The unimodularity of the induced toric tilings

Induced tilings $\mathcal T=\mathcal T|_\mathrm{Kr}$ of the $d$-dimensional torus $\mathbb T^d$, generated by the embedded karyon $\mathrm{Kr}$, are considered. The operations of differentiation are defined $\sigma\colon\mathcal T\to\mathcal T^\sigma$, as a result we get again induced partitions $\mathcal T^\sigma=\mathcal T|_{\mathrm{Kr}^\sigma}$ of the same torus $\mathbb T^d$, generated by the derived karyon $\mathrm{Kr}^\sigma$. In the language of the karyons $\mathrm {Kr}$ the derivations of $\sigma$ reduce to a combination of geometric transformations of the space $\mathbb R^d$. It is proved that if the karyon $\mathrm{Kr}$ is unimodular, then it generates an induced tiling $\mathcal T=\mathcal T|_\mathrm{Kr}$ and the derivative karyon $\mathrm{Kr}^\sigma$ is unimodular again. So there exists the corresponding derivative tiling $\mathcal T^\sigma=\mathcal T|_{\mathrm {Kr}^\sigma}$. Using unimodular karyons one can build an infinite family of induced tilings $\mathcal T=\mathcal T(\alpha,\mathrm{Kr}_*)$ depending on a shift vector $\alpha$ of the torus $\mathbb T^d$ and the initial karyon $\mathrm{Kr}_*$. Two algorithms are presented for constructing such unimodular karyons of $\mathrm{Kr}_*$.