Banach-Tarski Paradox Banach Space Banach Algebra

which states that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitely many pieces and, moving them using only rotations and translations, reassemble the pieces into two balls of the same radius as the original.

This paradox is very similar to earlier Hausdorff paradox, and its proof is based on the same idea. Therefore, it is more correctly called the Hausdorff-Banach-Tarski paradox.

Formally: Let A and B be two subsets of Euclidean space. We call them equi-decomposable if they can be represented as finite unions of disjoint subsets and such that, for any i, the subset is isometric to . Then, the paradox can be reformulated as follows:

For the ball, five pieces are sufficient to do this; it cannot be done with fewer than five. There is an even stronger version of the paradox:

In other words, a marble can be cut up into finitely many pieces and reassembled into a planet, or a telephone could be cut up and reassembled as a water lily. These transformations are not possible with real objects made of atoms, but it is possible with their geometric shapes.

It should be noted that, in general, it is well known that an infinite set (such as two spheres) can be transformed bijectively to a similarly infinite subset of itself (such as one sphere). However, such transformations in general are non-isometric or involve an uncountablyIn mathematics the term countable set is used to describe the size of a set, e. the number of elements it contains. Non-mathematicians can usually only measure the size of finite sets by counting and have an unclear concept of infinite sets and the differ infinite number of "pieces"—the surprising consequence of the Banach-Tarski paradox is that it can be done with only rotation and translation (isometric mapping) of a finite number of pieces (albeit infinitely convoluted/complicated pieces, which individually are not measurable).

Note that in the decomposition, the pieces won't be measurableMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura, and so they will not have "reasonable" boundaries nor a "volume" in the ordinary sense. It is impossible to carry out such a disassembly physically because disassembly "with a knife" can create only measurable sets. This pure existence statement in mathematics points out that there are many more sets than just the measurable sets familiar to most people.

The paradox also holds in all dimensions starting with three, and it does not hold for subsets of the Euclidean plane. (In three dimensions, a planar subset has an empty interior, therefore one can not apply the statement above.) Still, there are some paradoxical decompositions in the plane: a circle can be cut into finitely many pieces and reassembled to form a square of equal area; see Tarski's circle-squaring problemTarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a circle (including its interior) in the plane, cut it into finitely many pieces, and reassemble the pieces so as to get a square of equal area. This was proven to.

The paradox shows that it is impossible to define "volume" on all bounded subsets of Euclidean space such that equi-decomposable sets will have equal "volume".

Its proof is based on the earlier work of Felix HausdorffFelix Hausdorff ( November 8 1868 January 26 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory and functional analysis. He defined and studied partially ordered, who found a closely related paradox 10 years earlier; in fact Banach-Tarski paradox is a simple corollary of technique developed by Hausdorff.

Logicians most often use the term "paradox" for a statement in logic which creates problems because it causes contradictions, such as the Liar paradoxThe liar paradox is a concept from the fields of philosophy and logic. It refers to paradoxical statements such as: I am lying now. or This statement is false. To avoid having a sentence refer to its own truth value, one can also construct the paradox as or Russell's paradoxRussell's paradox is a paradox discovered by Bertrand Russell in 1901 which shows that the naive set theory of Cantor and Frege is contradictory. Consider the set M to be "The set of all sets that do not contain themselves as members". Formally: A is an e. The Banach-Tarski paradox is not a paradox in this sense but rather a proven theorem; it is a paradox only in the sense of being counter-intuitive. Because its proof prominently uses the axiom of choice, this counter-intuitive conclusion has been presented as an argument against adoption of that axiom.

1 A sketch of the proof

Essentially, the paradoxical decomposition of the ball is achieved in four steps:

Find a paradoxical decomposition of the free groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st-1 su-1ut-1 . in two generators.
Find a group of rotations in 3-d space isomorphic to the free group in two generators.
Use the paradoxical decomposition of that group and the axiom of choice to produce a paradoxical decomposition of the hollow unit sphere.
Extend this decomposition of the sphere to a decomposition of the solid unit ball.

We now discuss each of these steps in more detail.

Step 1 The free group with two generators a and b consists of all finite strings that can be formed from the four symbols a, a⁻¹, b and b⁻¹ such that no a appears directly next to an a⁻¹ and no b appears directly next to a b⁻¹. Two such strings can be concatenated and converted into a string of this type by repeatedly replacing the "forbidden" substrings with the empty string. For instance: abab^-1a^-1 concatenated with abab^-1a yields abab^-1a^-1abab⁻¹a, which gets reduced to abaab⁻¹a. One can check that the set of those strings with this operation forms a group with neutral element the empty string, here denoted ε. We will call this group G.

The group G can be "paradoxically decomposed" as follows: let S(a) be the set of all strings that start with a and define S(a⁻¹), S(b) and S(b⁻¹) similarly. Clearly,

G = {ε} ∪ S(a) ∪ S(a⁻¹) ∪ S(b) ∪ S(b⁻¹)

but also

G = a S(a⁻¹) ∪ S(a), and

G = b S(b⁻¹) ∪ S(b).

(The notation a S(a⁻¹) means take all the strings in S(a⁻¹) and concatenate them on the left with a.) Make sure that you understand this last line, because it is at the core of the proof. Now look at this: we cut our group G into four pieces (Forget about ε for now, it doesn't pose a problem.), then "rotated" some of them by multiplying with a or b, then "reassembled" two of them to make G and reassembled the other two to make another copy of G. That's exactly what we want to do to the ball.

Step 2 In order to find a group of rotations of 3-d space that behaves just like (or "isomorphic to") the group G, we take two orthogonal axes and let A be a rotation of arccos(1/3) about the first and B be a rotation of arccos(1/3) about the second. (This step cannot be performed in two dimensions.) It is somewhat messy but not too difficult to show that these two rotations behave just like the elements a and b in our group G. We'll skip it. The new group of rotations generated by A and B will be called H. Of course, we now also have a paradoxical decomposition of H.

Step 3 The unit sphere S² is partitioned into orbits by the action of our group H: two points belong to the same orbit if and only if there's a rotation in H which moves the first point into the second. We can use the axiom of choice to pick exactly one point from every orbit; collect these points into a set M. Now (almost) every point in S² can be reached in exactly one way by applying the proper rotation from H to the proper element from M, and because of this, the paradoxical decomposition of H then yields a paradoxical decomposition of S².

Step 4 Finally, connect every point on S² with a ray to the origin; the paradoxical decomposition of S² then yields a paradoxical decomposition of the solid unit ball. (The center of the ball needs bit more care, but we omit this part in the sketch.)

NB This sketch glosses over some details. One has to be careful about the set of points on the sphere which happen to lie on an axis of rotation of some matrix in H. On the one hand, there are countably many such points so they "do not matter", and on the other hand it is possible to patch up even those points. The same applies to the center of the ball.

2 Further reading

"Sur la décomposition des ensembles de points en parties respectivement congruentes", Fundamenta Mathematicae, 6, (1924), 244-277, the original paper of Banach and Taski (in French).
Layman's Guide to the Banach-Tarski Paradox (from Kuro5hin)
Francis E. Su, "The Banach-Tarski Paradox", http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf
S. Wagon, The Banach-Tarski Paradox, Cambridge University Press, 1986.

Theorems Group theory Paradoxes Measure theory

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Home > Banach-Tarski paradox

1 A sketch of the proof

2 Further reading

Topics: Banach-Tarski Paradox, Banach Space, Banach Algebra...