cube can be repeated in the whole space, we obtain a tiling of ${\mathbb{R}}^3$ with tetrahedra: $${\mathbb{R}}^3=\bigcup_{z\in{\mathbb{Z}}^3}\bigcup_{j=1}^{24}(\overline{{\mathbf{\Delta}}_j}+z)
 \label{eq:tiling}$$ and the corresponding partition of unity $$\sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24}{{\ensuremath {\mathds 1} }}_{\ell{\mathbf{\Delta}}_j}(x-\ell z)\equiv1,\qquad \text{for a.e. $x\in{\mathbb{R}}^3$},$$ for any fixed size $\ell>0$ of the tiles. Any ${\mathbf{\Delta}}_j$ can be written as ${\mathbf{\Delta}}_j=\mu_j{\mathbf{\Delta}}$ where ${\mathbf{\Delta}}$ is a reference tetrahedron with $0$ as its center of mass. Here $\mu_j=(z_j,R_j)\in C_1\times SO(3)$ is an appropriate translation and rotation, which acts as $\mu_j x=R_jx-z_j$, hence ${\mathbf{\Delta}}_j=R_j{\mathbf{\Delta}}-z_j$. In each tetrahedron we now place the regularized characteristic function $$\boxed{\chi_{\ell,\delta,j}:=\frac{1}{(1-\delta/\ell)^3}{{\ensuremath {\mathds 1} }}_{\ell\mu_j(1-\delta/\ell){\mathbf{\Delta}}}\ast \eta_\delta.} 
 \label{eq:def_chi_ell_tiling}$$ Here $\eta_\delta(x)=(10/\delta)^{3}\eta_1(10x/\delta)$ where $\eta_1$ is a fixed $C^{\infty}_c$ non-negative radial function with support in the unit ball and such that $\int_{{\mathbb{R}}^3}\eta_1=1$. Assuming that $\delta{\leqslant}\ell/2$, the function $\chi_{\ell,\delta,j}$ has its support well inside $\ell {\mathbf{\Delta}}_j$, at a distance proportional to $\delta$ from its boundary. The prefactor has been chosen to ensure that $$\int_{{\mathbb{R}}^3}\chi_{\ell,\delta,j}=\frac{\ell^3}{24}=|\ell{\mathbf{\Delta}}|.$$ The function $$\sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24} \chi_{\ell,\delta,j}(x-\ell z)$$ is equal to $(1-\delta/\ell)^{-3}>1$ inside the tiles but vanishes in a neighborhood of the boundary of the tiles. It is the incomplete partition of unity which we have mentioned above. We obtain a partition of unity after averaging over the translations of the tiling: $$\frac{1}{\ell^3}\int_{C_\ell}\sum_{z\in{\mathbb{Z}}^3}\sum_{j=1}^{24} \chi_{\ell,\delta,j}(x-\ell z-\tau)\,d\tau=1,\qquad \text{for a.e. $x\in{\mathbb{R}}^3$}.
