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.