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Marcus du Sautoy explains the idea of dimensions to Alan Davies, moving from a point to a line and on to an area and volume - and then into the fourth dimension. He explains how a 4D cube would now have 16 vertices. Travelling to Paris, they look at the La Defense building, which was built as a 3D representation of a 4D cube. Such a representation does retain much of the information about the theoretical 4D cube - 16 vertices, 32 edges and 24 faces. They then discuss the practical applications that 4D geometry have had in computing, even though the theory was discovered way before any application was thought of. As Alan Davies ponders at the end - Mathematics is actually very creative.