Geometrical investigations of honeycombs and speculations on how honeybees measure and construct the hexagons and rhombi of their cells are centuries old. Here we show that honeybees neither have to measure nor construct the highly regular structures of a honeycomb, and that the observed pattern of combs can be parsimoniously explained by wax flowing in liquid equilibrium. The structure of the combs of honeybees results from wax as a thermoplastic building medium, which softens and hardens as a result of increasing and decreasing temperatures. It flows among an array of transient, close-packed cylinders which are actually the self-heated honeybees themselves. The three apparent rhomboids forming the base of each cell do not exist but arise as optical artefacts from looking through semi-transparent combs.