This shows the subdivision of the prism $[0,1]\times\Delta_3$ into the $3$-simplices $\zeta_i(\Delta_4)$ for $i=0,\dotsc,3$. Each vertex of the prism has the form $(i,e_j)$ with $i\in\{0,1\}$ and $j\in\{0,1,2\}$; this is labelled $ij$ in the diagram above. The simplex $\zeta_0(\Delta_3)$ has vertices $\{(0,e_0),(1,e_0),(1,e_1),(1,e_2)\}$ or in other words $\{00,10,11,12\}$. The simplex $\zeta_1(\Delta_3)$ has vertices $\{00,01,11,12\}$, and the simplex $\zeta_2(\Delta_3)$ has vertices $\{00,01,02,12\}$.