  The geometry of Borromean three-body halos, such as two-neutron halo nuclei
or triatomic molecules close to dissociation, is investigated using a
three-body model. This model enables to analytically derive the universal
geometric properties found recently within an effective-field theory for halos
made of a core and two resonantly-interacting particles [Phys. Rev. Lett., 128,
212501 (2022)]. It is shown that these properties not only apply to the ground
three-body state, but also to all the excited (Efimov) states where the
core-particle interaction is resonant. Furthermore, a universal geometry
persists away from the resonant regime between the two particles, for any state
close to the three-body threshold. This "halo universality" is different from
the Efimov universality which is only approximate for the ground state. It is
explained by the separability of the hyper-radius and hyper-angles close to the
three-body dissociation threshold.
