In the last chapter, we investigated a special collection of groups we called ‘accidents’. These were the beautiful double cusp groups in which two symmetries are forced to be parabolic, corresponding to two different rational numbers of our choice. These groups lay right on the borderline between the relatively well-behaved quasifuchsian regime, and the total disorder of non-discreteness. As we are about to see, however, this is not the full tale. There are other yet stranger groups hovering on this same boundary between order and chaos.

Two millennia before the hesitant introduction of imaginary numbers, came the ancient discovery of another class of numbers whose name has acquired over time an even more negative [sic] connotation: irrational numbers, numbers, that is, which cannot be expressed as a ratio p/q. Legend has it that Pythagoras, founder of a religious cult circa 500 BC, discovered that is irrational, and that the revelation was so unnerving the fact was kept secret within the brotherhood. The irrationals lurk in the cracks between the rationals, a kind of invisible glue without which the line would fall apart.