The paper presents a universal karyon algorithm, applicable to an arbitrary collection of reals α = (α1, . . . , αd), which is a modification of the simplex-karyon algorithm. The main distinction is that instead of a simplex sequence, an infinite sequence T = T0,T1, . . . ,Tn, . . . of d-dimensional parallelohedra Tn appear. Every parallelohedron Tn is obtained from the previous one Tn−1 by differentiation, $$ {\mathbf{T}}_n={\mathbf{T}}_{n-1}^{\sigma n} $$ . The parallelohedra Tn are the karyons of some induced toric tilings. A certain algorithm (ϱ-strategy) for choosing infinite sequences σ|={σ1, σ2, …, σn, …} of differentiations σn is specified. This algorithm ensures the convergence ϱ(Tn) −→ 0 as n → +∞, where ϱ(Tn) denotes the radius of the parallelohedron Tn in the metric ϱ chosen as the objective function. It is proved that the parallelohedra Tn have the minimality property, i.e., the karyon approximation algorithm is the best one with respect to the karyon Tn-norms. Also an estimate for the rate of approximation of real numbers α = (α1, . . . , αd) by multidimensional continued fractions is derived.