  Encoding a dimension in the internal degree of freedom of an atom provides an
interesting tool for quantum simulation, facilitating the realization of
artificial gauge fields. We propose an extension of the synthetic dimension
toolbox, making it possible to encode two dimensions within a large atomic
spin. The protocol combines first- and second-order spin couplings, such that
the spin projection $m$ and the remainder $r=m$ (mod 3) of its Euclidian
division by 3 act as orthogonal coordinates on a synthetic cylinder. It is
suited for an implementation with lanthanide atoms, which feature a large
electronic spin and narrow optical transitions for applying the required spin
couplings. This method is useful for simulating geometries with periodic
boundary conditions, and engineering various types of topological systems
evolving in high dimensions.
