In his book "Flatland, a romance  of many dimensions", Edwin Abbott imagines a two-dimensional world inhabited by two-dimensional creatures.  The lack of a third dimension might appear to reduce possibilities all round but we will explain how a system of algebra based on two dimensional configurations is in fact very rich. It has even been proposed as the  basis for building a quantum computer. The talk, intended for those without much mathematics background, will be enhanced by a  certain number of more or less relevant anecdotes and images.