Complex Symmetries

This volume is a collection of essays on complex symmetries. It is curated, emphasizing the analysis of the symmetries, not the various phenomena that display those symmetries themselves. With this, the volume provides insight to nonspecialist readers into how individual simple symmetries constitute complex symmetry. The authors and the topics cover many different disciplines in various sciences and arts. Simple symmetries, such as reflection, rotation, translation, similitude, and a few other simple manifestations of the phenomenon, are all around, and we are aware of them in our everyday lives. However, there are myriads of complex symmetries (composed of a bulk of simple symmetries) as well. For example, the well-known helix represents the combination of translational and rotational symmetry. Nature produces a great variety of such complex symmetries. So do the arts. The contributions in this volume analyse selected examples (not limited to geometric symmetries). These include physical symmetries, functional (meaning not morphological) symmetries, such as symmetries in the construction of the genetic code, symmetries in human perception (e.g., in geometry education as well as in constructing physical theories), symmetries in fractal structures and structural morphology, including quasicrystal and fullerene structures in stable bindings and their applications in crystallography and architectural design, as well as color symmetries in the arts. The volume is rounded of with beautiful illustrations and presents a fascinating panorama of this interdisciplinary topic. COMPLEX SYMMETRIES CONTENTS COMPLEX SYMMETRIES Complex Symmetries in Repeating Hyperbolic Patterns 1 INTRODUCTION 2 1. THE SYMMETRIES OF REPEATING HYPERBOLIC PATTERNS 3 2. INVARIANTS OF SYMMETRIES OF REPEATING HYPERBOLIC PATTERNS 4 3. COMPLEX SYMMETRIES BUILT FROM SIMPLE SYMMETRIES 5 RESULTS REFERENCES IgnoTheory: A Compositional System for Intermedia Art Based on Tiling Patterns and Labelled Graphs 1 TILING PATTERNS 1.1 The Ignotiles: Rule-Based Tiling Patterns 1.2 Algorithmic Rules for Generating Arrays of Ignotiles 1.2.1 Twelve-point Solutions and Ignoquads 1.2.2 Permutations of an Ignoquad 1.2.3 Latin Squares 2 GRAPHS 2.1 Musical Composition 2.1.1 Mapping the Ignotiles to the Western Diatonic System 2.1.2 Symmetry under Rotation and Reflection of Tiles 2.1.3 Parametric Graphs Derived from the Ignotiles 2.2 Intermedia Art 2.2.1 IgnoTheory: Making Intermedia Art with Parametric Graphs 3 SYMMETRIES 3.1 Types of Symmetry in the IgnoTheory Generative System 3.2 Symmetries between Maps and Graphs 3.3 Mapping Arbitrary Values from Graphs to Maps REFERENCES Symmetries of Maps on Surfaces 1 DEFINITIONS AND PRELIMINARIES 2 THE SYMMETRY CONSIDERATIONS 3 COMPLEX SYMMETRIES 4 SOME RESULTS Acknowledgements REFERENCES Symmetry in Projection of 4-Dimensional Regular Polychora 1 INTRODUCTION 2 SYMMETRY 3 PROJECTION 4 REGULAR POLYCHORA 5 ORTHOGONAL 3-PROJECTION AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA 6 ORTHOGONAL 2-PROJECTION AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA 7 PLANAR TESSELLATIONS BY ORTHOGONAL 2-PROJECTION OF REGULAR POLYCHORA 8 ONE-POINT PERSPECTIVE AND ROTATIONAL SYMMETRY OF REGULAR POLYCHORA 9 ORTHOGONAL 3-PROJECTION AND REFLECTIONAL SYMMETRY OF SEMIREGULAR POLYCHORA 10 SPHERICAL STRUCTURES ON THE MODEL OF SEMIREGULAR POLYCHORA 11 CONCLUSIONS BIBLIOGRAPHY Morphic Polytopes and Symmetries 1 POLYHEDRA AND TOPOLOGY 1.1 Geometric polytopes 1.2 Abstract polytopes and realization 1.3 Morphic polytopes 1.4 Non-simple cells 1.5 Overlays and twists 2 ABSTRACT, MORPHIC AND CONCRETE SYMMETRIES 2.1 Isomorphisms as symmetries 2.2 Internal symmetries and realization 3 COMPLEX SYMMETRIES IN MORPHIC POLYTOPES AND COMPOUNDS 3.1 Automorphisms and kaleidoscopes 3.2 Regular morphic polytopes 3.3 Compounds as polyfolds REFERENCES Inducing the Symmetries Out of the Complexity: The Kepler Triangle and Its Kin as a Model Problem 1 INTRODUCTION 2 GENERALISING THE KEPLER TRIANGLE 2.1 Collecting the Data: State Description 2.2 Finding the Invariants: Aspect Description 2.3 Inducing the Laws: Process Description 3 EXPLORING THE NEW LINE OF A STUDY 4 CONCLUSION REFERENCES APPENDIX I APPENDIX II Synchronizing the Isotropic Vector Matrix with the Stellated Vector Matrix INTRODUCTION 1.1 Phenomenon 1.2 Simple Symmetries 2 STELLATED VECTOR MATRIX (SVM) 3 RESULTS 3.1 IVM Features (Fuller) 3-1-1 Tetrahedron edges 3.1.1 Sphere center 3.2 SVM Features (Lehman) 3-2-1 5-Fold Matrix 3.2.1 Offset Matrix Fig. (47 (L277-1)) 3.2.2 Dodecahedron inside spheres (Fig. 49 (L-276)) 3.3 IVM/SVM Comparisons 3-3-1 Two Vector Matrices 3.3.1 Vector Formation 3.3.2 IVM Tetrahedron Centered in 2 Frequency SVM Tetrahedrons 3.3.3 Two Elements of the Vector 4 DISCUSSION 5 CONCLUSION TAXONOMY REFERENCES An Analogy and Several Symmetries 1 PRELIMINARY – ELECTRIC NETWORKS 2 PRELIMINARY – PLANAR FRAMEWORKS 3 AN ANALOGY BETWEEN ELECTRIC NETWORKS AND PLANAR FRAMEWORKS 4 SOME REFLECTION-TYPE SYMMETRIES 5 SOME ROTATION-TYPE SYMMETRIES REFERENCES Discrete Lattices on the Single Bearing Spiral: From Geometry to Botany 1 INTRODUCTION 2 DEFINITIONS 3 THE CLASSIFICATION TABLE 4 LATTICE DEGENERATION AND THE TYPES OF BEARING SPIRALS 5 FIBONACCI-TYPE SPIRAL LATTICES AND CLASSIC PHYLLOTAXIS 6 GROWTH RINGS 7 CONCLUSION REFERENCES Symmetry and Invariance: Interdisciplinary Teaching 1 INTRODUCTION 2 HISTORICAL AND EPISTEMOLOGICAL INTERLUDE 2.1 Towards group theories 2.2 Mathematics and Physics Interplay 3 COMPLEX SYMMETRY IN EDUCATIONAL CONTEXT 3.1 Theoretical Background 3.2 Abstraction in Education 3.3 Measuring symmetries 4 COMPLEX SYMMETRIES IN MANIPULATION GAMES REFERENCES Dilative Rotation, Dilative Reflection in Mathematics, Nature, Art, and Education 1 INTRODUCTION, CLASSIFICATION OF ISOMETRIES AND SIMILARITIES 1.1 Isometries 1.2 Similitude 1.3 Dilative rotation and dilative reflection 1.3.1 Dilative rotation in the plane 1.3.2 Dilative reflection in the plane 1.3.3 Dilative rotation in space 1.4 Spirals and dilative rotation 1.4.1 The Archimedean spiral 1.4.2 Logarithmic spiral 1.4.3 Helix 2 SPIRALS AND DILATIVE ROTATIONS IN NATURE AND THE ARTS 3 COMPOSITE TRANSFORMATIONS IN EDUCATION REFERENCES Relationship of Symmetry and Combinatorics in the Poly-Universe Game 1 OPERATION OF THE POLY-UNIVERSE GAME 1.1 Structure and symmetry of the basic elements 1.2 The colour combinations 2 CONNECTIONS OF RULES AND SYMMETRY 2.1 Basic position, inverse (mirrored) position 2.2 Entire side joins of triangles 2.3 Entire side joins of circles 2.3.4 Partial circle joins 2.4 Entire side joins of squares 3 CLOSED SYMMETRIC CONFIGURATIONS 3.1 Closed configuration from 6 elements by the triangle 3.2 Closed configuration from 4 elements by the square 3.3 Closed ring configuration from 6 elements by the circle 4 SYMMETRY OF THE RING OF POLY-UNIVERSE CIRCLE 4.1 In case of the same basic colour and of the same size and colour connection 4.2 In case of the different base colour and of the same size connection 5 SUMMARY REFERENCES Symmetry in Arrow-Like Salt Crystals INTRODUCTION 1 SYMMETRIES IN DENDRITE- AND HOPPER CRYSTALS 2 TRANSLATIONAL SYMMETRY IN ARROW-LIKE CHAIN CRYSTALS 3 SELF-SIMILARITY AND COMPLEX SYMMETRIES IN THE ARROW-LIKE CHAINS OF CRYSTALS 4 DISCUSSION ACKNOWLEDGEMENTS REFERENCES Symmetries in Stellar, Galactic, and Extragalactic Astronomy INTRODUCTION 1 SYMMETRIES IN ASTRONOMY – HISTORICAL CONTEXT 2 SYMMETRIC CELESTIAL BODIES AND PHENOMENA 2.1 Spherical symmetry in modern astronomy 2.2 Celestial objects/phenomena with a lesser degree of symmetry 3 SCIENTIFIC USE OF THE OBSERVED SYMMETRIES ACKNOWLEDGEMENTS REFERENCES Symmetries and the Genetic Code 1 INTRODUCTION 2 THE SYMMETRIC SYSTEM OF BINARY-OPPOSITIONAL TRAITS IN GENETIC ALPHABETS 3 GENETIC MATRICES, DNA COMPLEMENTARY PAIRS, AND THE GOLDEN SECTION 4 SOME CONCLUDING REMARKS REFERENCES Global Symmetry Local Asymmetry in the Realized Buildings by the Innovation Studio ONL BV BUILDING BODY BODY PLAN AXES OF SYMMETRY TYPES OF SYMMETRY VECTORIAL BODIES LEMNISCATE LOCAL FOLD LINES TWISTED DODECAHEDRON GENERIC SYMMETRICAL COMPONENTS INFORMED BY LOCAL PARAMETERS CONCLUSION REFERENCES Theme, Motive, Structure and Symmetry – Pentasonata by Andrzej Panufnik 1 INTRODUCTION TO COMPLEX SYMMETRY 2 WHY IS IT CONSIDERED SYMMETRY? 3 HOW DO SIMPLE SYMMETRIES BUILD THE COMPLEX SYMMETRY? 3.1 Thematic material and methods of its development 3.2 Pentasonata as a “symmetrical” composition 3.3 Thematic material – observation of complex symmetry in the structure of the composition: from spatial to narrative symmetry 3.3.1 Spatial symmetry 3.3.2 Narrative symmetry 3.3.3 The axis of symmetry, the centre of the work 4 THE THEMATIC INTEGRATION OF THE FORM – TESTIMONY TO THE SIMPLE AND COMPREHENSIVE SYMMETRY REFERENCES Fluid Symmetry: Logical to Artists, Mesmerizing to Viewers 1 INTRODUCTION 2 THREE POSITIONS IN ART 3 FLUID SYMMETRY 3.1 Biological basics 3.2 Artistic execution leading the viewer’s experience 3.2.1 Decision process 3.2.2 Tintoretto’s brazen serpent 3.2.3 Biederman’s coloured relief 3.2.4 Fré Ilgen’s Crawlin’ King Snake Crawlin’ King Snake 4 CONCLUSIONS REFERENCES Complex Symmetries in Réograms 1 INTRODUCTION 2 THE ARTWORK FLORESCENCE 3 COMPOSITION OF THE SYMMETRIES OF THE DISCUSSED ARTWORK REFERENCES