  The transport on top of a periodic two-dimensional hexagonal magnetic pattern
of (i) a single macroscopic steel sphere, (ii) a doublet of wax/magnetite
composite spheres, and (iii) an immiscible mixture of ferrofluid droplets with
a perfluorinated liquid is studied experimentally and analyzed theoretically.
The transport of all these magnetic objects is achieved by moving an external
permanent magnet on a closed modulation loop around the two-dimensional
magnetic pattern. The transport of one and also that of two objects per unit
cell is topologically protected and characterized by discrete displacements of
the particles as we continuously scan through a family of modulation loops. The
direction and the type of transport is characterized by the winding numbers of
the modulation loops around special objects in control space, which is the
space of possible directions of the external magnetic field. The number of
winding numbers necessary for characterizing the topological transport
increases with the number of particles per unit cell. The topological character
of the transport is destroyed when transporting a large collection of particles
per unit cell like it is the case for a macroscopic assembly of magnetic nano
particles in a ferrofluid droplet for which the transport is geometrical and no
longer topological. To characterize the change in the transport from
topological to geometrical, we perform computer simulations of the transport of
an increasing number of particles per unit cell.
