Mathematics is beautiful : suggestions for people between 9 and 99 years to look at and explore / Heinz Klaus Strick.

Format:

Book

Author/Creator:

Strick, Heinz Klaus, author.

Language:

English

Subjects (All):

Mathematics.

Physical Description:

1 online resource (370 pages)

Place of Publication:

Berlin, Germany : Springer, [2021]

Summary:

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.

Contents:

Intro

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.

Notes:

Includes index.

Description based on print version record.

ISBN:

3-662-62689-6

OCLC:

1258386122