A stable decagonal single phase in conventionally solidified Al–Co–Ni and Al–Co–Cu alloys was found to form in each composition range of 15 to 20 at% Co and 10 to 15%Ni or 15 to 20%Cu. Coexisting decagonal and crystalline phases were also formed in conventionally solidified Al₇₀Rh₁₅Ni₁₅ and Al₆₅Rh₂₀Cu₁₅ alloys. The stable decagonal Al–Co–Ni (or Cu) alloys have a columnar morphology with a cross-sectional shape of five dendrite arms radiating from the dendrite centers. The unique morphology is thought to result from a preferential growth along the tenfold periodic axis. The stable decagonal alloys are formed in the criteria of the ratio of the diameter of the Brillouin zone boundary to that of the Fermi sphere (K_p⁄2k_F)\simeq1.0, the outer electron per atom ratio (e⁄a)=1.7 to 1.8 and the atomic size factor (λ)=0.10 to 0.11, indicating the importance of an electronic structure near the Fermi surface in the construction of the stable decagonal structure. The stable decagonal phase is important for a better understanding of the formation mechanism and fundamental properties of quasicrystals in comparison with the theoretical expectation which has been developed for the two-dimensional quasicrystals.