Hyperbolic Functions

with Configuration Theorems and Equivalent and Equidecomposable Figures

V. G. Shervatov, B. I. Argunov, L. A. Skornyakov, V. G. Boltyanskii

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486170053
• Topic: Mathematics
• Subtopic: Trigonometry
• Index: Mathematics

This single-volume compilation of three books centers on Hyperbolic Functions, an introduction to the relationship between the hyperbolic sine, cosine, and tangent, and the geometric properties of the hyperbola. The development of the hyperbolic functions, in addition to those of the trigonometric (circular) functions, appears in parallel columns for comparison. A concluding chapter introduces natural logarithms and presents analytic expressions for the hyperbolic functions.
The second book, Configuration Theorems, requires only the most elementary background in plane and solid geometry. It discusses several interesting theorems on collinear points and concurrent lines, showing their applications to several practical geometric problems, and thus introducing certain fundamental concepts of projective geometry. Equivalent and Equidecomposable Figures, the final book, discusses the mathematical conditions of dissecting a given polyhedron into a finite number of pieces and reassembling them into another given polyhedron.

Equivalent and Equidecomposable Figures

V. G. Boltyanskii

PREFACE TO THE AMERICAN EDITION

THIS BOOKLET deals with a class of problems fundamental to the theories of area and volume. In the case of plane figures the central problem considered stems from the following question: If two polygons have equal areas, is it possible to dissect one of them into a finite number of parts which can be rearranged to form the other? The second part of the booklet is concerned with the analogous problem for solid figures. In view of the elementary nature of the topics themselves, it may seem surprising that some of the theorems proved in this booklet are the product of comparatively recent research.

For the first four chapters, the reader needs only the background of one year of algebra and a half-year of plane geometry. Solid geometry and trigonometry are needed for the last two chapters. Some of the proofs toward the end of the booklet are rather difficult, and on first reading the student may wish to learn only the statement of these theorems.

CONTENTS

CHAPTER 1. The Bolyai-Gerwin Theorem; Equidecomposability of Polygons

1. Decomposition method

2. The Bolyai-Gerwin theorem

3. Complementation method

CHAPTER 2. The Hadwiger-Glur Theorem

4. Motions

5. The Hadwiger-Glur theorem

CHAPTER 3. Equidecomposability and the Concept of Additive Invariants

6. The additive invariant J_l(M)

7. T-equidecomposability

8. Properties of the invariant J_l(M)

9. Centrally symmetric polygons

CHAPTER 4. Equidecomposability and the Concept of Groups

10. Groups

11. Groups of motions

12. A property of the group S (optional)

CHAPTER 5. The Theorems of Dehn and Hadwiger for Polyhedra

13. Equidecomposable polyhedra

14. The theorem of Hadwiger

15. The theorem of Dehn

16. Proof of the theorem of Hadwiger

17. n-dimensional polyhedra

CHAPTER 6. Methods for Calculating Volumes

18. The method of limits

19. Equivalence of the decomposition and complementation methods (optional)

Appendix

1. Necessary and sufficient conditions for the equidecomposability of polyhedra

2. G-equidecomposability of polyhedra

Bibliography

1. The Bolyai-Gerwin Theorem; Equidecomposability of Polygons

1. DECOMPOSITION METHOD

Let us examine the two figures represented in Fig. 1. All line segments making up the cross-shaped figure are of equal length, and the side of the square is equal to the line segment AB. The dotted lines shown in the illustration divide these figures into the same number of congruent parts (corresponding parts in the two figures are marked by the same numbers). This fact is expressed in words as follows: the figures represented in Fig. 1 are equidecomposable. In other words, two figures are said to be equidecomposable if it is possible to decompose one of them into a finite number of parts which can be rearranged to form the second figure.

Fig. 1

It is clear that two equidecomposable figures have equal areas. On this is based a simple method for calculating areas, which is known as the decomposition method. This method (already known to Euclid more than 2,000 years ago) is as follows: In order to calculate the area of a figure, t...