Symmetry in Science and Art

Brossura

ISBN 10: 1468420690 ISBN 13: 9781468420692

Casa editrice: Springer Nature, 2012

The perception of symmetry in art and in nature has been appreciated since antiquity, with development of the underlying laws tracing back at least to Pythagorean times. By the end of the eighteenth century it was realized that the immense variety of natural crystal shapes could be accounted for on the basis of a rather small number of symmetry operations, of which some were equally applicable to biological systems. The mathematical theory of symmetry continued to mature throughout the last century, culminating in the independent discoveries in Russia, Germany, and England that a total of only 230 independent ways exist in which the operations of rotation, reflection, and translation can be combined to transform three-dimensional geometrical objects into themselves. Derivation of the 230 space groups depends ultimately on restricting the meaning of symmetry to that of a property of purely geometrical figures. A. V. Shubnikov and his collaborators, over the past three decades, expanded this concept of symmetry to include the sign of transformation operations.

Contenuti:

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.