Impossible cube ambiguity present in a Necker cube illustration, in which a cube is drawn with its edges as line segments, and can be interpreted as being in either of two different three-dimensional orientations. An impossible cube is usually rendered as a Necker cube in which the line segments representing the edges have been replaced by what are apparently solid beams. In Escher's print, the top four joints of the cube, and the upper of the two crossings between its beams, match one of the two interpretations of the Necker cube, while the bottom four joints and the bottom crossing match the
Doubling the cube at solving what they saw as an obstinate but soluble problem. However, the nonexistence of a solution was finally proven by Pierre Wantzel in 1837. In algebraic terms, doubling a unit cube requires the construction of a line segment of length , where ; in other words, . This is because a cube of side length 1 has a volume of , and a cube of twice that volume (a volume of 2) has a side length of the cube root of 2. The impossibility of doubling the cube is therefore equivalent to the statement that is not a constructible