As I was thinking about it, I thought about another way to prove it, inspired by Mamikon's visual calculus method.

If we think of a plane parallel to the axis of the bead and tangent to its inner cylindrical surface, the intersection of the bead and the plane is a disk of diameter h. If we now rotate the plane around the axis for a whole turn such that it remains tangent to the inner surface, the disk will also rotate a whole turn, as if it sweeps the volume of a sphere of diameter h.

Post scriptum. Afterwards I found a drawing of such a bead in Mamikon's original notes (see second page of his drawings).

Beautiful... why couldn't I see it.. Thanks for the inspiration, and then a brilliant solution

ReplyDeleteThanks for the mention on your blog. My first explanation in the comment on your first post was a bit confused. I came up with this tangent disk later. There are also nice related illustrations on Mamikon's site, for example http://www.its.caltech.edu/~mamikon/Sphere.html

ReplyDeleteHi Arjen,

ReplyDeleteThis idea of proof based on Mamikon's theorem is really cool!

Best regards,

Cristi