Banach-Tarski Paradox

Nicolas Monod teaches at the École polytechnique fédérale in
Lausanne and leads the Ergodic and Geometric Group Theory group
there. In May 2016 he was invited to give the Gauß lecture of the
German Mathematical Society (DMV) at the Technical University in
Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski
Paradox. The morning after his lecture we met to talk about
paradoxes and hidden assumptions our mind makes in struggling
with geometrical representations and measures.


A very well-known game is Tangram. Here a square is divided into
seven pieces (which all are polygons). These pieces can be
rearranged by moving them around on the table, e.g.. The task for
the player is to form given shapes using the seven pieces – like
a cat etc.. Of course the Tangram cat looks more like a flat
Origami-cat. But we could take the Tangram idea and use thousands
or millions of little pieces to build a much more realistic cat
with them – as with pixels on a screen. In three dimensions one
can play a similar game with pieces of a cube. This could lead to
a LEGO-like three-dimensional cat for example. In this
traditional Tangram game, there is no fundamental difference
between the versions in dimension two and three.


But in 1914 it was shown that given a three-dimensional ball,
there exists a decomposition of this ball into a finite number of
subsets, which can then be rearranged to yield two identical
copies of the original ball. This sounds like a magical trick –
or more scientifically said – like a paradoxical situation. It is
now known under the name Banach-Tarski paradox. In his lecture,
Nicolas Monod dealt with the question: Why are we so surprised
about this result and think of it as paradoxical?


One reason is the fact that we think to know deeply what we
understand as volume and expect it to be preserved under
rearrangements (like in the Tangram game, e.g.).Then the impact
of the Banach-Tarski paradox is similar for our understanding of
volume to the shift in understanding the relation between time
and space through Einstein's relativity theory (which is from
about the same time). In short the answer is: In our every day
concept of volume we trust in too many good properties of it.


It was Felix Hausdorff who looked at the axioms which should be
valid for any measure (such as volume). It should be independent
of the point in space where we measure (or the coordinate system)
and if we divide objects, it should add up properly. In our
understanding there is a third hidden property: The concept
"volume" must make sense for every subset of space we choose to
measure. Unfortunately, it is a big problem to assign a volume to
any given object and Hausdorff showed that all three properties
cannot all be true at the same time in three space dimensions.
Couriously, they can be satisfied in two dimensions but not in
three.


Of course, we would like to understand why there is such a big
difference between two and three space dimensions, that the naive
concept of volume breaks down by going over to the third
dimension. To see that let us consider motions. Any motion can be
decomposed into translations (i.e. gliding) and rotations around
an arbitrarily chosen common center. In two dimensions the order
in which one performs several rotations around the same center
does not matter since one can freely interchange all rotations
and obtains the same result. In three dimensions this is not
possible – in general the outcomes after interchanging the order
of several rotations will be different. This break of the
symmetry ruins the good properties of the naive concept of
volume.


Serious consequences of the Banach-Tarski paradox are not that
obvious. Noone really duplicated a ball in real life. But measure
theory is the basis of the whole probability theory and its
countless applications. There, we have to understand several
counter-intuitive concepts to have the right understanding of
probabilities and risk. More anecdotally, an idea of Bruno
Augenstein is that in particle physics certain transformations
are reminiscent of the Banach-Tarski phenomenon.


Nicolas Monod really enjoys the beauty and the liberty of
mathematics. One does not have to believe anything without a
proof. In his opinion, mathematics is the language of natural
sciences and he considers himself as a linguist of this language.
This means in particular to have a closer look at our thought
processes in order to investigate both the richness and the
limitations of our models of the universe.


References:


F. Hausdorff: Bemerkung über den Inhalt von Punktmengen.
Math. Ann. 75 (3), 428–433, 1914.

S. Banach and A.Tarski: Sur la décomposition des ensembles de
points en parties respectivement congruentes, Fundamenta
Mathematicae 6, 244–277, 1924.

J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta
Mathematicae 13, 73–116, 1929.

S. Wagon: The Banach–Tarski Paradox. Cambridge University
Press, 1994.

B.W. Augenstein: Links Between Physics and Set Theory, Chaos,
Solitons and Fractals, 7 (11), 1761–1798, 1996.

N. Monod: Groups of piecewise projective homeomorphisms, PNAS
110 (12), 4524-4527, 2013.

Vsauce-Video on the Banach-Tarksi Paradox