# Cube Shadow Theorem

Interesting discussions on hypocubic structures:

- https://ru-clip.com/video/rAHcZGjKVvg/the-cube-shadow-theorem-pt-1-prince-rupert-s-paradox.html
- https://ru-clip.com/video/cEhLNS5AHss/the-cube-shadow-theorem-pt-2-the-best-hypercube-shadows.html

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`

.