How to decompose a unit cube into tetrahedra? I was presented with the problem of breaking the unit cube $[0,1] \times [0,1] \times [0,1] $ into tetrahedron shapes.  The first two pieces are easy, but it's not so easy to visualize after that.  I found:


$\{ (0,0,0), (1,0,0), (0,1,0), (0,0,1) \}$
$\{  (1,1,1), (1,1,0), (1,0,1), (0,1,1) \}$


What remains is a triangular prism.  Then maybe I think it readily splits into 2 or 3 pieces.  In any case, they are difficult to draw and keep track of all the data.



Also I think I could have started differently with this other tetrahedron:


$\{ (0,0,0), (1,1,0), (0,1,1), (1,0,1) \}$




There should be lots of solutions, but I neither have the date to store into a computer nor do I have picture of even a single one.
The simplest way is a partition into $3! = 6$ tetrahedra:


$\{x,y,z : 0 \leq x \leq y \leq z \leq 1\}$.
$\{x,y,z : 0 \leq x \leq z \leq y \leq 1\}$.
$\{x,y,z : 0 \leq y \leq x \leq z \leq 1\}$.
$\{x,y,z : 0 \leq y \leq z \leq x \leq 1\}$.
$\{x,y,z : 0 \leq z \leq x \leq y \leq 1\}$.
$\{x,y,z : 0 \leq z \leq y \leq x \leq 1\}$.


The corresponding tetrahedra have the following vertices:


$(0,0,0),(0,0,1),(0,1,1),(1,1,1)$.
$(0,0,0),(0,1,0),(0,1,1),(1,1,1)$.
$(0,0,0),(0,0,1),(1,0,1),(1,1,1)$.
$(0,0,0),(1,0,0),(1,0,1),(1,1,1)$.
$(0,0,0),(0,1,0),(1,1,0),(1,1,1)$.
$(0,0,0),(1,0,0),(1,1,0),(1,1,1)$.