A paradoxical decomposition of the real line

In this paper we demonstrate how to partition the real number line into four subsets which may be reassembled, via “piecewise rigid functions” that preserve Lebesgue measure, into two copies of the line. We then employ a similar process to split the line into $2k$ pieces that yield $k$ copies of the line, or even into countably many subsets to obtain countably many copies of $\mathbb{R}$.