Let $d=2$, and consider the region surrounded by $x=0$, $y=0$ and $x+y=1$, and call it $T_2$. Consider the square surrounded by $x=0$, $y=0$, $x=1$, $y=1$, and call it $C_2$. The volume of the region $T_2$ is $\dfrac{1}{2}$, and volume of $C_2$ is $1$. Consider this map $R_2: C_2 \rightarrow T_2$: for every point $p$ if $p \in T_2$, then it returns $p$ itself. And if $p \notin T_2$, then it returns its reflection. The volume of $C_2$ is two times the volume of $T_2$. This map is a two-to-one map.
Let $d=3$, and consider the region surrounded by $x=0$, $y=0$, $z=0$ and $x+y+z=1$, and call it $T_3$. Consider the square surrounded by $x=0$, $y=0$, $z=0$, $x=1$, $y=1$, $z=1$ and call it $C_3$. The volume of the region $T_3$ is $\dfrac{1}{6}$, and volume of $C_3$ is $1$. The volume of $C_3$ is six times the volume of $T_3$. Is there a canonical map $R_3:C_3 \rightarrow T_3$, which is a six-to-one map?
Is there a canonical map $R_d: C_d \rightarrow T_d$, which is a $D$-to-one map, where $D$ is a number depending on $d$ (probably $D=d!$).