1.1. Proof
4.3.2. Theorem 4.3
4.3.2.1. Proof
4.3.3. Theorem 4.4
4.3.3.1. Proof
4.4. Pomegranate Texture
Conclusion
Chapter 5
“Inert” Substances as a Self-Regulating Medium Tending to Capture Space
Abstract
Introduction
5.1. Geometric Growth Models of Dissipative Systems
5.1.1. Theorem 5.1
5.1.1.1. Proof
5.1.2. Theorem 5.2
5.1.2.1. Proof
5.2. The Dimension of Clusters of Several Shells in the Form of Plato’s Bodies
5.2.1. Theorem 5.3
5.2.1.1. Proof
5.3. Filling the Space with Simplices of Increasing Dimension
5.4. Filling the Space with Cross-Polytopes of Increasing Dimension
5.4.1. Theorem 5.4
5.4.1.1. Proof
5.5. Clusters on an Octahedron
Conclusion
Chapter 6
Spatial Models of Sugars and Their Compounds
Abstract
Introduction
6.1. Spatial Structure of Stereoisomers of Glyceraldehyde and Dihydroxyacetone
6.2. The Dimension of Linear Molecules of Monosaccharides with a Carbon Length from 4 to 7
6.3. Functional Dimension of Monosaccharides with a Closed Carbon Chain with Trhee Chiral Carbon Atoms
6.4. Functional Dimension of Monosaccharides with a Closed Carbon Chain with Four Chiral Carbon Atoms
6.4.1. Theorem 6.1
6.4.1.1. Proof
6.5. 3D Simplified Image of Pyranose Monosaccharide Molecules
6.6. Monosaccharide Chains
Conclusion
Chapter 7
The Theory of the Folder and Native Structures of the Proteins
Abstract
Introduction
7.1. Dimensions of Protein Molecules
7.2. Linear Polypeptide Chain Structure
7.3. Turns of Polypeptie Chains
7.4. Spiral Polypeptide Chains
7.5. Folder Structures of the Amino Acids
7.6. Native Structure of Globular Proteins with Parallel Arrangement of Amino Acid Residues
7.7. Native Structure of Globular Proteins with Antiparallel Arangement of Aminoacide Residues
7.8. Native Structure of Globular Proteins with Parallel and Antiparallel Arragement of Amino Acid Residie, and α-Spirals
7.9. Globular Proteins as Molecular Machines
7.9.1. Theorem 7.1
7.9.1.1. Proof
7.9.2. Theorem 7.3
7.9.2.1. Proof
Conclusion
Chapter 8
Geometry of the Structure of Nucleic Acids in the Space of the Highest Dimension
Abstract
Introduction
8.1. The Dimension of Phosphoric Acid and Its Residue
8.2. The Dimension of the Molecules Deribose and Deoxyribose
8.3. The Structure of the α-D-Ribose and 2-Deoxy α-D-Ribose Nucleic Acids
8.4. The Three-Dimensional Model of the Nucleic Acid Molecule
Conclusion
Chapter 9
Interaction of Nucleic Acids in the Space of Higher Dimension and the Transmission of Hereditary Information
Abstract
Introduction
9.1. Polytopes with Antiparallel Edges
9.2. The Polytope of Hereditary Information
9.3. Hidden Nucliec Acid Bond Order
9.4. The Law of Conservation of Incidents in Polytope of Hereditary Information
9.4.1. Theorem 9.1
Proof
9.5. Methylated Polytope of Hereditary Information
9.6. Nucliec Acids Methylation
9.7. The Law of Conservation of Incidents in the Methylated Polytope of Hereditary Information
Conclusion
Chapter 10
Dimension of Substances and Life
Abstract
Introduction
10.1. The Structure of Water
10.2. The Dimensional of Biomolecules
10.3. The Memory of Water
10.4. Chelated Compounds
10.5. Dimension and Genes
Conclusion
Summary
References
About the Author
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