  In this paper, we look at the improvement of our knowledge on a family of
tilings of the hyperbolic plane which is brought in by the use of Sergeyev's
numeral system based on grossone. It appears that the information we can get by
using this new numeral system depends on the way we look at the tilings. The
ways are significantly different but they confirm some results which were
obtained in the traditional but constructive frame and allow us to obtain an
additional precision with respect to this information.
