Perovskite-like ABO₃ oxides A = (Ca, Sr, Ba) and B = (Ti, Mn, Fe, Co, Ni) show a large variety of structures and physical properties. Among them is BaTiO₃ (BTO), one of the most investigated and used perovskites. In a BTO film on Pt(111), the first oxide quasicrystal was discovered. Herein, by means of first-principle methods, the cubic and hexagonal phases of bulk BaFeO₃ (BFO) are investigated. Both phases show ferromagnetic order. Monolayers and double layers of BFO are studied on a Pt(111) surface. The double-layer configuration of the cubic and hexagonal phases is structurally inequivalent but both double-layer films show antiferromagnetic order. In analogy to the BTO quasicrystal approximant structure on Pt(111), a corresponding BFO structure is investigated. The Fe atoms are surrounded by three oxygen atoms and the resulting FeO₃ units are separated by barium atoms with the total stoichiometry Ba₅Fe₄O₁₂.