geometry what structure is derived from half a parallelepiped structure

Question

the question comes from vector and determinants.

we all know that two 2D vectors v, and w, is a parallelogram. and half this is an triangle. they can be found by the determinant.

i wanted to extend it to 3D, and there are no problem in that we just find the Volume instead of an area. the structure is now a parallelepiped structure. Parralelepiped

but what structure is derived from the half method for finding a triangle, now in 3D.

it became to trigy for me to justify it, since if one looks at the picture cutting from v to W (their arrows). then this is clear, but do I also need to cut between say u and v, or/and u and w.

another reasoning is cutting in half is equal to half the volume thus we can only cut it ones (do people get this)

and is this cutting the v and w, or one of the other combinations or are this actually not important.

another thing for 2D, 2 variables, we use 1/2 the area for a triangle. should I for 3D 3 variables do 1/3

if any body asking what are you trying to find. i just want to know if taking half the Determinant of a parallelepiped what structure comes.

and if it is not a triangle, and I want a triangle. then should I use 1/3 of the determiant to find its volume for the structure is this doing half of v and w, plus half of u and v, and/or u and w. so we cut 2 times or cur 3 times

lastly if 3D cutting equivalent of 2D, is this 1/3 of Determinat then is this extended to higher dimensions by Rⁿ needs 1/n of the determinant

another way of saying it: is if 2d is 1/2, is 3d 1/3, 4d 1/4,...

Answer 1

In $N$-dimensional Euclidean space the notion of triangle in 2D generalizes to a simplex. In 3D, the simplex is the tetrahedron.

If we have $N+1$ vertices in $N$ dimensions, they form a simple. If we name the vertices $\vec{v}_0, \vec{v}_1, \dots, \vec{v}_N$, the volume $V$ of the simplex they form is $$\begin{aligned} V &= \frac{1}{N!} \left\lvert \det\biggl( \begin{matrix} (\vec{v}_1 - \vec{v}_0) & (\vec{v}_2 - \vec{v}_0) & \dots & (\vec{v}_N - \vec{v}_0) \end{matrix} \biggr) \right\rvert \\ ~ &= \frac{1}{N!} \left\lvert \det\biggl( \begin{matrix} \vec{v}_0 & \vec{v}_1 & \dots & \vec{v}_N \\ 1 & 1 & \dots & 1 \\ \end{matrix} \biggr) \right\rvert \\ \end{aligned}$$ You can use either of the right sides you want, they both will always yield the same result.

Examples:

In 2D, if we have a simplex with vertices $(0,0)$, $(0,1)$, and $(1,0)$, then its volume is $$V ~=~ \frac{1}{2!} \left\lvert \det\biggl( \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \biggr) \right\rvert ~=~ \frac{1}{2!} \left\lvert \det\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 1 & 1 \\ \end{matrix}\right) \right\rvert ~=~ \frac{1}{2}$$

In 3D, if we have a simplex with vertices $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, and $(1,0,0)$, then its volume is $$V ~=~ \frac{1}{3!} \left\lvert \det\left( \begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \\ \end{matrix} \right) \right\rvert ~=~ \frac{1}{3!} \left\lvert \det\left( \begin{matrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 1 & 1 & 1 \\ \end{matrix} \right) \right\rvert ~=~ \frac{1}{6}$$

The generalisation of parallelogram and parallelepiped is parallelotope, and its volume is the magnitude of the determinant above, when $\vec{v}_0$ is some vertex, and $\vec{v}_k$ for $k = 1 \dots N$ are the vertices that share an edge with $\vec{v}_0$. The ratio of their volumes is $1/N!$ in $N$ dimensions.