The cone of positive‐definite quadratic forms is subdivided into aggregates of parallelohedra having certain properties. In E6, the Σ‐subcones are investigated and, in particular, the Σ0‐subcone will be described which first occurs in E6 and which is governed by the group . Further the ‐subcone which is governed by the group will be discussed for d = 6. The structures of quasicrystals can be considered as sections of lattices of translations in higher dimensions, which has greatly stimulated the investigation of lattices in arbitrary dimensions. The structure of the cone of positive definite quadratic forms in Ed×d is investigated for d = 6. A partition of the cone in Φ and Σ domains is performed and its local symmetries are determined.