Here is a nice schematic drawing of the device followed by an explanation.
From bhaaluu's description:
The hand-crank turns a 30-tooth peg-wheel gear which drives a 10-tooth peg wheel gear. The 10-tooth peg-wheel gear spins on a fixed vertical shaft. connected via a wooden tube to the spindle head, which rotates at a 3:1 ratio. At the top of the fixed vertical shaft is a 15-tooth spur gear. It is fixed to the vertical shaft and doesn't move. At the base of the spindle tube is a 42-tooth spur gear which rotates on it's own axis, which is tilted at an angle. As the spindle tube rotates around the vertical shaft, the 42-tooth spur gear rotates. The spindle tube, a heavy-walled cardboard tube, is connected to the 42-tooth spur gear, so every time it turns around the fixed vertical shaft, it rotates. This is what gives the diamond pattern. This string winding contraption is a prototype.
[ Thanks bhaaluu! ]