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:

• $(0,0,0),(0,0,1),(0,1,1),(1,1,1)$.

• $(0,0,0),(0,1,0),(0,1,1),(1,1,1)$.

• $(0,0,0),(0,0,1),(1,0,1),(1,1,1)$.

• $(0,0,0),(1,0,0),(1,0,1),(1,1,1)$.

• $(0,0,0),(0,1,0),(1,1,0),(1,1,1)$.

• $(0,0,0),(1,0,0),(1,1,0),(1,1,1)$.

StackExchange Computer Science Q#89910, answer score: 4