[en] Monoclinic τ²-Al₁₃Co₄, τ³-Al₁₃Co₄ and τ⁴-Al₁₃Co₄ phases with a and c parameters about τ², τ³ and τ⁴ times [τ = (1 + 5^1/2)/2 = 1.61803, which is the golden number], respectively, greater than their counterparts in Al₁₃Co₄ (space group Cm, a = 1.5183, b = 0.8122, c = 1.2340 nm, β = 107 54' [Hudd and Taylor (1962). Acta Cryst. 15, 441-442]) have been found by selected-area electron diffraction and high-resolution electron microscopy (HREM). The [010] HREM images showed that these τ-inflated phases have a similar pentagonal layer structure as Al₁₃Co₄, but the edge length of the pentagons is τ², τ³ and τ⁴ times larger, respectively, than that in Al₁₃Co₄. This explains why these τ-inflated phases have about the same β angle (≅108 ) and b representing a repeat unit of the stacking of these pentagonal layers. The τ² pentagon (1.23 nm) consists of six smaller pentagons (0.47 nm) existing in Al₁₃Co₄, one inverted and five of the same orientation as the τ² pentagon. Similarly, the τ³ pentagon (2.00 nm) consists of six τ pentagons (0.76 nm), and the τ⁴ pentagon (3.23 nm) of six τ² pentagons which in turn contain six 0.47 nm pentagons in each of them. The greater the inflation, the larger the unit-cell parameters, and the better the approximation of its structure to that of a decagonal quasicrystal (DOC). In other words, a DQC is considered to be the end member of this family of τ-inflated phases. (orig.)