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 Jl(M)
7. T-equidecomposability
8. Properties of the invariant Jl(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.
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...