  In the current study, the existence of periodic orbits around a fixed
homogeneous cube is investigated, and the results have powerful implications
for examining periodic orbits around non-spherical celestial bodies. In the two
different types of symmetry planes of the fixed cube, periodic orbits are
obtained using the method of the Poincar\'e surface of section. While in
general positions, periodic orbits are found by the homotopy method. The
results show that periodic orbits exist extensively in symmetry planes of the
fixed cube, and also exist near asymmetry planes that contain the regular Hex
cross section. The stability of these periodic orbits is determined on the
basis of the eigenvalues of the monodromy matrix. This paper proves that the
homotopy method is effective to find periodic orbits in the gravity field of
the cube, which provides a new thought of searching for periodic orbits around
non-spherical celestial bodies. The investigation of orbits around the cube
could be considered as the first step of the complicated cases, and helps to
understand the dynamics of orbits around bodies with complicated shapes. The
work is an extension of the previous research work about the dynamics of orbits
around some simple shaped bodies, including a straight segment, a circular
ring, an annulus disk, and simple planar plates.
