Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem

@article{Laczkovich1990EquidecomposabilityAD,
  title={Equidecomposability and discrepancy; a solution of Tarski's circle-squaring problem},
  author={Mikl{\'o}s Laczkovich},
  journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)},
  year={1990},
  volume={1990},
  pages={117 - 77}
}
  • M. Laczkovich
  • Published 1990
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
Tarski's circle-squaring problem asks whether a disc is equidecomposable to a square; that is, whether a disc can be decomposed into finitely many parts which can be rearranged to obtain a partition of a square [16]. The problem was motivated by the well-known Banach-Tarski theorem stating that in /S* any two sets are equidecomposable provided that they are bounded and have non-empty interior. In particular, any ball is equidecomposable to any cube. On the other band, the existence of a Banach… 

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References

SHOWING 1-10 OF 16 REFERENCES

Convex bodies equidecomposable by locally discrete groups of isometries

We show that if a polytope K 1 , in ℝ d can be partitioned into a finite number of sets, and these sets can be moved by isometries in a locally discrete group to form a convex body K 2 , then K 2 is

Distinct representatives of subsets

If we make this assumption it follows that 2JÎ annuls dA/dyir, where r is the order of A in yx. Let s be the order of A in y%. We form the resultant R of A and dA/dyir, considered as algebraic

On the product of three homogeneous linear forms and indefinite ternary quadratic forms

  • J. CasselsH. Swinnerton-Dyer
  • Mathematics
    Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences
  • 1955
Isolation theorems for the minima of factorizable homogeneous ternary cubic forms and of indefinite ternary quadratic forms of a new strong type are proved. The problems whether there exist such

Elementarer Beweis einer isoperimetrischen Ungleichung

Es sei bemerk t , dass bei dem tJbergang yon einem nicht konvexen Gebiet zur konvexen Hiille L verkleinert , F vergrtissert wird, wiihrend R unveriinclert bleibt. Daher geniigt es die Ungleichung (2)

Geometric Measure Theory

Introduction Chapter 1 Grassmann algebra 1.1 Tensor products 1.2 Graded algebras 1.3 Teh exterior algebra of a vectorspace 1.4 Alternating forms and duality 1.5 Interior multiplications 1.6 Simple

Scissor congruence

It is shown that certain simple figures can not be cut by scissors into pieces that can be reassembled to form certain other simple figures.

The Banach-Tarski paradox, Cambridge 1986. Department of Analysis, E tv s Lorand University, Muzeum krt

  • Budapest, Hungary Eingegangen 22. Dezember
  • 1988

FACTORIZATION OF EVEN GRAPHS

An Introduction to Diophantine Approximation