In this post, we will see the book Unsolved And Unsolvable Problems In Geometry

by H. Meschkowski.

# About the book

We shall discuss some famous problems which occupied the attention of mathematicians for thousands of years and which have been shown to be unsolvable in recent decades and, in addition, some very old classical problems which, up till now, have scarcely been mentioned in books on the subject. We have endeavoured to vary well-known methods of proof for classical problems and to generalise their formulation.

Thus, the object of this book is not the systematic representation of a well-defined section of geometry. But rather our aim is of a methodological nature: we want to develop the kind of arguments which are suitable for the solution of the geometrical problems considered and to discuss the questions implied by the existence of unsolvable problems.

In the experience of mathematical institutes and authors of mathematical publications attempts are still made to solve problems which are known to be unsolvable. And so we fear that there will be some readers who, after reading this book, will believe that, despite everything, they have found a solution to the problem of trisecting an angle. We should like to advise these readers as follows: Read through the proof showing that such a solution is impossible three times and then try to find the mistake in your argument. Better still, give up such attempts since they really cannot lead anywhere. There are plenty of open questions—many of which are stated in this book—to which mathematical intuition may be applied with real prospect of success. However, it should be noted that it is not particularly easy to solve the open questions stated. It is sometimes much simpler to discover new results in a new branch of mathematics rather than to solve one of the problems left open in elementary geometry. However, according to Erhard Schmidt, it is better to solve old problems with new methods than to solve new problems with old. Therefore we feel justified in encouraging work on the “ elementary ” problems stated in this book.

For the reader’s convenience there is, at the end of the Table of Contents, a diagram showing the interrelationships of the chapters.

The book was translated from Russian by Jane A. C. Burlak and was published in 1966.

Credits to original uploader.

You can get the book here.

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# Contents

PREFACE v

## Chapter I. THE PROBLEMS 1

## Chapter II. REGULAR PACKINGS IN THE PLANE AND ON THE SPHERE 5

1. Problems on the density of packing 5

2. Lemmas 8

3. Packing problems in the plane 13

4. Packing problems on the sphere 18

5. Further problems 25

## Chapter III. IRREGULAR PACKINGS AND COVERINGS 29

1. The existence theorem 29

2. The packing problem graph 30

3. A point system for n = 7 32

4. The packing problem for x = 7 34

5. The covering problem graph 38

6. The covering problem for n = 5 40

7. The covering problem for 1 = 7 45

8. An Outline of further problems 47

## IV. PACKING OF CONGRUENT SPHERES 50

1. The densest lattice packing 50

2. The finest rigid packing 54

## V. LEBESGUE’S “TILE” PROBLEM 57

1. Formulation of the problem 57

2. The hexagonal tile 58

3. The tiles of Pal and Sprague 60

4. Curves and domains of constant width 62

5. Applications 65

6. The covering problem in ℛ3 67

## VI. ON THE EQUIVALENCE OF POLYHEDRA BY DISSECTION 71

1. The plane problem 71

2. Dehn’s theorem 73

3. Examples on Dehn’s theorem 75

4. The algebra of polyhedra 82

5. The basis polyhedra 84

## VII. THE DECOMPOSITION OF RECTANGLES INTO INCONGRUENT SQUARES 91

1. The problem 91

2. The definition of a graph 95

3. Some calculations for decompositions 97

4. The interpretation in electrical engineering 100

5. The decomposition of rectangles with commensurable sides 101

## VIII. UNSOLVABLE EXTREMAL PROBLEMS

1. A theorem on the triangle 103

2. Besicovitch’s theorem 103

3. Sets of points at an integral distance apart 107

## IX. CONSTRUCTIONS WITH RULER AND COMPASSES 113

1. Some comments on the use of ruler and compasses 113

2. Some construction problems 117

3. The proof of the impossibility of (8a) and (8b) 119

4. The quinquesection of the angle 123

5. The uniqueness of the construction 126

## X. CONSTRUCTIONS ON THE SPHERE 128

1. Stereographic projection 128

2. Constructions with spherical compasses and spherical ruler 130

3. Squaring the circle on a sphere 133

4. A lemma 135

## XI. PROBLEMS IN THE GEOMETRY OF SETS 137

1. A new definition of equivalence by dissection 137

2. Hausdorff’s theorem 141

3. Paradoxical decompositions in #3 150

4. Paradoxical decompositions in the plane 155

## XII. THE METHODOLOGICAL SIGNIFICANCE OF “UNSOLVABLE” PROBLEMS 159

BIBLIOGRAPHY 163

INDEX 167