The two basic remanent domain structures (RDS) observed in magnetically uniaxial platelets and their variations as a function of the thickness of the platelet and the value of the rotational susceptibility of the material is discussed. The parallel-plate structure (PPS) first proposed by Landau and Lifshitz represents the lowest energy configuration or ground state for the given geometry, i.e., the easy axis normal to the platelets. The validity of their half-power law connecting the domain width with the thickness of the platelet is discussed in the light of subsequent theories and experiments and is shown to be correct over a limited thickness range only. The honeycomb domain structure (HCS) consisting of a closely packed array of circular cylindrical domains, found later experimentally, is shown by the latest calculations to be a RDS having a total free energy only about 0.4 percent higher than that of the PPS. The domain spacing is found to obey a similar thickness dependence as that in the PPS.