  We show that $\{0,\varphi+3,-3\varphi+2,-\varphi+\frac{5}{2}\}$ is the set of
slopes of nonexpansive directions for a minimal subshift in the Jeandel-Rao
Wang shift, where $\varphi=(1+\sqrt{5})/2$ is the golden mean. This set is a
topological invariant allowing to distinguish the Jeandel-Rao Wang shift from
other subshifts. Moreover, we describe the combinatorial structure of the two
resolutions of the Conway worms along the nonexpansive directions in terms of
irrational rotations of the unit interval. The introduction finishes with
pictures of nonperiodic Wang tilings corresponding to what Conway called the
cartwheel tiling in the context of Penrose tilings. The article concludes with
open questions regarding the description of octopods and essential holes in the
Jeandel-Rao Wang shift.
