Walser H. The Golden Ratio. Geometric and Number Theoretical Considerations 2024

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Walser H. The Golden Ratio. Geometric and Number Theoretical Considerations 2024

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Textbook in PDF format This book illustrates key mathematical aspects of the Golden Ratio: It particularly delves into geometric and number theoretical connections and examples, and makes further considerations and generalizations accessible. The book is primarily aimed at students, pupils, mathematics teachers, and interested laypeople. It is modular in structure, so the individual chapters can be read independently of one another. The reading is intended to encourage one's own geometric activities. Tips and procedural hints from the craft-creative realm are also provided. Supplementary animations can be accessed with the SN More Media App: simply download the SN More Media App for free, scan a picture or link with the play button, and immediately play the animation on your smartphone or tablet. The Author Dr. Hans Walser was a high school teacher and a lecturer at ETH Zurich, the University of Basel, and the University of Teacher Education Basel in the field of teacher training and mathematics for students of natural and engineering sciences. The translation was done with the help of artificial intelligence. A subsequent human revision was done primarily in terms of content. Preface Contents What is it about? What is the “Golden Ratio”? Designations Continuous Division Golden Geometry Constructions of the Golden Ratio The Classic Construction Construction with Bisectors Circle and Squares Circle and Triangles Circle, Squares and Triangles A Construction with the Compass Alone A Construction in the Right-Angled Triangle The Regular Pentagon and the Regular Decagon Approximate constructions for the regular pentagon Grid Geometry The Golden Rectangle Unit Square and Golden Rectangle Constructions of the Golden Rectangle Divisions of the Golden Rectangle Spirals in the Golden Rectangle Existence of irrational numbers Generalizations of the Golden Rectangle The Tall Golden Rectangle Golden Polygons The Golden Parallelogram Golden Triangles Golden Ellipses Square Root of the Golden Ratio Geometric Sequence Golden Staircase Double Spiral Area Divisions Dividing a Square Extending Sides Golden Angles Examples Golden Pie Cosine and Tangent The Steepest Line Rectangle in the Semicircle A Trigonometric Relationship Golden Trigonometry Formulas and Examples Fourier and the Golden Ratio Rosettes Folding and Cutting Paper Strip Construction of the Regular Pentagon From Strip to Knot Pre-Folding A Self-Correcting Algorithm Another Pentagon Heptagon Knot with Two Strips The Golden Rectangle Origami DIN Format Regular Pentagon Exact Folding Construction Approximate Construction Star Figures The Pentagram The Figure of Odom Number Sequences Linearization of Powers of the Golden Ratio Fibonacci Sequences Golden Trapezoid and Golden Star Family Tree of a Drone Approximation of the Golden Rectangle by Fibonacci Squares In Pascal’s Triangle Arbitrary Starting Values Continued Fractions Chain Roots Regular and Related Solids The Regular Solids Constructions Based on the Cube and Octahedron In the Cube In the Octahedron Diagonals Diagonals in the Icosahedron Diagonals in a Dodecahedron Rhombic Solids The rhombic triacontahedron Rhombic dodecahedron and rhombic icosahedron A star solid Examples Number Games Circles and semicircles Intersections Area and Volume Constructions with Equilateral Triangles Optimization Golden Probabilities Literatur