Combinatoric of the karyon tilings

In this article, we study the combinatorial properties of the karyon tilings $\mathcal{T}$ of the torus $\mathbb{T}^d$ of an arbitrary dimension $d$. Our main results are the following statements: 1) the karyon corona $\mathbf{Cr}$ contains all types of polyhedral stars of the $\mathcal{T}$ tilings; 2) the number of all faces of dimension $a$ of the tiling $\mathcal{T}$ is equal to $md!/((d-a)!a!)$, where $m$ is the order of tilling.