Interesting discussions on hypocubic structures:

It’s kind of hard believe that two values with different dimensions are tied to each other closely. How can we prove it get a deeper understanding of it. Googling “cube shadow theorem” doesn’t find anything useful else except these two videos. (Not sure why there’s no wikipedia entry for this.)

Let `u`, `v`, `w` to be three unit vectors spanning the unit cube. Then, `u + v + w` gives us the vector connecting these two opposing vertexes. Its projecting onto z-axis is the height of the cube, `(u + v +w) . z`.

The area of the shadow on xy-plane is the sum of three plane, `uv`, `vw`, `wu`, projected onto xy-plane. Let’s focus on `uv` firstly; `u x v` gives us a vector perpendicular to `uv` facet with the magnitude of its area, which is `w` in the unit cube case. Projecting `uv` facet onto xy-plane is equivalent to calculate the dot product of these two normal vectors, `w` and `z`, which is equivalent to projecting `w` to z-axis. If we do the same for the other two facets, and sum all three values, we get `(u + v + w) . z`.