  Felix Klein in course of his study of the regular icosahedron and its
symmetries encountered a highly symmetric configuration of $60$ points in
${\mathbb P}^3$. This configuration has appeared in various guises, perhaps
post notably as the configuration of points dual to the $60$ reflection planes
in the group $G_{31}$ in the Shephard-Todd list.
  In the present note we show that the $60$ points exhibit interesting
properties relevant from the point of view of two paths of research initiated
recently. Firstly, they give rise to two completely different unexpected
surfaces of degree $6$. Unexpected hypersurfaces have been introduced by Cook
II, Harbourne, Migliore, Nagel in 2018. One of unexpected surfaces associated
to the configuration of $60$ points is a cone with a single singularity of
multiplicity $6$ and the other has three singular points of multiplicities
$4,2$ and $2$. Secondly, Chiantini and Migliore observed in 2020 that there are
non-trivial sets of points in ${\mathbb P}^3$ with the surprising property that
their general projection to ${\mathbb P}^2$ is a complete intersection. They
found a family of such sets, which they called grids. An appendix to their
paper describes an exotic configuration of $24$ points in ${\mathbb P}^3$ which
is not a grid but has the remarkable property that its general projection is a
complete intersection. We show that the Klein configuration is also not a grid
and it projects to a complete intersections. We identify also its proper
subsets, which enjoy the same property. \
