Let \(\mathcal{T}\) (m, υ) be a universal d-dimensional karyon tiling. Its parameters, the weight vector m and the star υ, belong to the dual module space ∆d×∆d that is the direct product of two d-dimensional simplexes. The star v defines a geometry of the parallelepipeds T0, T1, . . . , Td, of which the tiling \(\mathcal{T}\) (m, υ) consists, and the weight vector m sets the local rules and frequency distribution of the parallelepipeds of the tiling. Knowing the parameters m and v, one can construct the entire tiling \(\mathcal{T}\) (m, υ) by a local algorithm \(\mathcal{A}\) . It is proved that the differentiation of the karyon tiling \(\mathcal{T}\) (m, υ) → \({\mathcal{T}}^{\sigma }\) (m, υ) is equivalent to some explicitly defined elementary transformation of the centered unimodular basis u.
