Symmetries structure of karyon tilings

In this article, we study the symmetry properties of the karyon tilings $\mathcal{T}$ of the torus $\mathbb{T}^d$ of arbitrary dimension $d$. Its main results are the following statements: 1) The tilings $\mathcal{T}$ are translation invariant relative to the canonical shift $S$ of the torus $\mathbb{T}^d$. This is a fundamental property of the karyon tilings. 2) Nondegenerate karyon tilings $\mathcal{T}$ have $2^d$ central symmetries.