The cube surface area is 4 times the average size of the shadow across all rotations, according to the video.

In fact, the video states this for all convex shapes.

I've been trying something similar for 2D, but there it doesn't quite seem to hold.

Consider a very thin rectangle of size 1 by epsilon. Then it has circumference 2 (ignoring the epsilon). The shadow it casts at angle phi has size |sin phi|. Now, if we average |sin phi| from 0 to 180 degrees, (or 0 to 360 or 0 to 90) we get (2 / pi).

I haven't checked whether this average holds for things other than thin rectangles, but I'd imagine so. I then find it weird we get a trancendental number in 2D but an integer in 3D.

Not a surprise to see pi in there really, since we're averaging over "surfaces" of circles/spheres. In general (spoiler), it does generalize to arbitrary dimensions. We get a rational factor for odd dimensions, and 1/pi * rational for even.

See sections 6-9 here for demonstration: https://arxiv.org/pdf/1109.0595.pdf

I guess that only works if you are considering convex shapes (which sphere and cube obviously are)

and Graham's number is a correct but very weak upper bound -- as usual.