Unearthing the Banach–Tarski paradox

Archimedes could move the earth no matter – what the girth. Banach–Tarski broke it in pieces, rotated them and showed size increases. How on earth? This is “anarth !”

Abstract

Greeks used the method of cutting a geometric region into pieces and recombining them cleverly to obtain areas of figures like parallelograms. In such problems, the boundary is ignored. However, in our discussion, we will take every point of space into consideration. The human endeavour to compute lengths, areas, and volumes of irregular complicated shapes and solids created the subject of ‘measure theory’. The paradox of the title can be informally described as follows. Consider the earth including the inside stuff. It is possible to decompose this solid sphere into finitely many pieces and apply three-dimensional rotations to these pieces such that the transformed pieces can be put together to form two solid earths! The whole magic lies in the word ‘pieces’. The pieces turn out to be so strange that they cannot be ‘measured’.