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Banach–Tarski paradox

The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in three-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1]

For the book about the paradox, see The Banach–Tarski Paradox (book).

An alternative form of the theorem states that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".


The theorem is called a paradox because it contradicts basic geometric intuition. "Doubling the ball" by dividing it into parts and moving them around by rotations and translations, without any stretching, bending, or adding new points, seems to be impossible, since all these operations ought, intuitively speaking, to preserve the volume. The intuition that such operations preserve volumes is not mathematically absurd and it is even included in the formal definition of volumes. However, this is not applicable here because in this case it is impossible to define the volumes of the considered subsets. Reassembling them reproduces a set that has a volume, which happens to be different from the volume at the start.


Unlike most theorems in geometry, the mathematical proof of this result depends on the choice of axioms for set theory in a critical way. It can be proven using the axiom of choice, which allows for the construction of non-measurable sets, i.e., collections of points that do not have a volume in the ordinary sense, and whose construction requires an uncountable number of choices.[2]


It was shown in 2005 that the pieces in the decomposition can be chosen in such a way that they can be moved continuously into place without running into one another.[3]


As proved independently by Leroy[4] and Simpson,[5] the Banach–Tarski paradox does not violate volumes if one works with locales rather than topological spaces. In this abstract setting, it is possible to have subspaces without point but still nonempty. The parts of the paradoxical decomposition do intersect a lot in the sense of locales, so much that some of these intersections should be given a positive mass. Allowing for this hidden mass to be taken into account, the theory of locales permits all subsets (and even all sublocales) of the Euclidean space to be satisfactorily measured.

Obtaining infinitely many balls from one[edit]

Using the Banach–Tarski paradox, it is possible to obtain k copies of a ball in the Euclidean n-space from one, for any integers n ≥ 3 and k ≥ 1, i.e. a ball can be cut into k pieces so that each of them is equidecomposable to a ball of the same size as the original. Using the fact that the free group F2 of rank 2 admits a free subgroup of countably infinite rank, a similar proof yields that the unit sphere Sn−1 can be partitioned into countably infinitely many pieces, each of which is equidecomposable (with two pieces) to the Sn−1 using rotations. By using analytic properties of the rotation group SO(n), which is a connected analytic Lie group, one can further prove that the sphere Sn−1 can be partitioned into as many pieces as there are real numbers (that is, pieces), so that each piece is equidecomposable with two pieces to Sn−1 using rotations. These results then extend to the unit ball deprived of the origin. A 2010 article by Valeriy Churkin gives a new proof of the continuous version of the Banach–Tarski paradox.[13]

2000: Von Neumann's paper left open the possibility of a paradoxical decomposition of the interior of the unit square with respect to the linear group SL(2,R) (Wagon, Question 7.4). In 2000, proved that such a decomposition exists.[15] More precisely, let A be the family of all bounded subsets of the plane with non-empty interior and at a positive distance from the origin, and B the family of all planar sets with the property that a union of finitely many translates under some elements of SL(2, R) contains a punctured neighborhood of the origin. Then all sets in the family A are SL(2, R)-equidecomposable, and likewise for the sets in B. It follows that both families consist of paradoxical sets.

Miklós Laczkovich

2003: It had been known for a long time that the full plane was paradoxical with respect to SA2, and that the minimal number of pieces would equal four provided that there exists a locally commutative free subgroup of SA2. In 2003 constructed such a subgroup, confirming that four pieces suffice.[16]

Kenzi Satô

2011: Laczkovich's paper left open the possibility if there exists a free group F of piecewise linear transformations acting on the punctured disk D \{0,0} without fixed points. Grzegorz Tomkowicz constructed such a group,[18] showing that the system of congruences ABCB U C can be realized by means of F and D \{0,0}.

[17]

2017: It has been known for a long time that there exists in the hyperbolic plane H2 a set E that is a third, a fourth and ... and a -th part of H2. The requirement was satisfied by orientation-preserving isometries of H2. Analogous results were obtained by [19] and Jan Mycielski[20] who showed that the unit sphere S2 contains a set E that is a half, a third, a fourth and ... and a -th part of S2. Grzegorz Tomkowicz[21] showed that Adams and Mycielski construction can be generalized to obtain a set E of H2 with the same properties as in S2.

John Frank Adams

2017: Von Neumann's paradox concerns the Euclidean plane, but there are also other classical spaces where the paradoxes are possible. For example, one can ask if there is a Banach–Tarski paradox in the hyperbolic plane H2. This was shown by Jan Mycielski and Grzegorz Tomkowicz.[23] Tomkowicz[24] proved also that most of the classical paradoxes are an easy consequence of a graph theoretical result and the fact that the groups in question are rich enough.

[22]

2018: In 1984, Jan Mycielski and Stan Wagon constructed a paradoxical decomposition of the hyperbolic plane H2 that uses Borel sets. The paradox depends on the existence of a properly discontinuous subgroup of the group of isometries of H2. Similar paradox is obtained by Grzegorz Tomkowicz [26] who constructed a free properly discontinuous subgroup G of the affine group SA(3,Z). The existence of such a group implies the existence of a subset E of Z3 such that for any finite F of Z3 there exists an element g of G such that , where denotes the symmetric difference of E and F.

[25]

2019: Banach–Tarski paradox uses finitely many pieces in the duplication. In the case of countably many pieces, any two sets with non-empty interiors are equidecomposable using translations. But allowing only Lebesgue measurable pieces one obtains: If A and B are subsets of Rn with non-empty interiors, then they have equal Lebesgue measures if and only if they are countably equidecomposable using Lebesgue measurable pieces. Jan Mycielski and Grzegorz Tomkowicz extended this result to finite dimensional Lie groups and second countable locally compact topological groups that are totally disconnected or have countably many connected components.

[27]

Hausdorff paradox

Nikodym set

Paradoxes of set theory

 – Problem of cutting and reassembling a disk into a square

Tarski's circle-squaring problem

Von Neumann paradox

Churkin, V. A. (2010). "A continuous version of the Hausdorff–Banach–Tarski paradox". Algebra and Logic. 49 (1): 91–98. :10.1007/s10469-010-9080-y. S2CID 122711859.

doi

Edward Kasner & James Newman (1940) , pp 205–7, Simon & Schuster.

Mathematics and the Imagination

Stromberg, Karl (March 1979). "The Banach–Tarski paradox". The American Mathematical Monthly. 86 (3). Mathematical Association of America: 151–161. :10.2307/2321514. JSTOR 2321514.

doi

"The Banach–Tarski Paradox" (PDF).

Su, Francis E.

(1994). The Banach–Tarski Paradox. Cambridge: Cambridge University Press. ISBN 0-521-45704-1.

Wagon, Stan

Wapner, Leonard M. (2005). . Wellesley, Massachusetts: A.K. Peters. ISBN 1-56881-213-2.

The Pea and the Sun: A Mathematical Paradox

; Wagon, Stan (2016). The Banach–Tarski Paradox 2nd Edition. Cambridge: Cambridge University Press. ISBN 9781107042599.

Tomkowicz, Grzegorz

at ProofWiki

Banach–Tarski paradox

by Stan Wagon (Macalester College), the Wolfram Demonstrations Project.

The Banach-Tarski Paradox

by David Morgan-Mar provides a non-technical explanation of the paradox. It includes a step-by-step demonstration of how to create two spheres from one.

Irregular Webcomic! #2339

Vsauce. – via YouTube gives an overview on the fundamental basics of the paradox.

"The Banach–Tarski Paradox"

Banach-Tarski and the Paradox of Infinite Cloning