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Comment by quickthrower2 | Hacker News Reader

Haven't watched yet but an obvious lower bound is 2 times it's shadow (one for each side) - in fact a coin achieves this.

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I don't see why this is getting downvoted. It's a perfectly valid argument that's very much in the spirit of 3b1b.

And 6x is a good upper bound (achieved by a cube). It's nice that the correct answer is halfway between these.

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

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

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

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