Modular curve stitching

Consider the points of a clock face, numbered 1 up to 12, and for each point k, draw a straight line between k and 2k, taking remainders modulo 12. For example, there is a line between 5 and 10 because 10=5×2, and there is a line between 9 and 6, because 18=9×2 and 18 and 6 leave the same remainder when divided by 12.