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Explaining the Banach–Tarski Problem

Banach–Tarski Problem

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Causing much controversy in the mathematical community, the Banach-Tarski paradox remains one of the most debated and intriguing line of mathematics. For the purpose of this article, it is recommended that the reader first understands the term “infinity”.

In short, the Banach-Tarski paradox shows that you can duplicate a 3-dimensional set of points without adding any new points. Basically, an object can double its volume without additional data.

This theory is obviously counter-intuitive and is not seemingly possible in the real world. This is due to the fact that in the physical world, everything is measurable and there can only be a finite amount of ‘things’ in a space. However, using the concepts of unmeasurable data sets and the axiom of choice, this theory can be proved within the realms of mathematics.

The Banach-Tarski paradox uses the fact that a sphere can divided into a finite set of data points which can then be rotated in order to reconstruct the shape into two identical shapes which are the same as the original. It has been found that this can work with as little as 5 pieces, and works without stretching, bending or adding new points.

It should be noted that in order for this to work, the sphere must have an infinite volume. As it is noted in the previous article, infinity divided by 2 is still infinity and therefore 2 spheres can be created from the same space. Basically, one plus one can still equal one.

Obviously, this cannot be true in the real world because we are bound by physical laws and do not have an infinite density but there is a simple real-world analogy that can help visualise the problem.

Taking a simple balloon with some amount of air inside and releasing that air into a container, it is possible to fill two balloons with this gas. The two balloons will now contain half the volume of the original balloon as it has been shared equally between the two. By using the fact that pressure and volume and inversely related, it is possible to reduce the pressure in the room by half, causing the balloons to expand to double their size. They are both the same as the original without any air being added into the system.

It is clear to see that this analogy is not quite the same as the mathematical proof due to the balloons not having an infinite density, however, it in theory, this would be the case.

Another theory implies that any two solid objects can be dissected and reassembled into each other. An extreme example of this thinking this that "a pea can be chopped up and reassembled into the Sun".

The paradox is something that has been shied away from due to its complexity and the fact that it, at least currently, has no real-world applications. The acceptance of this strange theory also leads to some unusual consequences in how we view the universe. It is for these reasons that this paradox has been highly controversial in the field.

Sources and Further Reading

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