Symmetries of Universal Karyon Tilings

Universal karyon tilings 𝒯 (υ, μ, ρ) are generated by the parallelepipeds T₀, T₁,…, T_d dividing the space ℝ^d. The tilings 𝒯 (υ, μ, ρ) are parameterized by triples (υ, μ, ρ) running over the infinite cylinder ∆ × ∆ ×ℝ with the base ∆ ×∆ that is the direct product of two simplices ∆ of dimension d. The parameter v defines a geometry of the parallelepipeds T_k and the parameters μ, ρ define the symmetry of tiling T (υ, μ, ρ). The usual and generalized symmetries of the karyon tilings T ( υ, μ, 0) are considered. The generalized symmetries are quasi-symmetries that map the tilings T (υ, μ, 0) to their dual tilings 𝒯^* (v, μ, 0).