New information on the phase diagrams of Ti-Fe-Si-O and Ti-Zr-Ni alloys near the quasicrystal and rational approximant compositions is presented. α(TiFeSiO), the 1/1 rational approximant, is shown to form in a peritectic mode from the liquid, indicating the possibility to produce single-crystal samples. Very long duration annealing studies demonstrate unambiguously that the TiZrNi i-phase and 1/1 approximant form at low temperatures by a solid-state transformation; their phase fields do not extend to the liquidus temperatures. The first undercooling measurements of electrostatically-levitated droplets of the Ti-Zr-Ni alloys are presented. These nucleation studies provide new information on the structural relations between polytetrahedral phases and the undercooled liquid, and on the phase transformation processes. The reduced undercooling for the polytetrahedral phases in these alloys is less than for crystal phases of a similar composition, demonstrating a low interfacial energy between the polytetrahedral phase and the liquid.