Finding the dimensions of a box (cuboid) given a hexagon filled in to look like the box

Question

Suppose I have hexagons that like the ones below but I know the area and the points of each hexagon that represent a cuboid of dimensions g,h,d. How can I find the values for g, h, and d? Any pointers on finding a solution?

Answer 1

Think of it this way. Let's consider the plane of the page to be the $xy$-plane. You're given these three $2$ dimensional vectors $u,v,w\in\mathbb{R}^3$ that don't have any $z$-component. In $3d$ space, these are $3$ dimensional vectors that are $u,v,w$ respectively with some unknown $z$ component. I.e. they are $u+az,v+bz,w+cz$ for some constants $a,b,c\in\mathbb{R}$ while $z\in\mathbb{R}^3$ is the unit vector in the $z$-direction.

We are given that these $3d$ vectors are the edges of a box. So we know that their respective dot products are $0$. We can then do the calculations and get the $$\begin{cases}u\cdot v+ab=0\\v\cdot w+bc=0\\w\cdot u+ca=0\end{cases}$$ This is very symmetrical and hence fairly easy to solve. We can calculate that $(abc)^2=-(u\cdot v)(v\cdot w)(w\cdot u)$ so $a=\frac{abc}{-v\cdot w}$ and etc. for the other two.

Some observations are that we require that the RHS of the equation for $(abc)^2$ to be positive. Else I guess this means there is no cube. Our formulations of course also require the assumption that none of the possible denominators are $0$. If any of them are, we can see that actually at least $2$ of them must be and moreover there are of course infinite solutions.