Symmetry in Science and Art

1 Introduction • From Intuitive Concepts to the Definition of Symmetry.- Relative Equality • Equality as a Basis for Geometric Regularity and the Theory of Symmetry.- Geometric Regularity.- Symmetry as a Special Kind of Geometric Law.- Symmetry, Beauty of Form, and Harmony.- 2 Symmetry of One-Sided Rosettes.- Plane of Symmetry • Symmetry of Animals, Plants, Machines, and Other Objects.- Symmetry Axis • Principle of Rotation and the Symmetry of Processes Occurring in Time.- Enantiomorphism • Rightness and Leftness of Figures.- Symmetry Axis Combined with Symmetry Planes.- Formation of Symmetrical Rosettes • Cutting Rosettes from Paper • Role of Physical Factors.- Polar and Nonpolar Planes and Axes.- Singular Points, Lines, and Planes • Multiplicity of Points.- Exact Definition of a One-Sided Rosette.- 3 Symmetry of Figures with a Singular Point.- Mirror-Rotation Axis and Center of Symmetry • The Symmetry of Crystals • Parallel and Antiparallel Segments and Planes.- Symmetry Axis with a Perpendicular Plane m • Rotating Parts of Machines • Crystals • Symmetry of an Electric Voltaic Pile and a Cylindrical Magnet.- Principal Axis Combined with Longitudinal and Transverse Planes m • Snowflakes, Machine Parts, and Everyday Objects.- Principal Axis Combined with Two-Fold Transverse Axes • Twisted Shapes • Rotation of the Plane of Polarization.- Principal Axis Combined with Planes and Two-Fold Axes.- Regular Polyhedra.- The Two Symmetry Classes of the Sphere • Optically Rotating Liquids • Spherulites.- Review of the Symmetry Classes of Figures with a Singular Point • Spherical and Stereographic Projections of Symmetry Elements.- Two Types of Figures with a Singular Point • One-Sided and Two-Sided Rosettes.- Comparison of the Symmetry of Crystals and Organisms • Coordinate and Noncoordinate Notation for Symmetry Classes.- Fedorov Kaleidoscopes for Producing Figures with a Singular Point.- Systems of Equivalent Points • Molecules.- Symmetrical Pencils of Straight Lines and Polyhedra • Simple Forms.- Symmetry and the Structural Formulas of Molecules.- Symmetry of Directed Quantities • Vectors and Tensors.- Concluding Remarks.- 4 Symmetry of One-Sided Bands.- Translation Axis as a Necessary Symmetry Element of Bands • Border Decorations for Subway Passages and Intersections.- Glide-Reflection Plane.- Translation Axis with Transverse Two-Fold Axes • Border Decorations for Passages with Two-Way Traffic.- Other Symmetry Classes of One-Sided Bands.- Kaleidoscopes for Forming One-Sided Bands.- Review of the Seven Symmetry Classes of One-Sided Bands.- 5 Symmetry of Two-Sided Bands.- The Second-Order Screw Axis.- The 31 Symmetry Classes of Bands.- Cutting Bands from Paper.- 6 Symmetry of Rods.- Rational and Irrational Screw Symmetry Axes • Screws.- Basis for the Derivation of the Symmetry Classes of Rods.- Rods Generated by Figures with One Symmetry Axis.- Rods Generated by Figures with One Mirror-Rotation Axis.- Rods Generated by Figures with Symmetry n : m.- Rods Generated by Figures with Symmetry n m.- Rods Generated by Figures with Symmetry n : 2.- Rods Generated by Figures with Symmetry 2ñ m.- Rods Generated by Figures with Symmetry m n : m.- Review of Rod Symmetry Types with Finite and Infinite Translations.- Limiting Symmetry Classes for Rods • Shafts with Pulleys • Screws • One-Dimensional Continua and Discontinua.- Some Generalizations • Unified Principle of Symmetry Transformations in Three-Dimensional Space.- 7 Symmetry of Network Patterns • Two-Dimensional Continua and Semicontinua.- Plane Nets.- The 17 Symmetry Classes of Network Patterns (The Plane Space Groups) • Examples of Patterns in Folk Art.- Projections of Symmetry Elements for Network Patterns • Coordinate and Noncoordinate Notation for Symmetry Classes.- Network Patterns in Nature, Technology, and Art.- Superposition of Net work Patterns • Technical Applications • The Bragg Law • Beats.- Cutting Network Patterns from Paper.- Kaleidoscopes for Network Patterns.- Parallelogons and Planigons • Their Use in Parquets.- Regular Systems of Points • Law of Conservation of the Products of the Multiplicities of Points and Their Relative Numbers.- Plane Isogons and Isohedra • Parquets.- Symmetry Mixing • The Perception of Vertical Planes.- One-Sided Plane Continua.- One-Sided Plane Semicontinua.- 8 Symmetry of Layers.- Symmetry Elements of Layers.- Derivation of the Symmetry Classes of Layers • Representations and Notation.- The 80 Symmetry Classes of Layers.- Two-Sided Plane Continua and Semicontinua.- Systemization of Symmetry Groups.- 9 Symmetry of Three-Dimensional Spaces • Discontinua and Continua.- Kaleidoscopes for Three-Dimensional Periodic Discontinua of the Highest Symmetry.- Space Lattices and Groups of Parallel Translations.- The 230 Space Groups of a Discontinuum • Structure of Crystals.- Close Packing of Spheres • Its Significance for Crystallography and Building Technology.- Fedorov Parallelohedra and Stereohedra.- Law of Multiple Proportions in Structural Crystallography and Chemistry.- Spatial Semicontinua with Two Axes of Continuous Translations.- Spatial Semicontinua with One Axis of Continuous Translations.- Symmetry of Three-Dimensional Continua.- 10 Elements of Group Theory • The Classical Crystallographic Groups.- Definition of a Group • Groups of Transformations of Geometric and Physical Objects • Abstract Groups.- Example: The Crystallographic Group 2/m • Groups of Permutations and Orthogonal Matrices Isomorphic with the Group 2/m.- Some Properties of Groups • Subgroups • Factor Groups • Homomorphic Relationships Between Groups.- Extension of Groups by Means of Direct, Semidirect and Quasi-Products • Crystallographic Groups as Extensions of Rotation Groups.- Space (Fedorov) Groups ? as Extensions of the Translation Groups by Means of the Crystallographic Point Groups and Their Isomorphic Groups by Modulus.- 11 Groups of Generalized Symmetry • Antisymmetry and Colored Symmetry.- Crystallographic Antisymmetry Point Groups as Extensions of the Classical Crystallographic Groups by Means of the Groups 1‘, 2’, m’, 1’, 4(mod 2), 4’(mod 2).- Antisymmetry Space (Shubnikov) Groups III as Extensions of the Classical Space (Fedorov) Groups ? or as Extensions of the Translation Groups T.- Crystallographic Point Groups of Colored Symmetry as Extensions of the Classical Crystallographic Groups by Means of the Groups of Color Permutations P and G(p)*.- The Colored Symmetry Space (Belov) Groups ? as Extensions of the Classical Space (Fedorov) Groups ? or as Extensions of the Translation Groups T.- Limits to Symmetry Theory • Other Generalizations.- 12 Symmetry in Science and Art • Conservation Laws • Symmetrization and Dissymmetrization of Physical Systems • Principle of Symmetry for Composite Systems.- Symmetry and Structure • Symmetry as a Structural Law of Integral Systems and as a Method of Studying Structural Regularities.- Transformation Laws and Symmetry of Physical Quantities (in the Approximation of a Homogeneous Continuum) • Limiting Groups of Antisymmetry and Colored Symmetry.- Transformation Laws and Symmetry of Physical Quantities (in the Approximation of a Periodic Discontinuum) • Space Tensors in Colored Groups.- Composite Systems • Principle of the Superposition of Symmetry Groups • Laws Governing Changes and Conservation of Symmetry.- Relation Between the Symmetries and Properties of Systems • Symmetry of Physical Equations and Laws • Conservation Laws and Phase Transitions.- Symmetry and Dissymmetry in Art • Laws of Composition • Structure-System Methods of Analyzing Artistic Creations.- Conclusion • Heuristic Significance of the Principles of Symmetry • Symmetry as a Philosophical Concept.- Resumé.