Dürer’s cubes

In the second book of his Underweysung der Messung, printed in 1525, Albrecht Dürer sets figures together into floors. Of lozenges with “zwen weyt winckel” and “zwen eng”, two wide angles and two narrow ones, he writes: set two side by side and push the third in across them, “so sicht solichs wie ein gefirter wuerffel”, so it looks like a cube, and set them so on and on. His woodcut shows them in a square. He gives no angle; three lozenges close round a point only at sixty degrees, so that is the lozenge here, and each cube is a hexagon of one opening cut in three. The net is our own, and it is exact. A chain of circles of one opening steps along the upright; neighbouring circles cross beside it, and the lines through those crossings and the pins fall at thirty degrees both ways. Every further line runs through two crossings already found, the uprights too, until the net fills the square. The sides of the square are uprights of the net; swung about the point of the net level with the first pin, at its distance from that pin, each side is cut at its corners. The three tones are ours.