  For a crystallographic root system, dominant regions in the Catalan
hyperplane arrangement are in bijection with antichains in a partial order on
the positive roots. For a noncrystallographic root system, the analogous
arrangement and regions have importance in the representation theory of an
associated graded Hecke algebra. Since there is also an analogous root order,
it is natural to hope that a similar bijection can be used to understand these
regions. We show that such a bijection does hold for type $H_3$ and for type
$I_2(m)$, including arbitrary ratio of root lengths when $m$ is even, but does
not hold for type $H_4$. We give a criterion that explains this failure and a
list of the 16 antichains in the $H_4$ root order which correspond to empty
regions.
