Mathematics Is Beautiful : Suggestions for People Between 9 and 99 Years to Look at and Explore

In 17 chapters, this book attempts to deal with well-known and less well-known topics in mathematics. This is done in a vivid way and therefore the book contains a wealth of colour illustrations. It deals with stars and polygons, rectangles and circles, straight and curved lines, natural numbers, square numbers and much more. If you look at the illustrations, you will discover plenty of exciting and beautiful things in mathematics. The book offers a variety of suggestions to think about what is depicted and to experiment in order to make and check your own assumptions. For many topics, no (or only few) prerequisites from school lessons are needed. It is an important concern of the book that young people find their way to mathematics and that readers whose school days are some time ago discover new things. The numerous references to internet sites and further literature help in this respect. "Solutions" to the suggestions interspersed in the individual sections can be downloaded from the Springer website. The book was thus written for everyone who enjoys mathematics or who would like to understand why the book bears this title. It is also aimed at teachers who want to give their students additional or new motivation to learn. This book is a translation of the original German 2nd edition Mathematik ist schön by Heinz Klaus Strick, published by Springer-Verlag GmbH, DE, part of Springer Nature in 2019. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). In the subsequent editing, the author, with the friendly support of John O'Connor, St Andrews University, Scotland, tried to make it closer to a conventional translation. Still, the book may read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. Preface Contents 1 Regular Polygons and Stars 1.1 Properties of Regular Stars 1.2 Drawing Stars 1.3 Diagonals in a Regular n-Sided Figure 1.4 Vertex Angle in a Regular n-Pointed Star 1.5 Compounded n-Pointed Stars 1.6 Regular n-Sided Figures in the Complex Plane 1.7 Setting up Game Schedules Using Regular n-Sided Figures 1.8 References to Further Literature 2 Patterns of Colored Stones 2.1 Sum of the First n Natural Numbers 2.2 The Sum of the First n Odd Natural Numbers 2.3 Quotients of Sums of Odd Natural Numbers 2.4 Representation of a Natural Number as the Sum of Consecutive Natural Numbers 2.5 Sum of the First n Square Numbers of Natural Numbers 2.6 Sum of the First n Cubes of the Natural Numbers 2.6.1 Proof of the Formula for the Sum of the First n Cube Numbers by Al-Karaji 2.6.2 Proof of the Formula for Cube Numbers by Wheatstone 2.7 Pythagorean Triples 2.7.1 Simple Types of Pythagorean Triples 2.7.2 Further Pythagorean Triples 2.7.3 General Method for the Determination of all Pythagorean Triples 2.7.4 Formula for Generating all Pythagorean Triples 2.8 References to Further Literature 3 Dissection of Rectangles into Largest Possible Squares 3.1 A Game with a Rectangle 3.2 Mathematical Analysis of the Game—Description Using Continued Fractions 3.3 Relationship Between the Continued Fraction Expansion and Rectangles 3.4 Dissection of Special Rectangles—Fibonacci Rectangles 3.5 The Sequence of Fibonacci Numbers 3.6 Relationship with the Euclidean Algorithm 3.7 Examples of Infinite Sequences of Rectangle Dissections 3.8 Determination of Continued Fractions of Square Roots 3.9 References to Further Literature 4 Circles and Circular Rings 4.1 The Number π—The Circumference and Area of a Circle 4.2 Circular Rings (Annuli) 4.3 Shifted Semicircles 4.4 Braided Bands 4.5 Tracks 4.6 References to Further Literature 5 Pentominoes and Similar Puzzles 5.1 Simple Polyominoes 5.2 Pentominoes 5.2.1 Tessellation of Rectangles with Pentominoes 5.2.2 Tessellation of Enlarged Pentomino Figures by Pentominoes 5.2.3 Tessellation of Triangular Figures Using Pentominoes 5.3 Hexominoes 5.4 References to Further Literature 6 Curve Stitching 6.1 Circle as Basic Figure—Sides and Diagonals in Regular Polygons 6.2 Square as Basic Figure 6.2.1 Special Star Figures in a Square 6.2.2 Parabolas in a Square 6.3 Digression: Envelope of a Family of Curves 6.3.1 Examples of Families of Straight Lines 6.3.2 Determining the Equation of the Enveloping Parabola 6.4 Curves of Pursuit 6.5 Circle as Basic Figure: Epicycloid 6.6 Perpendicular Axes as Basic Figure: Astroids 6.7 References to Further Literature 7 Calculating with Square Numbers—Number Cycles 7.1 Calculating with Square Numbers 7.1.1 Calculating with Square Numbers: From One Square Number to the Next 7.1.2 Calculating with Square Numbers: A Special Rule for Square Numbers with the Final Digit 5 7.1.3 Calculating with Square Numbers: Using Equidistant Numbers 7.1.4 Calculating with Square Numbers: Checking the Final Digits 7.1.5 Calculating with Square Numbers: Comparison of Methods 7.2 Number Cycles 7.2.1 Number Cycles Ending After One or Two Steps 7.2.2 Periodic Cycles 7.3 Number Cycles Modulo n 7.4 Number Cycles for Higher Powers 7.4.1 Analyzing the Last Two Final Digits of Cubic Numbers 7.4.2 Investigations of the Last Three Final Digits of a Cubic Number 7.5 References to Further Literature 8 Partitions of Regular Polygons 8.1 Continued Bisection 8.2 Continued Trisection 8.3 Continued Quadrisection 8.4 Continued Dissection into Five Equal Parts 8.5 Continued Dissections into n Subareas of Equal Size 8.6 Geometric Sequences and Series 8.7 Dissection of Regular Polygons into Subareas of Equal Size 8.8 References to Further Literature 9 Weighing in the Ternary Numeral System 9.1 Solving the Simple Cases of the Weighing Problem 9.2 Solution of the Other Cases of the Weighing Problem 9.3 Representation of Natural Numbers in the Ternary Numeral System 9.4 Relationship Between the Two Representations 9.5 References to Further Literature 10 Tessellation of Regular 2n-Sided Figures with Rhombi 10.1 Tessellation of a Regular 10-Sided Figure 10.2 Applying the Method of Tessellation to Other Regular 2n-Sided Figures 10.3 Generalizations of the Tessellation Properties 10.4 Instructions for Making the Diamond Puzzles 10.5 Alternative Tessellation Designs of the Regular 10-Sided Figure with Rhombi 10.6 Symmetrical Tessellation of Regular 2n-Sided Figures 10.7 Symmetrical Tessellation of the Regular 2n-Sided Figure From Outside to Inside 10.8 Rhombus Tessellation for Regular 5-Sided Figures, 7-Sided Figures, 9-Sided Figures, etc. 10.9 References to further literature 11 Geometric Figures on Grid Paper 11.1 Rectangles with a Given Area 11.2 Rectangles of Equal Perimeter 11.3 Special Rectangles: The 4 × 4-Rectangle and the 3 × 6-Rectangle 11.4 Variations of Rectangular Figures 11.5 Investigations on Pick’s Theorem 11.6 A Rule for Rectangular Polygons 11.7 Checking Pick’s Theorem for Triangles 11.8 Considerations on a General Proof of Pick’s Theorem 11.9 References to Further Literature 12 Sum of Spots 12.1 Sum of Spots When Rolling two Regular Hexahedrons 12.2 Sums of dice When Rolling Several Regular Hexahedrons 12.3 An Erroneous Notion of Sums of Spots 12.4 A Fair Game of Dice 12.5 The Sicherman Dice 12.6 Other Devices with Random Output for Double Throwing 12.7 Algebraic Background for the Different Display Options 12.8 Probability Distribution of Sums of Spots for Rolling n Dice 12.9 Probability Distributions of the Platonic solids 12.10 Comparison of Probability Distributions with Equal Sums of Spots 12.11 An Example of the Central Limit Theorem 12.12 Determining Sums of Dice Using Markov Chains 12.13 References to Further Literature 13 The Missing Square 13.1 Apparently Congruent Figures 13.2 The Paradox of the Missing Square and the Right Angle Altitude Theorem of Euclid 13.3 The Missing Square and Other Methods of Euclid 13.3.1 Application of Euclid’s Theorem 13.3.2 Application of Areas 13.4 Other Properties in Connection with Fibonacci Numbers 13.5 Arrangement by Sam Loyd 13.6 Other Appropriate Triples of Numbers 13.7 The Missing Square and the Pythagorean Theorem 13.8 References to Further Literature 14 Dissection of Rectangles into Squares of Different Sizes 14.1 Rectangles which can be Dissected into Nine or Ten Squares of Different Sizes 14.2 Determining the Side Lengths for a given Tessellation 14.3 Introduction of the Bouwkamp notation to describe a tessellation 14.4 Squares, which can be Dissected into Squares of Different Sizes 14.5 Connection with Electrical Circuits 14.6 A Game with Rectangular Dissections 14.7 References to Further Literature 15 Kissing Circles 15.1 Examination of Touching Circles using Trigonometric Methods 15.2 Descartes’ Theorem 15.3 Examples with Integral Radii 15.4 Pappus chains 15.5 Touching circles with curvature 0 15.6 Sangaku 15.7 References to Further Literature 16 Sums of Powers of Consecutive Natural Numbers 16.1 Derivation of Sum Formulas using Arithmetic Sequences of Higher Order 16.2 Determination of Coefficients by Comparing Consecutive Elements in the Sum Sequence 16.3 Alhazenʼs Derivation of the Sum Formulas for Higher Powers 16.4 Thomas Harriot Discovers a Connection between Triangular and Tetrahedral Numbers 16.5 Fermat’s Discovery 16.6 Pascal’s Method for Determining Formulas for the Sum of Powers 16.7 Representation of the Sum Formulas using Bernoulli Numbers 16.8 Determination of Sum Formulas using Lagrange Polynomials 16.9 References to Further Literature 17 The Pythagorean Theorem 17.1 The Pythagorean Theorem and the Classical Proofs of Euclid 17.1.1 First Proof by Euclid 17.1.2 Euclid’s second proof 17.2 “Beautiful” Proofs of the Pythagorean Theorem 17.3 Proofs of the Pythagorean Theorem by Dissection 17.3.1 Perigal’s Proof by Dissection 17.3.2 Göpel’s Proof by Dissection 17.3.3 Gutheil’s Proof by Dissection 17.3.4 Epstein’s and Nielsen’s Proof by Dissection 17.3.5 Dobriner’s and Thieme’s Proof by Dissection 17.4 Presentation of Proofs by Means of Tile Patterns 17.5 Some Proofs of Historical Significance 17.6 Infinite Pythagorean Sequences 17.7 Generalization of the Pythagorean Theorem 17.8 The Lune of Hippocrates of Chios and Other Circle Figures 17.9 Application of the Pythagorean Theorem to Quadrilaterals 17.10 Integral Pythagorean partners and special Pythagorean sequences 17.11 Heronian Triangles 17.12 Stamps of Pythagoras and the Pythagorean Theorem 17.13 References to further literature General References to Appropriate Literature Index Strick demonstrates the exciting and aesthetically pleasing aspects of mathematics to help entice young people to become interested in the subject and to help those whose school days are some time ago, to remember, and to discover new things. There are numerous references to internet sites and further reading to help further the mathematical exploration. "Solutions" to the mathematical games and brain teasers are interspersed throughout the text. --Adapted from publisher description